Harmonic Analysis, Geometric Measure Theory and Additive ...thiele/workshop/proceedings.pdf ·...

Harmonic Analysis, Geometric MeasureTheory and Additive Combinatorics

Summer School∗, Catalina Island

Jun 24th - Jun 29th 2012

Organizers:

Izabella Laba, University of British Columbia, Vancouver

Malabika Pramanik, University of British Columbia, Vancouver

Christoph Thiele, University of California, Los Angeles

∗supported by NSF grant DMS 1001535

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Contents

1 Arithmetic progressions in sets of fractional dimension 6Gagik Amirkhanyan, Georgia Tech . . . . . . . . . . . . . . . . . . 61.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6

2 Local estimates for exponential polynomials and their appli-cations to inequalities of the uncertainty principle type, PartII: Applications 11Michael Bateman, UCLA . . . . . . . . . . . . . . . . . . . . . . . . 112.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . 112.2 Establishing the uncertainty principle from Turan’s lemma:

an idealized scenario . . . . . . . . . . . . . . . . . . . . . . . 132.2.1 Proof of Theorem 6 given wishlist above . . . . . . . 14

2.3 Averaging . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15

3 The endpoint case of a Stein-Tomas theorem for subsets ofthe real line 17Marc Carnovale, UBC . . . . . . . . . . . . . . . . . . . . . . . . . 173.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . 173.2 Lions-Peetre Interpolation Spaces . . . . . . . . . . . . . . . . 193.3 Skirting the Triangle Inequality via Interpolation . . . . . . . 213.4 Finishing up . . . . . . . . . . . . . . . . . . . . . . . . . . . . 22

4 Necessary conditions for Lp(Rn)-Fourier multipliers 26Vincent Chan, UBC . . . . . . . . . . . . . . . . . . . . . . . . . . 264.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . 264.2 Idempotents case . . . . . . . . . . . . . . . . . . . . . . . . . 264.3 A generalization . . . . . . . . . . . . . . . . . . . . . . . . . . 28

5 Salem-Bluhm’s construction of Salem sets 31Xianghong Chen, UW-Madison . . . . . . . . . . . . . . . . . . . . 315.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . 315.2 The main result . . . . . . . . . . . . . . . . . . . . . . . . . . 315.3 The set . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 325.4 The measure . . . . . . . . . . . . . . . . . . . . . . . . . . . . 335.5 The Fourier transform . . . . . . . . . . . . . . . . . . . . . . 335.6 Randomization . . . . . . . . . . . . . . . . . . . . . . . . . . 33

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5.7 From average decay to deterministic decay . . . . . . . . . . . 335.8 The key estimate . . . . . . . . . . . . . . . . . . . . . . . . . 345.9 Proof of the average decay . . . . . . . . . . . . . . . . . . . . 355.10 The dimensions . . . . . . . . . . . . . . . . . . . . . . . . . . 36

6 Buffon’s needle estimates for rational product Cantor sets 37Kyle Hambrook, UBC . . . . . . . . . . . . . . . . . . . . . . . . . 376.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . 376.2 The SSV property of φ′t . . . . . . . . . . . . . . . . . . . . . . 406.3 The SLV structure of φ′′t . . . . . . . . . . . . . . . . . . . . . 416.4 Reduction to lower bounds on integrals; required upper bounds 436.5 The main argument . . . . . . . . . . . . . . . . . . . . . . . . 44

7 Projecting the One-Dimensional Sierpinski Gasket 47Edward Kroc, UBC . . . . . . . . . . . . . . . . . . . . . . . . . . . 477.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . 477.2 The measure of the projections . . . . . . . . . . . . . . . . . 487.3 Bounds on the dimension of the projections . . . . . . . . . . 50

8 Wiener’s ‘closure of translates’ problem and Piatetski-Shapiro’suniqueness phenomenon 53Allison Lewko, University of Texas at Austin, Microsoft Research

New England . . . . . . . . . . . . . . . . . . . . . . . . . . . 538.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . 538.2 Tools . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 55

9 Bounded orthogonality systems and the Λ(p)-set problem II 57Mark Lewko . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 579.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . 57

10 Local estimates of exponential polynomials and their appli-cations to inequalities of uncertainty principle type - partI 62Christoph Marx, UCI . . . . . . . . . . . . . . . . . . . . . . . . . . 6210.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6210.2 Nazarov’s theorem . . . . . . . . . . . . . . . . . . . . . . . . 64

10.2.1 Bernstein-type estimates and order reduction . . . . . 6410.2.2 The role of “zero counting” . . . . . . . . . . . . . . . 65

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10.3 Extensions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 66

11 Salem sets and restriction properties of Fourier transforms 68Eyvindur Ari Palsson, University of Rochester . . . . . . . . . . . . 6811.1 Classical restriction . . . . . . . . . . . . . . . . . . . . . . . . 6811.2 Two notions of dimension . . . . . . . . . . . . . . . . . . . . 6811.3 Main result . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6911.4 Salem sets . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7011.5 Sketch of proof of theorem . . . . . . . . . . . . . . . . . . . . 71

12 Maximal operators and differentiationtheorems for sparse sets: Part II 73Alex Rice, UGA . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7312.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7312.2 Main results . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7412.3 Motivation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7512.4 Linearization and discretization . . . . . . . . . . . . . . . . . 7512.5 Transverse correlations . . . . . . . . . . . . . . . . . . . . . . 76

13 Maximal operators and differentiation theorems for sparsesets, part I 79Pablo Shmerkin, Surrey . . . . . . . . . . . . . . . . . . . . . . . . 7913.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7913.2 Construction of the sets Sk . . . . . . . . . . . . . . . . . . . . 8013.3 Internal tangencies and transversal intersections . . . . . . . . 8113.4 Main result . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8213.5 A martingale argument, and conclusion of the proof . . . . . . 83

14 Bounded orthogonality systems and the Λ(p)-set problem I 85Stefan Steinerberger, University Bonn . . . . . . . . . . . . . . . . . 8514.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8514.2 Sketch of the general proof . . . . . . . . . . . . . . . . . . . . 87

15 On a problem of Erdos on sequences and measurable sets, &Infinite patterns that can be avoided by measure. 89Krystal Taylor, UMN . . . . . . . . . . . . . . . . . . . . . . . . . . 8915.1 Introduction: known classes of non-universal sets . . . . . . . 8915.2 Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 90

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15.2.1 Sequences with ’slow decay’ are universal . . . . . . . . 9015.2.2 Infinite sets are ’almost everywhere’ universal . . . . . 91

15.3 Sketch of proofs . . . . . . . . . . . . . . . . . . . . . . . . . . 9115.3.1 Construction of the set E in Theorem 2 . . . . . . . . 9115.3.2 Construction of the set E in Theorem 3 . . . . . . . . 92

16 Averages in the plane over convex curves and maximal op-erators 95Joshua Zahl, UCLA . . . . . . . . . . . . . . . . . . . . . . . . . . . 9516.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . 95

16.1.1 Background . . . . . . . . . . . . . . . . . . . . . . . . 9516.1.2 New Results . . . . . . . . . . . . . . . . . . . . . . . . 96

16.2 Proof the Theorem 1 . . . . . . . . . . . . . . . . . . . . . . . 9716.2.1 Reduction to a geometric problem . . . . . . . . . . . . 97

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1 Arithmetic progressions in sets of fractional

dimension

after Izabella Laba and Malabika Pramanik [1]A summary written by Gagik Amirkhanyan

Abstract

Let E ⊂ R be a closed set of Hausdorff dimension α. We prove thatif α is sufficiently close to 1, and if E supports a probabilistic measureobeying appropriate dimensionality and Fourier decay conditions, thenE contains non-trivial 3-term arithmetic progressions.

1.1 Introduction

From the introduction of the summarized paper: “

Definition 1. Let A ⊂ R be a set. We will say that A is universal for aclass E of subsets of R if any set E ∈ E contains an affine (i.e. translatedand rescaled) copy of A.

If E is the class of all subsets of R of positive Lebesgue measure, then itfollows from Lebesgue’s theorem on density points that every finite set A isuniversal for E . Namely, let E have positive Lebesgue measure, then E hasdensity 1 at almost every x ∈ E. In particular, given any δ > 0, we maychoose an interval I = (x− ε, x+ ε) such that |E ∩ I| ≥ (1− δ)|I|. If δ waschosen small enough depending on A, the set E ∩ I will contain an affinecopy of A.

An old question due to Erdos is whether any infinite set A ⊂ R can beuniversal for all sets of positive Lebesgue measure. It is known that not allinfinite sets are universal: for instance, if A = an∞n=1 is a slowly decayingsequence such that an → 0 and an−1

an→ 1, then one can construct explicit

Cantor-type sets of positive Lebesgue measure which do not contain an affinecopy of A [2]. There are no known examples of infinite sets A which areuniversal for the class of sets of positive measure. In particular, the questionremains open for A = 2−n∞n=1.

The purpose of this paper is to address a related question: if A ⊂ R isa finite set and E ⊂ [0, 1] is a set of Hausdorff dimension α ∈ [0, 1], mustE contain an affine copy of A? In other words, are finite sets universal for

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the class of all sets of Hausdorff dimension α? This more general statementalready fails if A = 0, 1, 2 and E is a set of Hausdorff dimension 1 butLebesgue measure 0. This is due to Keleti [3], who actually proved a strongerresult: there is a closed set E ⊂ [0, 1] of Hausdorff dimension 1 such that Edoes not contain any “rectangle” x, x+ r, y, y + r with x 6= y and r 6= 0.

Hence one may ask if there is a natural subclass of sets of fractionaldimension for which a finite set such as 0, 1, 2 might be universal. Thisquestion is addressed in Theorem 2, which is the main result of this article.We define the Fourier coefficients of a measure µ supported on [0, 1] as

µ(k) =

∫ 1

0

e−2πikxdµ(x).

Theorem 2. Assume that E ⊂ [0, 1] is a closed set which supports a proba-bilistic measure µ with the following properties:

(A) µ([x, x+ ε]) ≤ C1εα for all 0 < ε ≤ 1,

(B) |µ(k)| ≤ C2(1− α)−B|k|−β2 for all k 6= 0,

where 0 < α < 1 and 2/3 < β ≤ 1. If α > 1 − ε0, where ε0 > 0 is asufficiently small constant depending only on C1, C2, B, β, then E contains anon-trivial 3-term arithmetic progression.

We note that if (A) holds with α = 1, then µ is absolutely continuous withrespect to the Lebesgue measure, hence E has positive Lebesgue measure.This case is already covered by the Lebesgue density argument.

In practice, (B) will often be satisfied with β very close to α. It willbe clear from the proof that the dependence on β can be dropped from thestatement of the theorem if β is bounded from below away from 2/3, e.g.β > 4/5; in such cases, the ε0 in Theorem 2 depends only on C1, C2, B.

The assumptions of Theorem 2 are in part suggested by number-theoreticconsiderations, which we now describe briefly. A theorem of Roth states thatif A ⊂ N has positive upper density, i.e.

limN→∞#(A ∩ 1, . . . , N)

N> 0, (1)

then A must contain a non-trivial 3-term arithmetic progression. Szemeredi’stheorem extends this to k-term progressions. It is well known that Roth’stheorem fails without the assumption (1). However, there are certain natu-ral cases when (1) may fail but the conclusion of Roth’s theorem still holds.

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For example, there are variants of Roth’s theorem for random sets and setssuch as primes which resemble random sets closely enough. The key conceptturns out to be linear uniformity. It is not hard to prove that if the Fouriercoefficients A(k) of the characteristic function of A are sufficiently small, de-pending on the size of A, then A must contain 3-term arithmetic progressionseven if its asymptotic density is 0. The Roth-type results mentioned abovesay that the same conclusion holds under the weaker assumption that A hasan appropriate majorant whose Fourier coefficients are sufficiently small (thisis true for example if A is a large subset of a random set).

If the universality of A = 0, 1, 2 for sets of positive Lebesgue mea-sure is viewed as a continuous analogue of Roth’s theorem, then its lower-dimensional analogue corresponds to Roth’s theorem for integer sets of den-sity 0 in N. The above considerations suggest that such a result might holdunder appropriately chosen Fourier-analytic conditions on E which could beinterpreted in terms of E being “random.” We propose Assumptions (A)-(B)of Theorem 2 as such conditions.

To explain why Assumptions (A)-(B) are natural in this context, we give abrief review of the pertinent background. Let dimH(E) denote the Hausdorffdimension of E. Frostman’s lemma (see e.g. [2]) asserts that if E ⊂ R is acompact set then

dimH(E) = supα ≥ 0 : ∃ a probabilistic measure µ supported on E

such that (A) holds for some C1 = C1(α).

We also define the Fourier dimension of E ⊂ R as

dimF (E) = supβ ≥ 0 : ∃ a probabilistic measure µ supported on E

such that |µ(ξ)| ≤ C(1 + |ξ|)−β/2 for all ξ ∈ R,

where µ(ξ) =∫e−2πiξxdµ(x). Thus (A) implies that E has Hausdorff dimen-

sion at least α, and (B) says that E has Fourier dimension at least 2/3.It is known that

dimF (E) ≤ dimH(E) for all E ⊂ R; (2)

in particular, a non-zero measure supported on E cannot obey (B) for anyβ > dimH(E). It is quite common for the inequality in (2) to be sharp: forinstance, the middle-thirds Cantor set has Hausdorff dimension log 2/ log 3,

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but Fourier dimension 0. Nonetheless, there are large classes of sets suchthat

dimF (E) = dimH(E).

Such sets are usually called Salem sets.Assumptions (A)-(B) are closely related, but not quite equivalent, to the

statement that E is a Salem set. On the one hand, we do not have to assumethat the Hausdorff and Fourier dimensions of E are actually equal. It sufficesif (B) holds for some β, not necessarily equal to α or arbitrarily close to it.On the other hand, we need to control the constants C1, C2, B, as the rangeof α for which our theorem holds depends on these constants.

Thus we need to address the question of whether measures obeying thesemodified assumptions can actually exist. We prove that given any C1 > 1,C2 > 0 and 0 < β < α < 1, there are subsets of [0, 1] which obey (A)-(B) withB = 0 and with the given values of C1, C2, α, β. Our construction is based onprobabilistic ideas similar to [4], but simpler. Salem’s construction [4] doesnot produce explicit constants, but we were able to modify his argument soas to show that, with large probability, the examples in [4] obey (A)-(B) withB = 1/2 and with C1, C2 independent of α for α close to 1.

The key feature of our proof is the use of a restriction estimate. Restric-tion estimates originated in Euclidean harmonic analysis.”

Our proof of Theorem 2 extends the approach of [5], [6] to the continuoussetting of sets of fractional dimension for which a restriction estimate isavailable. We use of the trilinear form Λ in a Fourier representation and adecomposition of the measure µ into “random” and “periodic” parts.

References

[1] I. Laba, M. Pramanik Arithmetic progressions in sets of fractional di-mension. Geom. Funct. Anal. 19 (2009), 429-456.

[2] K. Falconer, On a problem of Erdos on sequences and measurable sets,Proc. Amer. Math. Soc. 90 (1984), 77-78.

[3] T. Keleti, A 1-dimensional subset of the reals that intersects each of itstranslates in at most a single point, Real Anal. Exchange 24 (1998/99),no. 2, 843–844.

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[4] R. Salem, On singular monotonic functions whose spectrum has a givenHausdorff dimension, Ark. Mat. 1 (1950), 353–365.

[5] B. Green, Roth’s theorem in the primes, Ann. Math. 161 (2005), 1609-1636.

[6] B. Green, T. Tao, Restriction theory of the Selberg sieve, with applica-tions, J. Theor. Nombres Bordeaux 18 (2006), 147–182.

Gagik Amirkhanyan, Georgia Techemail: [email protected]

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2 Local estimates for exponential polynomi-

als and their applications to inequalities of

the uncertainty principle type, Part II: Ap-

plications

after F. Nazarov [1]A summary written by Michael Bateman

Abstract

We use Turan’s lemma as proved by Nazarov to prove a quan-titative version of the uncertainty principle, generalizing a result ofAmrein and Berthier.

2.1 Introduction

Nazarov’s paper [1] proves a variant of Turan’s lemma and then uses thislemma to establish several forms of the uncertainty principle. The uncer-tainty principle can be loosely stated as

Theorem 1 (Uncertainty principle, fuzzy version). A function cannot besimultaneously localized in space and in frequency.

An easy-to-establish version of this principle is

Theorem 2. For a smooth function f ∈ L2(R) with ||f ||2 = 1, we have∫|xf(x)|2dx ·

∫|ξf(ξ)|2dξ & 1. (1)

Already we can see from this inequality that if a function has spatialsupport in an ε neighborhood of the origin, then its Fourier support cannotbe contained in a 1

Cεneighborhood of the origin. The inspiration for the

version given by Nazarov is due to Amrein and Berthier:

Theorem 3 (Amrein-Berthier). Suppose f ∈ L2(R). If supp(f) and supp(f)have finite measure, then f = 0 almost everywhere.

This summary focuses on Nazarov’s proof of the following quantitativeversion of this last theorem:

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Theorem 4 (Nazarov). Let E,Σ ⊆ R have finite Lebesgue measure. Then

||f ||22 . eC|E||Σ|(∫

R\E|f |2 +

∫R\Σ|f |2). (2)

This theorem can be applied to obtain precise estimates on the simulta-neous rates of decay for a function and its Fourier transform. For example:

Corollary 5. Fix p and q satisfying 1p

+ 1q

= 1. Suppose f : R→ R satisfies

|f(x)| . eC1|x|p and |f(ξ)| . eC2|x|q . Then if C1 and C2 are small enough,f = 0 almost everywhere.

This is essentially a result of Morgan, who established optimal values forC1 and C2. We can obtain corollary by applying Nazarov’s theorem to setsof the form E = x : |f(x)| ≥ λ1 and Σ = ξ : |f(ξ)| ≥ λ2. Before we beginwith details of the proof, we present a simple reduction. It suffices to prove

Theorem 6. Let E ⊆ R have finite measure. Then for any function f ∈L2(R) supported on E, we have∫

Σ

|f |2 . eC|E||Σ|∫R\Σ|f |2 (3)

Proof that Theorem 6 implies Theorem 4 . Let f ∈ L2(R) be arbitrary, andwrite fE := f1E. Then

||f ||22 =

∫|fE|2 +

∫R\E|f |2 (4)

=

∫Σ

|fE|2 +

∫R\Σ|fE|2 +

∫R\E|f |2 (5)

. eC|E||Σ|∫R\Σ|f |2 +

∫R\Σ|fE|2. (6)

The second equality is by Plancherel and the inequality follows from Theorem6.

We now focus our attention on proving Theorem 6.

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2.2 Establishing the uncertainty principle from Turan’slemma: an idealized scenario

In this section we establish Theorem 6 using Turan’s lemma, provided wehave several favorable hypotheses at our disposal. Technically, these hypothe-ses are not justified; nevertheless we will be able to arrange for essentiallyequivalent hypotheses that are justified. To do this we will use the randomperiodization procedure described in Section 2.3. The goal is to understandthe form of the argument rather than all of the details.

To begin with, we fix a function f . We define the periodization g definedon the circle as follows:

g(t) =∑k∈R

f(k + t). (7)

An important fact is the following:

Proposition 7. With g defined as above, we have f(m) = gm. (Here we usethe notation gm to emphasize the different domains of g and f .).

Remark 8. This relationship is an important reason the function g is ofinterest to us. Another reason is the applicability of Turan’s lemma to ex-ponential polynomials. The function f , by Fourier inversion, is an integralof exponential functions, but g is a sum of exponential functions. By con-sidering an appropriately defined truncation of this sum, we will find a newfunction p, still related to f , that is a finite sum of exponential functions.Because of this form, we will be able to apply Turan’s lemma to p.

We decompose the function g into two pieces: g = p+ q, where

p(t) =∑

m∈Σ∪0

gme2πimt. (8)

In other words, p is the projection of g onto frequencies in Σ ∪ 0. Ofcourse this means that q is projection of g onto frequencies not in Σ ∪ 0;this, together with the proposition above, suggests that q should have somerelationship with the quantity

∫R\Σ |fE|

2 on the right-hand side of Theorem6.

We now write down a list of assumptions under which we can prove anestimate of the form in Theorem 6:

• WISH 1: |E| ≤ 110

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• WISH 2: #(Z ∩ Σ) ∼ |Σ|

• WISH 3: ||q||22 .∫R\Σ |fE|

2.

The first assumption is rather mild; it will be handled essentially by a rescaledversion of the argument below. The second and third assumptions are serious.The second says that the spectrum Σ should essentially be independent ofthe integers; the third is saying that the behavior of f on the integers shouldcapture the behavior of f on R. As previously stated, this is quite a wishlist.We hint at how to realize these wishes in the final section.

2.2.1 Proof of Theorem 6 given wishlist above

We will actually prove the following stronger claim:

Proposition 9. For all y ∈ R, we have

|f(y)|2 . C |Σ|∫R\Σ|fE|2. (9)

Remark 10. Integrating this inequality over y ∈ Σ is enough to establish theresult, since integrating over Σ costs a factor of |Σ|, which is easily absorbedby a quantity of the form eC|Σ|. In fact, we will establish the proof for y = 0,but applying the same proof to a modulation of f (and hence a translation inFourier space) yields the claim for general y).

Lemma 11 (Simple case of Turan’s lemma). Suppose p(t) =∑n

j=1 e2πiλjt,

λj ∈ Z. Suppose |p(t)| ≤ X for t in a set of measure ≥ α. Then

|p0| ≤∑k∈Z

|p0| . eCαX. (10)

Now define

F = t : g(t) = 0 ∩ t : |p(t)| .√∫

R\Σ|fE|2. (11)

Notice on this set, p(t) = −q(t). Hence By Turan’s lemma, we have

|f |2 = |g0|2 = |p0|2 (12)

≤

(∑k

|p0|

)2

≤ Cdeg(p)|F | sup

t∈F|p(t)|2 (13)

≤ C|Σ||F |

∫R\Σ|fE|2. (14)

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The second-to-last inequality follows from Turan’s Lemma, and the last in-equality follows from WISH 2, which guarantees that the degree of the poly-nomial p is approximately |Σ|. (Specifically, deg(p) = #Z ∩ Σ ∼ |Σ|,by WISH 2.) Additionally, we have used the definition of F to estimatesupt∈F |p(t)|2. The inequality established immediately above is close to ourgoal: we need only establish |F | ≥ 1

4.

To see this, recall that when g(t) = 0, we have p(t) = −q(t). Then WISH3 guarantees ∫

g=0|p|2 =

∫g=0

|q|2 (15)

.∫R\Σ|fE|2. (16)

Appealing to Chebyshev’s inequality yields that

|t ∈ g = 0 : |p(t)| ≥√∫

R\Σ|fE|2| ≤

1

C. (17)

The final step is to show that |g = 0| ≥ 12. But this follows immediately

from WISH 1, which says that |E| ≤ 110

, and the observation that |suppg| ≤|suppf |. Hence

|F | ≥ 1

2− 1

C≥ 1

4(18)

for C large enough.

2.3 Averaging

A key component of Nazarov’s proof of Theorem 4 is a “random periodiza-tion” of a given function f . We consider instead functions like gv(t) =∑

k∈Z f(k+tv

), where v is a random variable uniformly distributed in the in-terval (1, 2). Then following averaging lemma is used to find a periodizationof f satisfying the wishlist with high probability. Specifically, it helps usestablish WISH 3 above with high probability. WISH 1 can be establishedwith high probability using an even simpler argument. The lemma is appliedto |f |21R\Σ.

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Lemma 12. Fix ε > 0 and suppose φ : R → R is positive and integrable.Then ∫ 2

1

∑06=k∈Z

φ(εkv)dv ≤ 1

ε

∫φ (19)

(20)

Proof. We split the sum into ranges of positive and negative k. Then wechange variables to obtain∫ 2

1

∑k>0

φ(εkv)dv =∑k>0

∫ 2εk

εk

φ(u)dv

εk(21)

=1

ε

∫u≥0

∑k>0: u∈[εk,2εk]

φ(u)dv

k. (22)

But note that∑

k>0: u∈[εk,2εk]1k≤ 1, so the last display is controlled by

1

ε

∫u≥0

φ(u)du. (23)

copying the proof for negative k finishes the proof.

References

[1] F. Nazarov, Local estimates for exponential polynomials and their ap-plications to inequalities of the uncertainty principle type . (Russian)Algebra i Analiz 5 (1993), no. 4, 3–66; translation in St. PetersburgMath. J. 5 (1994), no. 4, 663-717.

Michael Bateman, UCLAemail: [email protected]

16

3 The endpoint case of a Stein-Tomas theo-

rem for subsets of the real line

after J. Bak and A. Seeger [1]A summary written by Marc Carnovale

Abstract

We prove the end point case of a Stein-Tomas theorem for restric-tions to singular sets of high Fourier dimension

3.1 Introduction

In 1999, Mockenhaupt [6] showed that the restriction phenomena is muchmore general than had been previously realized, by demonstrating that Fouriertransforms of certain Lp functions may be meaningfully restricted to lie inL2(µ) for µ a (singular) measure of high enough Hausdorff and Fourier dimen-sions. This is of interest not only because of the deep connections betweenrestriction theorems and a hierarchy of other problems and conjectures, fromlocal smoothing to Kakeya to the Falconer distance set conjecture, with ap-plications to PDE, number theory, geometric measure theory, and, of course,harmonic analysis, (which suggest that anything which sheds light on thetraditional question of Fourier restriction is highly significant) - but also be-cause singular sets of given Hausdorff (and, at times, Fourier) dimensionscome up, say, in regularity theory of PDE and in Geometric Measure Theory- fields where Harmonic Analysis has already had a powerful impact.

And, the result of Mockenhaupt (and later [5]) has seen use, for instance,in [4].

So since the Stein-Tomas theorem includes the end point - that is, theyconclude that ‖f‖L2(dσ) ≤ C‖f‖Lp(Rn) for σ the surface measure on the sphereSn−1 and 1 ≤ p ≤ 2n+2

n+3- it is no surprise that one should like to obtain an

end point result for Mockenhaupt’s generalization

‖f‖L2(dµ) ≤ C‖f‖Lp(Rn)

for1 ≤ p <2(2n− α + β)

4(n− α) + β

It is this at-the-time unresolved endpoint case which was the primaryfocus of Bak and Seeger’s paper. Although Mockenhaupt was able to borrow

17

many of the techniques used by Stein and Tomas, the classical endpointresult follows from embedding the surface measure on the sphere into an(explicit) analytic family and utilizing complex interpolation. This is notobviously available in the general setup of singular measures of given Fourierand Hausdorff dimensions, and so Bak and Seeger were forced to develop analternative approach. Before describing their contributions, however, let ustake a moment to recall the approach to the non-endpoint bounds.

Following Tomas, in order to obtain bounds for∫|f |2 dµ, it is enough to

obtain bounds on the convolution operator f 7→ f ∗ µ. Let us record this factas Lemma 1.

Lemma 1. Suppose that T denotes the operator with kernel µ and that‖T‖p→p′ ≤ C. Then

‖f‖L2(dµ) ≤ C ′‖f‖p (1)

Proof. By applying Plancheral, we have∫|f |2 dµ =

∫f(fµ)∨ =

∫f(f ∗ µ) ≤

C‖f‖p‖Tf‖p′ ≤ C‖f‖p‖f‖p = C‖f‖2p.

As important for us is that the reverse is also true.

Proposition 2. Suppose that the bound 1 holds. Then the operator T definedabove obeys the bound ‖T‖p→p′ ≤ C.

Proof. We have (being somewhat sloppy with our Plancheral and Fourierinversion)

‖Tf‖p′ =

(

∫|f ∗ µ|p′)

1p′ =

sup‖g‖p=1

∫g(f ∗ µ) =

sup‖g‖p=1

∫gfdµ ≤

sup‖g‖p=1

‖g‖L2(dµ)‖f‖L2(dµ) ≤

sup‖g‖p=1

‖g‖p‖f‖p = ‖f‖p

where we used Cauchy-Schwartz in the penultimate line and the restrictionesimate in the final line.

18

So nothing is lost by targetting the operator T rather than the restrictionestimate directly. In order to obtain bounds on T , we decompose its kernelµ into essentially disjoint frequency regimes by choosing a Schwartz χ0 oftotal mass 1 supported in [−1, 1] with χ|[− 1

2, 12

] ≡ 1, define χj := χ0(2−j−1·)−χ0(2−j·) (which is supported in |x| ∈ [2j, 2j+1], and define µj by µj := χjµso that µ =

∑j≥0 µj. We may then set Tjf := f ∗ µj, so that T =

∑Tj.

Then the two crucial estimates in [6] and [5] are

‖Tjf‖1 ≤ C2jα0‖f‖∞ (2)

‖Tjf‖2 ≤ C2jα1‖f‖2 (3)

where α0 = b is the Fourier dimension of µ and α1 = n − a, with a theHausdorff dimension of µ. Given our data, these are the best estimateswe can obtain for Tj as a mapping between these particular spaces. If weinterpolate these bounds using, say, Holder’s inequality, we obtain an esimate

‖Tj‖p→p′ ≤ C2k 2n−2α+β

p′ −β2 , which decays in k for p′ ≥ 22n−2α+β

β. As we used

the best possible estimates in 2 and 3, one might conclude that we won’t getbetter than this for ‖Tj‖p→p′ . Of course, since T =

∑Tj, we may apply the

triangle inequality to conclude an Lp → Lp′

bound on T .This is similar to the case in the Stein-Tomas theorem, where one cannot

obtain the endpoint bound through an application of the triangle inequalityhere - it is the one inefficient point in the proof. Instead, a more clevermeans of combining the bounds on the pieces Tj of T is necessary, and itis here that the solution introduced by Stein (complex interpolation) has noapparent analogue. This is the context in which Bak and Seeger’s resultenters.

3.2 Lions-Peetre Interpolation Spaces

One popular (and useful) solution to the abstract question of how to inter-polate two (Banach) spaces takes the name of Lions-Peetre interpolation.Suppose that A0 and A1 are two spaces, embeddable in some larger spaceso that sums of the form a0 + a1, ai ∈ Ai make sense. Suppose further thatwe have spaces B0 and B1, and a bounded mapping T : Ai → Bi definedon A0 + A1 sending A0 + A1 into B0 ∩ B1. We would like to interpolate thebounds on T between the Ai to bounds on some intermediary space.

The (or rather, a) solution for interpolating between spaces X = (X0, X1)is to first set K(t, x, X) = inf ‖x‖x0 + t‖x1‖x1 : x = x0 + x1, xi ∈ xi. This

19

gives us some weighted combination of the Xi norms - if for instance X1 ⊂ X0

with ‖ · ‖X0 < ‖ · ‖X1 , for t large this recovers the X0 norm, while for t = 0it yields the X1 norm.

If K(t, x, X) is a type of size estimate up to level t, then the expression

‖x‖Xθ,q := (∫

(t−θK(t, x, X))q dtt)

1q gives a weighted Lq average of these size

estimates (with the parameter θ controlling the weight). In fact, if we are inan Lp space and replace K(t, x, X) by the decreasing rearrangement x∗ of x,then this is precisely the Lorentz norm ‖x‖Lθ,q . This fact will take on greaterrelevance in a moment.

We call the space endowed with the ‖ · ‖Xθ,q norm, unsurprisingly, Xθ,q.It’s definition bears some resemblance to that of the Lorentz spaces Lp,q, withnorms

‖f‖Lp,q = (

∫(f ∗(t)t

1p )q

dt

t)

1q ) (4)

where f ∗ denotes the decreasing rearrangement of f - that is,

f ∗(t) = inf α ∈ R : λ(x : |f(x)| > α) ≤ t (5)

(i.e., the function f ∗ has the same distribution of values as f does, but thepoints which map to these values do so in such a way that f ∗ is monotonicdecreasing - morally, f ∗ maps 0 to whatever the most popular value of f is,then the next point to the next most popular value, and so on). Somethingmust also be said for the case q =∞ - as is natural, we take the supremum

of the integrand in this case, ‖f‖Lp,∞ = supt f∗(t)t

1p .

These Lorentz spaces have the useful properties that Lp,1 is the restrictedLp space (essentially Lp restricted to indicator functions of sets), Lp,p = Lp,and Lp,∞ is weak Lp.

They are also useful because they carry more information than Lp normsalone, and so are in many cases the most natural spaces in which to look foroptimal bounds on operators. They relate to the Lions-Peetre interpolationspaces in the following manner.

Write the space Xθ,q as [X0, X1]θ,q. Then the following is true: if 1p

=1−θp0

+ θp1

, then [Lp0,s, Lp1,s]θ,q = Lp,q.

Because when dealing with Lebesgue and Lorentz spaces, Xθ,q is not onlyconcrete but again a Lebesgue or Lorentz space, Lions-Peetre interpolationprovides us with useful alternative descriptions of essential spaces. In the

20

following section, we will see one of the ways that this description may beused.

3.3 Skirting the Triangle Inequality via Interpolation

The reason we look to Lions-Peetre interpolation is that it affords us astraightforward means of combining two different spaces’ bounds on piecesof an operator to obtain bounds on that operator in an intermediate space,without incurring the losses of a direct application of the triangle inequal-ity. To make this precise, suppose that an operator T =

∑Tj is defined on

A0 + A1, and maps Ai to the space Bi.We assume the bounds

‖Tj‖Ai→Bi = Mi2jαi (6)

where α0 < 0 < α1.For a given f that is in A0 and in Ai, these bounds can be combined as

‖m∑j=0

Tjf‖B1 + ‖∑j>m

Tjf‖B0 ≤ (7)

m∑j=0

M12jα1‖f‖A1 +∑j>m

M02jα0‖f‖A0 (8)

for arbitrary positive integers m.This situation is perfectly adapted to the Lions-Peetre interpolation since

it is defined in terms of the K functional K(Tf, t, B), which is a weightedsum of the Bi norms of Tf - exactly what we have control over.

To see this, recall that K(Tf, t, B) = infg0+g1=Tf ‖g0‖B0 +t‖g1‖B1 . We usethe decomposition of T =

∑Tj to bound this. Let m be a positive integer

to be chosen in a moment, and let g0 =∑

j>m Tjf , g1 =∑m

j=0 Tjf . Then asin 7, this is bounded by

t

m∑j=0

M12jα1‖f‖A1 +∑j>m

M02jα0‖f‖A0 ≤ (9)

C(tM12mα1‖f‖A1 +M02mα0‖f‖A0) (10)

21

We choose m to minimize this - which is achieved when both terms areabout equal to each other, or tM12mα1‖f‖A1 ≈ M02mα0‖f‖A0 , which is thesame as 2m(α0−α1) ≈ tM1‖f‖A1(M02mα0‖f‖A0)−1.

If we set θ = α1

α1−α0, this minimum is about equal to CM θ

0M1−θ1 ‖f‖θA0

‖f‖1−θA1

.

SinceK is defined via an infimum, we have a uniform bound onK(Tf, t, B),so we have found that

supt>0

K(Tf, t, B) ≤ CM θ0M

1−θ1 ‖f‖θA0

‖f‖1−θA1

(11)

Or in other words, we have obtained

‖Tf‖θ,∞ ≤ CM θ0M

1−θ1 ‖f‖θA0

‖f‖1−θA1

(12)

The applications of this idea ( first appearing in [2] and made explicit in[3]) to the problem at hand are obvious in light of how Lions-Peetre interpo-lation interacts with Lp and Lp,q spaces.

We address these in the next section.

3.4 Finishing up

We can now summarize what the ideas in the Bak-Seeger paper are. Thefirst has straightforward origins - to obtain an optimal result, one must becareful whenever introducing an inequality to be sure that it is the optimalone. We have already shown that switching focus from a bound on ‖f‖L2(dµ)

to the operator T : g 7→ g ∗ µ is optimal in this sense. And for our data,the bounds on Tj : g 7→ g ∗ µj in 2 and 3 are the best possible for Tj asmappings from the spaces considered. But what is not optimal is preciselywhich spaces are being considered in those bounds. So the first idea is this -one must replace the optimal bounds within the class of Lp mappings by theoptimal Lorentz space bounds. One can get a better estimate on ‖Tjg‖2, forinstance, than C2jα1‖g‖2 if one is willing to replace the norm on the right bya Lorentz space norm - we obtain strictly stronger information when workingwith these, and to use any other bounds is to throw away so much that theendpoint becomes unobtainable.

The second idea, and perhaps the one which proceeds the above, is to usethe Lions-Peetre interpolation to combine the bounds we have available and

22

achieve the optimal Lorentz space estimates. After optimal Lortenz boundsare in hand, these may be interpolated to yield the optimal Lp bounds.

To wit, applying the machinery of the previous section to the bounds 2and 3

‖Tjf‖1 ≤ C2jα0‖f‖∞‖Tjf‖2 ≤ C2jα1‖f‖2

(with (A0, A1) = (L1, L∞) and (B0, B1) = (L1, L∞), α0 = −b, α1 = n−a).Since θ = α1

α1−α0, Aθ,1 = Lp0,1 where 1

p0= 1−θ

2+ θ

1= 2α1−α0

2(α1−α0). Similarly,

Bθ,∞ = Lp′0,∞. By 12, this means

‖Tf‖p′0,∞ ≤ C‖f‖p0 (13)

for any f ∈ A0 ∩ A1 = L1 ∩ L2. Since we are working over R, this isn’teverything - but we can say that it includes indicator functions for sets, andso can concluded the restricted (weak) type estimate ‖T‖

Lp0,1 7→Lp′0,∞

.At this point, we can already show (recalling the discussion in Section

3.1) using the bound on T = (g 7→ g ∗ µ) that

‖f‖L2(dµ) = (

∫f(x)f ∗ µ(−x) dx)

12 ≤ (14)

C(‖f‖Lp0,1‖f ∗ µ‖12

Lp′0,∞≤ (15)

C‖f‖Lp′0,∞

(16)

And so we’re done.Except we are not done. This is the best restriction estimate from Lp

′0,∞

to L2(dµ). But this bound is not optimal in the sense that we can get abetter norm on the right, because we used the input ‖Tjf‖2 ≤ C2jα1‖f‖2

which is weaker information than the best Lorentz bound on ‖Tjf‖2, whichwe did not have available.

But the restriction estimate that we’ve just shown will actually give usthe best Lorentz bound on ‖Tjf‖2 - since

23

‖g ∗ µj‖22 =

∫|f |2|µj|2 ≤ (17)

C2jα1

∫|f(x)|2χj ∗ µ(x) dx = C2jα1

∫∫|f(x)|2χj(x− t) dµ(t) dx = (18)

C2jα1

∫χj(x)(

∫|f(x+ t)|2 dµ(t)) dx = (19)

(20)

The restriction estimate that we have already obtained tells us thatthe integral over t is bounded by ‖f‖2

Lp0,1, so the whole is bounded by

C2jα1∫|χj|‖f‖2

Lp0,1, and by the scaling on χj, this is bounded independent

of j.So we replace the naive 3 by the optimal

‖f ∗ µj‖2 ≤ C2jα02 ‖f‖Lp0,1 (21)

Now we run the argument again using this together with 2. We haveeffectively replaced our α1 = d− a by α′1 = d−a

2while changing the space A1

from L2 to Lp0,1. The argument for bounding T then gives

‖T‖Lρ,1→Lσ,∞ ≤ C (22)

where if one calculates the numbers carefully, they find

ρ =2(α′1 − α0)(α1 − α0)

α21 − 3α1α0 + α2

0

and (23)

σ =2(α0 − α′1)

α0

(24)

Since the operator T is self adjoint, we obtain also the bound

‖T‖Lσ′,1→Lρ′,∞ ≤ C (25)

Interpolating between these two bounds gives a range of other optimalLebesgue and Lorentz space estimates (since interpolating two weak typeinequalities yields a strong type, we can move outside of Lp,∞ - in particular,we obtain a bound on T : Lp

′0(Rn)→ Lp0,2).

24

The restriction estimate that this last is equivalent to is ‖f‖L2(dµ) ≤C‖f‖p0,2. We can leave the Lorentz space once and for all if we interpolatethis with the (trivial) optimal L1 → L∞ bound, obtaining

‖f‖Lq(dµ) ≤ C‖f‖p (26)

for p ∈ [1, p0] and q = α1

α1−α0p′.

References

[1] Bak, J. and Seeger, A., Extensions of the Stein-Tomas theorem. Math.Res. Lett. (2011), no. 18, 767–781;

[2] Bourgain, J., Estimations de certaines fonctions maximales. C. R. Acad.Sci. Paris Ser. I Math. (1985), no.10, 499-502;

[3] Carbery, A. and Seeger, A. and Wainger, S. and Wright, J., Classesof singular integral operators along variable lines. Journal of GeometricAnalysis (1999), no. 9, 583-605;

[4] Laba, I., and Pramanik, M., Arithmetic progressions in sets of fractionaldimension. Geom. Funct. Anal. (2008), no.2, 429-456;

[5] Mitsis, T., A Stein-Tomas restriction theorem for general measures.Publ. Math. Debrecen (2002), no.1-9, 89-99;

[6] Mockenhaupt, G., Salem sets and restriction properties of Fourier trans-forms. Geom. Funct. Anal. (2000), no.6, 1579–1587;

Marc Carnovale, UBCemail: [email protected]

25

4 Necessary conditions for Lp(Rn)-Fourier mul-

tipliers

after V. Lebedev and A. Olevskiı [1], [2]A summary written by Vincent Chan

Abstract

We find a necessary condition for the indicator function 1E of ameasurable set E ⊆ Rn to be an Lp(Rn)-Fourier multiplier for p 6= 2.This is then improved to a necessary condition for a general functionf instead of merely indicator functions.

4.1 Introduction

Let G be a locally compact Abelian group and let Γ be its dual group. Let mbe a bounded, measurable function on Γ, and define an associated operatorT = Tm by

T f = mf (1)

for f ∈ (Lp ∩ L2)(G), a dense subset of Lp(G) (1 ≤ p ≤ ∞). If T isa bounded operator on Lp(G), then m is called an Lp(G)-multiplier. Wedenote by Mp(Γ) the space of all these multipliers, and furnish it with thenorm

‖m‖Mp(Γ) = ‖T‖Lp(G)→Lp(G).

Then Mp(Γ) is a Banach space under pointwise multiplication. It is clearthat any operator satsfying (1) commutes with translation or is translation-invariant, that is, we have TSx0 = Sx0T for every translation operator(Sx0f)(x) = f(x+x0), x0 ∈ G. It is known that the converse is also true; thuswe have an easier way to determine if an operator gives rise to a multiplier.A natural question to ask is now, given a function m ∈ L∞(G) and a fixed p,how do we check if m ∈ Lp(G)? It should be said that the cases p = 1 andp = 2 are already well-understood, but in general this is a difficult question.

4.2 Idempotents case

We will simply things by considering the Γ = Rn and examining only indi-cator functions.

26

Indicator functions belonging to Mp(Γ) play an important role in sev-eral areas of mathematics. For instance, they yield a characterization oftranslation invariant, complemented subspaces of Lp(G). Let Bp(Γ) = E ⊆Γ : 1E ∈ Mp(Γ); notice this is an algebra of measurable sets. It can beshown that B1(Rn) = ∅,Rn, and B2(Rn) consists of every measurable set.In the case 1 < p < ∞, it is known that Bp(R) contains all intervals, andthus also all finite unions of intervals. It is natural to ask whether Bp(Γ) is infact a σ-algebra. In particular, the following question was posed: does thereexist a nowhere dense set E of positive measure that belongs to Bp(R) forp 6= 2?

Lebedev and Olevskiı [1] prove this is not the case; if E has positivemeasure then it must contain an entire interval. This is a simple consequenceof their theorem:

Theorem 1 (Lebedev, Olevskiı). If E ∈ Bp(Rn) for p 6= 2 (1 < p < ∞),then E is equivalent to an open set.

Here, equivalence means the symmetric difference of the sets has Lebesguemeasure 0; we denote Lebesgue measure by |·|. We use the following notation:B(x, r) is the ball in Rn with centre x and radius r > 0. A point x ∈ Rn iscalled a density point for E if

limr→0+

|E ∩B(x, r)|B(x, r)

= 1.

We use Ed to denote the set of its density points. It is well known that Ed isequivalent to E. For a set E ⊆ Rn, E is its closure and Ec is its complement.We define the essential boundary of E to be

∂∗E = Ed ∩ (Ec)d.

The following can be shown directly:

Lemma 2. Both E and Ec are equivalent to open sets if and only if |∂∗E| =0.

The next lemma is the key tool used in the theorem.

Lemma 3. Let |∂∗E| > 0. Then for every N ∈ N and any vector εk = 0, 1,1 ≤ k ≤ N , there exist vectors x0, h ∈ Rn such that the arithmetic progressionxk = x0 + kh (1 ≤ k ≤ N) satisfies xk ∈ Ed if εk = 1 and xk ∈ (Ec)d ifεk = 0.

27

We will prove this lemma by induction, building up the desried vectorsone step at a time. Essentially, we begin with vectors x

(0)0 , h(0) such that

x(0)0 +kh(0) ∈ (∂∗E)d, made possible by the hypothesis. For the inductive step,

we assume we have already found vectors x(j)0 , h(j) such that x

(j)k = x

(j)0 +kh(j)

satisfy

x(j)k ∈

Ed if εk = 1, k ≤ j

(Ec)d if εk = 0, k ≤ j

(∂∗E)d if j < k ≤ N.

We then make a small perturbation of these vectors so that the new pro-gression will satisfy the same conditions, with one crucial modification: the(j + 1)st term will be moved from (∂∗E)d to Ed if εj+1 = 1 or to (Ec)d ifεj+1 = 0. To do so, we make use of the definition of denisty points and theessential boundary, and work out the measure theory. To prove the theoremfrom this lemma will be a proof through contradiction, employing the resultof Lemma 2.

4.3 A generalization

12 years later, Lebedev and Olevskiı were able to say much more about theseLp-multipliers, by establishing the essential continuity property for multipli-ers (again, for the case p 6= 2). We say that a point x ∈ Rn is a point ofessential continuity of a function f if, for any ε > 0, there is a neighborhoodUx,ε of x such that |f(y)− f(x)| < ε for almost all y ∈ Ux,ε, and say f is es-sentially continuous if every point is a point of essential continuity. Lebedevand Olevskiı [2] showed:

Theorem 4 (Lebedev, Olevskiı). If f ∈ Mp(Rn) for p 6= 2 (1 0. Then for every N ∈ N and any vectorεk = ±1, 1 ≤ k ≤ N , there exist vectors x0, h ∈ Rn such that the arithmeticprogression xk = x0 + kh (1 ≤ k ≤ N) satisfies xk ∈ Ed

1 if εk = −1 andxk ∈ Ed

2 if εk = 1.

28

For a complex-valued function f ∈ L∞(Rn), we define its essential oscil-lation Ω(f, x) at a point x as

Ω(f, x) = limδ→0+

ess supy:|y−x|<δ

|f(y)− f(x)|.

Notice if x is a point of essential continuity of f , then Ω(f, x) = 0.To prove the theorem, we assume the set of points for which f is not

essentially continuous has positive measure. Then

E = x ∈ L(f) : Ω(f, x) > ε

has positive measure for some ε > 0, where L(f) is the set of Lebesgue pointsof f . Let c ∈ R be such that

E1 = x ∈ E : |f(x)− c| < ε/3

has positive measure, and let

E2 = x ∈ L(f) : |f(x)− c| > 2ε/3 .

Since |Ed1 ∩ Ed

2 | > 0, we can apply Lemma 5 to these sets; the specialvector εk = ±1, 1 ≤ k ≤ N , will be chosen so that the polynomialP (x) =

∑Nk=1 εke

ikx satisfies ‖P‖Lq(T) ≤ cp‖P‖L2(T) = cp√N . Some in-

equality manipulations will yield a contradiction as we take N sufficientlylarge, to complete the proof.

There are two things of note at this stage: first, it is easy to check thatthis is indeed a strengthening of Theorem 1. Secondly, the following is true:

Lemma 6. The following are equivalent:

(a) Almost every point x ∈ Rn is a point of essential continuity of f .

(b) There is a function g such that f = g almost everywhere, and g iscontinuous almost everywhere.

Then with this lemma, the main theorem can be restated as

Theorem 7 (Lebedev, Olevskiı). If f ∈Mp(Rn) for p 6= 2 (1 < p <∞), thenf coincides almost everywhere with a function which is continuous almosteverywhere.

29

References

[1] Lebedev, V. and Olevskiı, A., Idempotents of Fourier multiplier algebraGeom. Funct. Anal. 4 (1994), pp. 539-544;

[2] Lebedev, V. and Olevskiı, A., Fourier Lp-multipliers with bounded pow-ers (Russian) Izv. Ross. Akad. Nauk Ser. Mat. 70 (2006) pp 129–166;translation in Izv. Math 70 (2006) n0. 3 , 549–585 ;

Vincent Chan, UBCemail: [email protected]

30

5 Salem-Bluhm’s construction of Salem sets

after R. Salem [1] and C. Bluhm [2]A summary written by Xianghong Chen

Abstract

Given α ∈ (0, 1), we construct a random Cantor set whose Fourierand Hausdorff dimensions equal α almost surely.

5.1 Introduction

Recall that in the construction of the standard 13

Cantor set there are threeingredients: the dissection number 2, the dissection ratio 1

3and the positions

of the subintervals 0 and 23

(which we will call translations).Given α ∈ (0, 1), we are interested in constructing a Cantor-type set

whose Fourier dimension (see below for definition) and Hausdorff dimensionare both equal to α. Such sets are called Salem sets and were first constructedby Salem [1]. They are special in the sense that they close the gap betweenthe Fourier and Hausdorff dimensions (it is a general fact that the formercan not exceed the latter).

Salem achieved this by randomizing the dissection ratios and picking in-commensurable translations in the construction of Cantor set. On the otherhand one can also instead randomize the translations in order to obtain Salemsets. Both approaches increase the dissection number at each step in orderto make up for the ε-loss of decay in the case without such increments. Wewill follow the second approach which was introduced by Bluhm [2].

In what follows, we will restrict ourselves to R1. All measures are definedon Borel σ-algebra. The Fourier transform of a finite measure µ is definedby µ(ξ) =

∫eiξtµ(dt).

5.2 The main result

Theorem 1. Given α ∈ (0, 1), there exists a compact set K ⊂ [0, 1] and aprobability measure µ supported on K, such that(i) K has Hausdorff dimension α(ii) for all β < α, µ(ξ) = O(|ξ|−β/2) as |ξ| → ∞(iii) µ(I) . |I|α for all interval I.

31

5.3 The set

In fact we will construct a class of K most of which will have the propertiesstated in the theorem.

The construction will start with the nominal second step. In theN -th stepof the construction, the dissection number will be precisly N , the dissectionratio will be denoted by θN , the translations by XN,j, where j = 1, · · · , N .

More precisely, for N ∈ N, N ≥ 2, let θN = N−1α . Notice that

N−1 − θN = N−1(1−N−( 1α−1)) > cαN

−1.

Here we put cα = [1− 2−( 1α−1)]/3. Hence cα gives a uniform lower bound for

the portion of the gap that an interval of length θN can not fill in an intervalof length N−1.

For each N , pick XN,j ∈ [ j−1N

+ cαN, j−1N

+ 2cαN

], j = 1, · · · , N . Then we cancorrespondingly “dissect” [0, 1] into N disjoint intervals [XN,j, XN,j + θN ].

Now start with N = 2, we “dissect” [0, 1] into two intervals [X2,j2 , X2,j2 +θ2], j2 = 1, 2. Then perform the dissection with N = 3 to each [X2,j2 , X2,j2 +θ2], we get six intervals [X2,j2 + θ2X3,j3 , X2,j2 + θ2X3,j3 + θ2θ3], j2 = 1, 2, j3 =1, 2, 3. Continue the procedure, after the N -th step, we get 2 ·3 · · ·N disjointclosed intervals of the form

[X2,j2 + · · ·+ θ2 . . . θN−1XN,jN , X2,j2 + · · ·+ θ2 . . . θN−1XN,jN + θ2 . . . θN−1θN ].

32

Denote by KN the union of these intervals and set KX = ∩NKN , wherethe index

X = (XN,jN ) N=2,3,···jN=1,··· ,N

Then KX is a compact set.

5.4 The measure

Equip KN with the uniform probability measure µN and let FN be its dis-tribution function. Since FN is continuous and ‖FN − FN+1‖∞ ≤ (N !)−1,FN converges uniformly to a continuous distribution function F . DenoteµX = dF , the probability measure corresponding to F , then µX(KX) = 1.

5.5 The Fourier transform

Since µN converges weakly to µX , in particular, µN(ξ) → µX(ξ), ∀ξ. Noticethat for ξ 6= 0,

µN(ξ) =eiξθ2···θN − 1

iξθ2 · · · θN1

N · · · 2∑

j2,··· ,jN

eiξ(X2,j2+···+θ2···θN−1XN,jN )

=eiξθ2···θN − 1

iξθ2 · · · θN

N∏k=2

(1

k

k∑jk=1

eiξθ2···θk−1Xk,jk )

Let N →∞ we get

µX(ξ) =∞∏k=2

(1

k

k∑jk=1

eiξθ2···θk−1Xk,jk ),∀ξ.

5.6 Randomization

Now we randomize X such that XN,jN , N = 2, 3, · · · , jN = 1, · · · , N areindependent and eachXN,jN is uniformly distributed on [ jN−1

N+ cα

N, jN−1

N+ 2cα

N].

In what follows we suppress the subscript X.

5.7 From average decay to deterministic decay

It now suffices prove the Fourier decay estimate in the average sense. Pre-cisely, for any q,m ∈ N, q,m ≥ 1, we will show that for some constant

33

C = C(α,m, q),

E[|µ(ξ)|2q] ≤ |ξ|−(1− 1m

)αq,∀|ξ| ≥ C.

Assuming this is proven, we can choose q > 2mα−1 and let ξ = n ∈Z, |n| ≥ C in the above inequality, then

E[|n|(1−2m

)αq|µ(n)|2q] ≤ |n|−1mαq

Summing over n, we get

E[∑|n|≥C

|n|(1−2m

)αq|µ(n)|2q] ≤∑|n|≥1

|n|−2 <∞

Henceµ(n) = O(|n|−(1− 2

m)α

2 ), a.s.

In order to pass from the integers to the reals, notice the following

Lemma 2 (cf. [3] p.252). Let µ be a probability measure supported on [0, 1]and β > 0 such that µ(n) = O(|n|−β), then µ(ξ) = O(|ξ|−β).

Applying this lemma we see that almost surely we have

µ(ξ) = O(|ξ|−(1− 2m

)α2 ),∀m.

5.8 The key estimate

To prove the average decay estimate we first estimate

E[|1k

k∑j=1

eiηXk,j |2q] =1

k2qE[(

k∑j1,··· ,jq=1

eiη(Xk,j1+···+Xk,jq ))(k∑

i1,··· ,iq=1

e−iη(Xk,i1+···+Xk,iq ))]

=1

k2qE[

k∑j1,··· ,jq=1

∑i1,··· ,iq

=j1,··· ,jq

1] +1

k2qE[

∑n1,··· ,nk∈Z

(n1,··· ,nk)6=0

eiη(n1Xk,1+···+nkXk,k)]

≤ q!

kq+ sup

j=1,··· ,kn∈Z,n6=0

|E[eiηnXk,j ]|

≤ qq

kq+ 2c−1

α k|η|−1

34

E[|µ(ξ)|2q]1q ≤ (2

1q q)N

N !

≤ cα,q|ξ|−α(N + 1)αq+α(21q q)N

≤ cα,q|ξ|−α[(N − 1)!]1m

≤ cα,q|ξ|−αc− 1m

α,q |ξ|αm

= cα,m,q|ξ|−(1− 1m

)α

Since m can be arbitrarily large, one can rid of the constant by choosinglarger |ξ|. Raise both sides to the power q, we get for some C = C(α,m, q)

E[|µ(ξ)|2q] ≤ |ξ|−(1− 1m

)αq,∀|ξ| ≥ C.

5.10 The dimensions

Let K be a compact set in R1, define the Fourier dimension of K by

dimF (K) = supβ ∈ [0, 1] : ∃µ ∈ P(K), s.t. µ(ξ) = O(|ξ|−β/2)

where P(K) denotes the space of probability measures on K.

Lemma 3 (cf. [3] p.133). For any compact set K in R1, dimF (K) ≤ dimH(K).

Here dimH(K) denotes the Hausdorff dimension of K. Finally, one canshow that in the above construction, for anyK = KX and µ = µX , dimH(K) =α and µ(I) . |I|α for all interval I.

References

[1] Salem, R., On singular monotonic functions whose spectrum has a givenHausdorff dimension. Ark. Mat. 1 (1951), no. 4, 353–365;

[2] Bluhm, C., Random recursive construction of Salem sets. Ark. Mat. 34(1996), no. 1, 51–63;

[3] Kahane, J.-P., Some Random Series of Functions. 2nd ed, 1985, Cam-bridge University Press, Cambridge.

Xianghong Chen, University of Wisconsin-Madisonemail: [email protected]

36

6 Buffon’s needle estimates for rational prod-

uct Cantor sets

after M. Bond, I. Laba, and A. Volberg [2]A summary written by Kyle Hambrook

Abstract

The probability that Buffon’s needle intersects an e−n neighbour-hood of a one-dimensional self-similar rational product set, where thefactors are defined by at most 6 similarities, is at most Cn−p/ log logn

for some p > 0.

6.1 Introduction

The Buffon needle probability, or Favard length, of a compact set S ⊂ C is

Fav(S) :=1

π

∫ π

0

|projθ(S)|dθ.

Here projθ denotes orthogonal projection onto the line through the ori-gin making the angle θ with positive real axis. Pointwise, projθ(re

iθ′) :=r cos(θ′ − θ). We use |F | to denote the Lebesgue measure of F ⊂ R.

A set C is called a self-similar set if there is a positive integer L and dis-tinct, non-collinear points z1, . . . , zL ∈ C such that C is the unique compactset for which C =

⋃Lj=1 Tj(C). Here Tj : C → C are the so-called similarity

maps defined by Tj(z) = 1Lz + zj.

We are interested in Fav(SN), where SN is an L−N -neighbourhood of aself-similar set S∞. S∞ is of Hausdorff dimension ≤ 1 and of finite H1 mea-sure. Since the zj are not collinear, S∞ is unrectifiable and the Besicovitchtheorem says that |projθ(S∞)| = 0 for almost every θ (see [6]). It follows that

limN→∞

Fav(SN) = Fav(S∞) = 0. (1)

We are concerned with the rate of decay in (1). The main result of thispaper concerns the rational product set case where zjLj=1 = A×B for someA,B ⊂ Q. Without loss of generality, we may assume that A,B ⊂ Z andmin(A) = min(B) = 0. Define SN = AN ×BN +

z ∈ C : |z| < L−N

, where

A1 := A, AN+1 := AN + L−N−1A, B1 := B, and BN+1 := BN + L−N−1B.This is inconsistent with the general definition of SN that we gave above, butit is equivalent to it up to constants and more convenient to use.

37

Theorem 1. If SN = AN×BN and |A|, |B| ≤ 6, then Fav(SN) . N−p/ log logN

for some p > 0.

The proof of Theorem 1 is based on a new method of estimating so-called“Riesz products” of trigonometric polynomials. We now outline the proof.The arguments of [7] with the additional modifications of [3], [4], [5] reducethe proof to the problem of proving lower bounds on integrals of the form∫ 1

L−m

n∏j=1

|φt(Ljξ)|2dξ, (2)

where t = tan(θ) and

φt(ξ) :=1

L

∑(a,b)∈A×B

e2πi(a+tb)ξ.

We describe the reduction in more detail in Section 6.4.The s-th cyclotomic polynomial, for s ∈ N, is

Φs(x) :=∏

1≤d≤s(d,s)=1

(x− e2πid/s).

Definition 2. Let F (x) ∈ Z[x]. We write F (x) =∏4

i=1 F(i)(x), where each

F (i)(x) as a product of irreducible factors of A(x) in Z[x] defined as follows.

• F (1)(x) =∏

s∈S(1)F

Φs(x), S(1)F = s ∈ N : Φs(x) | F (x), (s, L) 6= 1,

• F (2)(x) =∏

s∈S(2)F

Φs(x), S(2)F = s ∈ N : Φs(x) | F (x), (s, L) = 1,

• F (3)(x) is the product of those irreducible factors of F (x) that have atleast one root of the form e2πiξ0 , ξ0 ∈ R \Q,

• F (4)(x) is the product of those irreducible factors of F (x) that have noroots on the unit circle.

Define

F ′(x) := F (1)(x)F (3)(x)F (4)(x), F ′′(x) := F (2)(x).

38

We make the following sequence of definitions

A(x) =∑a∈A

xa, φA(ξ) =1

|A|A(e2πiξ),

B(x) =∑b∈B

xb, φB(ξ) =1

|B|B(e2πiξ),

φ′A(ξ) = A′(e2πiξ), φ′B(ξ) = B′(e2πiξ), φ′t(ξ) = φ′A(ξ)φ′B(tξ),

φ′B(ξ) = B′(e2πiξ), φ′′B(ξ) = B′(e2πiξ), φ′′t (ξ) = φ′′A(ξ)φ′′B(tξ).

Then

φA(ξ) =1

|A|φ′A(ξ)φ′′A(ξ), φB(ξ) =

1

|B|φ′B(ξ)φ′′B(ξ),

φt(ξ) =1

Lφ′t(ξ)φ

′′t (ξ) = φA(ξ)φB(tξ).

We further define

P1(ξ) =n∏

j=m+1

φt(Ljξ), P ′1(ξ) =

n∏j=m+1

φ′t(Ljξ), P ′′1 (ξ) =

n∏j=m+1

φ′′t (Ljξ),

P2(ξ) =m∏j=1

φt(Ljξ), P ′2(ξ) =

m∏j=1

φ′t(Ljξ), P ′′2 (ξ) =

m∏j=1

φ′′t (Ljξ).

Since the integrand in (2) is unchanged under the reflection ξ → −ξ, we canwrite (2) as∫ 1

L−m

n∏j=1

|φt(Ljξ)|2dξ =1

2L−m

∫[−1,1]\[−L−m,L−m]

|P1(ξ)|2|P ′2(ξ)|2|P ′′2 (ξ)|2dξ

Our plan to deduce a lower bound for this integral is as follows. We first findlarge subsets V ′ and V ′′ of [−1, 1] on which P ′2 and P ′′2 are large. This leavesus to bound the integral∫

V ′⋂V ′′\[−L−m,L−m]

|P1(ξ)|2dξ.

It is easy to see that this integral is

≥∫V ′′|P1|2 −

∫(V ′

⋃[−L−m,L−m])c

|P1|2 −∫

[−L−m,L−m]

|P1|2. (3)

39

Good upper bounds for the last two integrals are established following themethods of [1], [3], [4], [5], and [7]; Lemmas 10 and 11 below are the precisestatements of the upper bounds we use. The method for proving a sufficientlystrong lower bound on

∫V ′′|P1|2 is called Salem’s trick and requires that V ′′

have a specific structure. Namely, we will require that V ′′ is the support ofa function with a non-negative Fourier transform. To meet this requirement,we will choose V ′′ to be a difference set Γ−Γ. Indeed, this is why we exploitthe symmetry of the integrand with respect to the reflection ξ → −ξ.

6.2 The SSV property of φ′t

We now state the terminology and results needed for the choice of V ′.

Definition 3. Let ϕ : R → C and ψ : N → [0,∞). We say that ϕ has theSSV property with SSV function ψ if there exist c2, c3 > 0 with c3 c2

such that

SSV :=

ξ ∈ [0, 1] :

m∏k=1

|ϕ(Lkξ)| . ψ(m)

is contained in Lc2m intervals of size L−c3m. If ψ(m) equals L−c1m, L−c1m logm,or L−c1m

2for some c1 > 0, we say that ϕ has the SSV, log-SSV, or square-

SSV property, accordingly.

In our application, the function ϕ will be either φ′t or one of its factors(recall P ′2(ξ) =

∏mk=1 φ

′t(L

kξ)), and we will need the constants ci to be uniformin t.

Proposition 4. φ′A has the log-SSV property. If A(x) has no roots e2πiξ0

with ξ0 ∈ R \Q, then φ′A has the SSV property. We can arrange for c3/c2 tobe as large as we want at the cost of increasing c1. The same assertions holdwhen φ′A and A(x) are replaced by φ′B(t · ) and B(x).

Observe that if ϕ1, ϕ2 have the SSV property with SSV function ψ, then sodoes ϕ1·ϕ2. One consequence of this observation is that we may consider eachfactor of A separately in proving Proposition 4. The proof that A(1)(e2πiξ)has the SSV property is straightforward argument relying on the fact thatΦs(x) and Φs(x

Lk) have common zeroes when (s, L) 6= 1. The proof thatA(3)(e2πiξ) has the log-SSV property relies on Baker’s theorem from the theory

40

of Diophantine approximation. Of course, if A(x) has no roots e2πiξ0 withξ0 ∈ R\Q, then A(3)(e2πiξ) ≡ 1 has the SSV property. As A(4)(e2πiξ) is neverzero, its contribution to the SSV set can absorbed into constants. The proof

As another consequence of our observation, Proposition 4 implies thatφ′t(ξ) = φ′A(ξ)φ′B(tξ) has at least the log-SSV property, and has the SSVproperty if both φ′A and φ′B(t·) do. The complement of the set

SSV (t) :=

ξ ∈ [0, 1] :

m∏k=1

|φ′t(Lkξ)| . ψ(m)

(for a particular value of t) will be the set playing the role of V ′.

6.3 The SLV structure of φ′′t

In this section, we give the terminology and results needed to describe thechoice of V ′′.

Definition 5. Let ϕ : R → C. We say that ϕ is SLV-structured if thereis a Borel set Γ ⊂ [0, 1] and constants C1, C2 such that

Γ− Γ ⊂

ξ ∈ R :

m∏k=1

|ϕ(Lkξ)| ≥ L−C1m

,

|Γ| ≥ C2KL−m

We call Γ an SLV set for ϕ.

The set V ′′ will be Γ− Γ, where Γ is an SLV set for φ′′t .To construct Γ requires that we understand the structure of the set where

P ′′2 is large, and to understand this we study which s ∈ N have Φs(x) | A(x)(and do similarly for B(x)). Since Φs(x) | A(x) if and only if

∑a∈A ζ

a = 0for each primitive s-th root of unity ζ, we are motivated to study so-calledlinear-multi polygon relations: A finite set of the form z1ζ1, . . . , zJζJ, wherez1, . . . , zJ ∈ Z and ζ1, . . . , ζJ are roots of unity, is called a linear-multipolygon relation (LMPR) if it satisfies

J∑j=1

zjζj = 0.

41

The proofs of Propositions 6 and 7 below rely on the theory of LMPRs (and,for the latter, a calculation exploiting projective relationships of the form(ζrs)

r = ζs).The following proposition implies that we can take Γ = [0, 1] in case

|A| 6= 5, |B| 6= 5

Proposition 6. If |A| = 2, 3, 4, 6, then A′′(x) ≡ 1. The same holds if A isreplaced by B.

The case |A| = |B| = 5 requires significantly more effort. It is handledby the combination of the next two propositions.

Proposition 7. Let SA = r : Φr(x) | A(x), (r, |A|) = 1. Suppose |A| = 5.There are j0, k0 ∈ N, depending only on A, such that any s ∈ SA has the forms = 2j03k0M for some M with (M, 6) = 1. In particular, if s0 = lcm(SA),then for each q | s0

2and each q | s0

3the set

ζaqa∈A, where ζq is a primitive

q-th root of unity, is not a LMPR. Analogous assertions hold when A isreplaced by B.

Set sA = 1 if q ∈ N : Φq | A, (q, L) = 1 is empty; otherwise, set sA =lcm(q ∈ N : Φq | A, (q, L) = 1). Define sB analogously. Let

K =

N ε0 if φ′t has the SSV property,

N ε0/ log logN if φ′t has the log-SSV property

where ε0 > 0 is a parameter.

Proposition 8. Suppose that we can write sA = s1,As2,A for integers s1,A,s2,A > 1 such that s2,A < |A| and such that Φq(x) does not divide A(x) forany q | s1,A with (q, L) = 1. Suppose that the analogous thing can be donewith B in place of A. Then there is a set Γ that is a finite union of intervalssuch that

Γ− Γ ⊂

ξ ∈ R :

m∏k=1

|ϕ(Lkξ)| ≥ L−C1m

,

|Γ| ≥ C2KL−m.

The zeroes of φ′′A and φ′′B are discrete subgroups of R with coarser sub-groups removed from them. The idea of Proposition 8 is to arrange for Γ−Γto be contained in an intersection of neighbourhoods of rescaled copies of

42

these coarser subgroups. This will keep Γ − Γ separated from the zeroesof P ′′2 . In order to ensure that the intersections are generically-sized, henceensuring that |Γ| is large enough, we use pigeonholing to choose appropriatecosets rather than use groups at each stage.

6.4 Reduction to lower bounds on integrals; requiredupper bounds

In this section, we describe how the proof of Theorem 1 is reduced to estab-lising lower bounds on integrals of the form (2). We also state precisely theupper bounds we need for the last two integrals in (3).

We make the change of variable t = tan(θ). This does no harm as weuse symmetry to consider only the case θ ∈ [0, π/4]. After rescaling, forz ∈ An×Bn, we may write projθ(z) = a+ tb for some a ∈ An, b ∈ Bn. Definethe counting function

fn,t :=∑

z∈An×Bn

1projθ(z+[0,L−n]2).

fn,t(x) counts the number of squares (of side L−n) that lie “above” or “below”x when the ray forming the angle θ = arctan(t) with the real axis is regardedas the positive “horizontal” direction. Using that supp(fn,t) = projθ(SN)and self-similarity, we can establish a quantitative version of the statement:|projθ(SN)| is small if and only if ‖fn,t‖2 is large. Set

νn = ∗nk=1νk, νk =1

L

∑(a,b)∈A×B

δL−ka+tL−kb.

Then

fn,t = Ln1[0,L−n] ∗ νn, fn,t(ξ) = Ln1[0,L−n](ξ) ·n∏k=1

φt(L−kξ).

Since ‖fn,t‖2 = ‖fn,t‖2, and since we can use a pigeonholing argument to effec-

tively neglect the decay factor Ln1[0,L−n](ξ), the task of bounding |projθ(SN)|from above is reduced to the problem of proving a lower bound on‖∏n

k=1 φt(L−kξ)‖2. The following proposition states the reduction precisely;

the set E appearing below is a set of directions t = tan(θ) which we do notdefine here.

43

Proposition 9. Theorem 1.2 is implied by the following statement: If ε0 > 0is sufficiently small, and if |E| ≥ 1/2K1/2, there is a t ∈ E such that∫ 1

L−m

n∏j=1

|φt(Ljξ)|2dx ≥ cKL−nN−αε0 .

The following two lemmas furnish the upper bounds we need on the lasttwo integrals in (3).

Lemma 10. For t ∈ E, we have∫ L−m

−L−m|P1|2 ≤ 4C0KL

−n. (4)

Lemma 11. Assume φt has the log-SSV property. If ε0 > 0 is sufficientlysmall, and if |E| ≥ 1/2K1/2, there exists a t0 ∈ E such that∫

SSV(t0)⋂

[L−1,1]

|P1,t0(ξ)|2dξ ≤ C0KL−n.

6.5 The main argument

Now we prove the statement which, according to Proposition 9, implies The-orem 1.

Proposition 12. If ε0 > 0 is sufficiently small, and if |E| ≥ 1/2K1/2, thereis a t ∈ E such that∫ 1

L−m

n∏j=1

|φt(Ljξ)|2dξ ≥ cKL−nN−αε0 . (5)

Proof. By Proposition 4, φt has at least the log-SSV property. Assume |E| ≥1/2K1/2, and assume ε0 > 0 is small enough that the hypothesis of Lemma11 is satisfied. Let t = t0 be the direction in E that Lemma 11 furnishes.Write the intergral in (5) as

1

2L−m

∫[−1,1]\[−L−m,L−m]

|P1(ξ)|2|P ′2(ξ)|2|P ′′2 (ξ)|2dξ. (6)

44

According to Section 6.2, we can find a SLV set Γ for P ′2. So, since |P ′′2 | ≥L−C1m on Γ− Γ, (6) is

≥ 1

2L−2C1m−m

∫Γ−Γ\[−L−m,L−m]

|P1|2|P ′2|2

By the definition of SSV (t0), |P ′2| & ψ(m) on SSV(t0)c. Hence (6) is

& L−2C1m−mψ(m)2

∫(Γ−Γ\[−L−m,L−m])

⋂SSV(t0)c

|P1|2

≥ L−2C1m−mψ(m)2

(∫Γ−Γ

|P1|2 −∫

[−L−m,Lm]

|P1|2 −∫

[L−m,1]⋂

SSV(t0)

|P1|2).

To bound∫

Γ−Γ|P1|2, we employ Salem’s trick on difference sets. Write

P1(ξ) = L−n−m∑

α∈A e2πiαξ, and note |A| = Ln−m. Let h = |Γ|−11Γ ∗ 1−Γ.

Then 0 ≤ h ≤ 1 and h = |Γ|−1|1Γ|2 ≥ 0. Hence∫Γ−Γ

|P1(ξ)|2dξ ≥∫

Γ−Γ

|P1(ξ)|2h(ξ)dξ

= L−2(n−m)∑α,α′∈A

∫Γ−Γ

h(ξ)e2πi(α−α′)ξdξ

= L−2(n−m)∑α,α′∈A

h(α− α′)

≥ L−2(n−m)∑α∈A

h(0) = L−2(n−m)|Γ| · |A|

≥ C2KL−mL−(n−m) = C2KL

−n.

Since C2 > 5C0, by combining the bound for∫

Γ−Γ|P1|2 with Lemmas 10 and

11 we find that (6) is

& L−2C1m−mψ(m)2KL−n & KL−nN−αε0

for some α > 0. The last inequality is true by the SSV or log-SSV propertyof φ′t and an appropriate choice of m

References

[1] Bond, M., Combinatorial and Fourier Analytic L2 Methods ForBuffon’s Needle Problem.

45

http://bondmatt.wordpress.com/2011/03/02/thesissecond-complete-draft/

[2] Bond, M., Laba, I., and Volberg, A., Buffon’s needle estimates forrational product Cantor sets.

[3] Bond, M. and Volberg, A., Buffon needle lands in ε-neighbourhooodof a 1-dimensional Sierpinski Gasket with probability at most | log ε|−c.Comptes Rendus Mathematique, 348 (2010), Issues 11-12, 653-656.

[4] Bond, M. and Volberg, A., Buffon’s needle landing near Besicovitchirregular self-similar sets. http://arxiv.org/abs/0912.5111

[5] Laba, I. and Zhai, K., The Favard length of product Cantor sets. Bull.London Math. Soc., 42 (2010), 370-377.

[6] Mattila, P., Geometry of Sets and Measures in Euclidean Spaces. Cam-bridge University Press, 1995.

[7] Nazarov, I., Peres, Y., and Volberg, A., The power law for the Buffonneedle probability of the four-corner Cantor set. Algebra i Analiz, 22(2010), 82-97; translation in St. Petersburg Math. J., 22 (2010), 6172.

Kyle Hambrook, UBCemail: [email protected]

46

7 Projecting the One-Dimensional Sierpinski

Gasket

after Richard Kenyon [3]A summary written by Edward Kroc

Abstract

We define the so-called one-dimensional Sierpinski gasket, calcu-late the measure of its linear projections in any direction and obtainbounds on the Hausdorff dimension of these projections.

7.1 Introduction

Define

S =

∞∑i=1

αi3−i | αi ∈ (0, 0), (1, 0), (0, 1)

.

It is easy to see that S may be described as the set

S = (x, y) ∈ C × C | x+ y ∈ C

where C is the ordinary “middle third” Cantor set constructed on [0, 1/2].Equivalently, S is the attractor in R2 for the three linear maps (x, y) 7→ (x

3, y

3),

(x, y) 7→ (x+13, y

3), (x, y) 7→ (x

3, y+1

3).

The set S resembles the Sierpinski gasket, obtained by replacing all oc-currences of 3 with 2 in the first (or third) description. We call S the one-dimensional Sierpenski gasket since S is a union of three copies of itself, eachscaled by a factor of 1/3, and thus has Hausdorff dimension 1.

Our main objects of study are the various linear projections of S onto thex-axis: define Su = πu(S), where

πu =

(1 u0 0

).

Notice that S0 = C and S1/2 = [0, 1/2]. In general, Su is the set ofreal numbers which have an expansion using negative powers of 3 and digits0, 1, u. Note that diam(Su) = 1/2 if 0 1,and diam(Su) = (1 + u)/2 if u < 0.

47

We will be concerned with both the linear measure and the dimension ofthe various projections Su. Througout this summary, we let µ denote one-dimensional Lebesgue measure and dim(E) denote the Hausdorff dimensionof the set E.

A theorem of Besicovitch [1] states that the projection in almost everydirection of an irregular set of Hausdorff dimension 1 in R2 has Lebesgueone-dimensional measure 0. We compute the measure of Su for every u.

Theorem 1. µ(Su) > 0 iff u is a rational of the form p/q in lowest termswith p+ q ≡ 0 mod 3. In this case, µ(Su) = 1/q.

This result gives a simple criterion for deciding when a set of 3 non-negative numbers represents a set of positive measure in base 3, a questionformulated by Odlyzko [5].

Concerning the dimension of our projections, a theorem of Marstrand[4] states that almost every linear projection of a set of dimension 1 in R2

has dimension 1. Furstenberg has conjectured that dim(Su) = 1 for everyirrational u; this remains open. We obtain bounds for the dimension of Suwhen u is close to a single rational (or is well-approximated by an appropriatesequence of rationals). A uniform bound on dim(Su) of 1− log(5/3)/2 log 3 >0.767 for any irrational u has since been obtained by Swiatek and Veerman[6] using energy estimates for certain natural measures supported on theprojections Su.

7.2 The measure of the projections

We prove the forward direction of Theorem 1 via a chain of lemmas. Theopposite direction follows from our lemmas and the fact that dim(Su) < 1 ifu = p/q with p+ q 6≡ 0 mod 3 (see [3]).

Lemma 2. If µ(Su) > 0, then Su contains an interval.

Sketch of proof. By definition,

Su =

∞∑i=1

ai3−i | ai ∈ 0, 1, u

;

thus,3Su = Su ∪ (Su + 1) ∪ (Su + u), (1)

48

where for A ⊂ R and x ∈ R, xA = xa | a ∈ A and x+A = x+a | a ∈ A.By subadditivity and the scaling property of Lebesgue measure, we see thatthe three translated copies of Su which cover 3Su are disjoint in measure.Similarly, for any n ≥ 0, 3nSu is covered by at most 3n translates of Supairwise disjoint in measure.

Since µ(Su) > 0, we may arrange these translates to “fill out” any intervalthat contains a Lebesgue point of Su. Since Su is closed, we invoke the Bairecategory theorem to conclude that Su contains an interval.

Lemma 3. If Su contains an interval, then u is rational.

Sketch of proof. This can be seen by appealing to the self-similarity of the setSu. We may tile the real line in a periodic manner using the translates of Sufrom 3nSu (we call these translates “tiles”). The possibility of such a tilingfollows from the fact that if Su has interior, then it is the closure of its interiorand that the boundary of Su has measure 0. Periodicity is imposed on anysuch tiling of R by virtue of the invariance of the tiling under expansion (bya factor of 3) and subdivision of the tiles (as in (1)). A simple calculationshows that such periodicity is only possible if u is rational.

Lemma 4. If u = p/q in lowest terms with p + q ≡ 0 mod 3, then µ(Su) =1/q.

Sketch of proof. Let 0 < p < q. Define Snp,q = ∑n

i=1 ai3−i | ai ∈ 0, p, q, so

that S∞p,q = qSu. Now Snp,q consists of 3n triadic rationals which are distinctsince p+ q ≡ 0 mod 3.

Define probability measures µn = 3−n∑

x∈Snp,qδ(x) where δ(x) is the unit

point mass at x. For each triadic interval I = [p3−k, (p + 1)3−k) with k ≤ nwe have µn(I) ≤ µ(I). Since S∞p,q is closed, we take a weak limit µ∞ of µn tofind that 1 = µ∞(S∞p,q) ≤ µ(S∞p,q).

We tile R as in the proof of Lemma 3 using translates of S∞p,q. Showingthat the period of this tiling is 1 will yield µ(S∞p,q) = 1, and thus µ(Su) = 1/q.

Let R ∈ Z+ be a period of the tiling. Then there is a set W ⊆ [0, R) ∩ Zsuch that each tile is of the form x+ S∞p,q, where x ∈ W +RZ, and there aretiles at each point of W +RZ. By the self-similarity of S∞p,q, the set W takenmodulo R is invariant under the three maps x 7→ 3x, x 7→ 3x+p, x 7→ 3x+q.

Let G be the directed graph with vertices V = [0, R)∩Z and edges fromx to (3x + d) mod R for each d ∈ 0, p, q. Let f be an eigenvector for theadjacency matrix T with eigenvalue 3:

Tf(x) = f(3x) + f(3x+ p) + f(3x+ q) = 3f(x), (2)

49

where the arithmetic in the arguments is done mod R. Note that we mayalways choose R large enough so that T has an eigenvalue of 3. Assume that3 - R (the alternative case is similar). Identifying V with ZR, we write thenth Fourier coefficient of 3φ(R)f and simplify using (2). This gives that thenth Fourier coefficient is nonzero only if n = 0. Consequently, f is constanton W (its support) and so W = [0, R) ∩ Z.

7.3 Bounds on the dimension of the projections

If u is rational, dim(Su) = limk→∞

1

klog3 |Sku|, where Sku =

∑ki=1 ai3

−i | ai ∈0, 1, u; i.e. dim(Su) equals the Minkowski dimension of Su. This followsfrom a counting argument and a theorem of Falconer [2] that guarantees theMinkowski and Hausdorff dimensions of self-similar sets agree.

Theorem 5. Let u be a real number and pi/qi a sequence of rationals suchthat pi+qi ≡ 0 mod 3, qi →∞, and such that there exists constants C, α > 0for which ∣∣∣∣u− pi

qi

∣∣∣∣ < C

qαi.

Then dim(Su) ≥ 1 − 1/α. In particular, if u is an appropriate Liouvillenumber, then dim(Su) = 1. This is a residual subset of R.

Sketch of proof. Suppose |u−p/q| < (2/3)3−k/q. The set Skp/q ⊂ Sp/q consists

of 3k distinct points on the lattice (3−k/q)Z. So we require at least 3k intervalsof length 3−k/q to cover Sp/q.

For each x ∈ Sp/q, let x′ be the point with the same sequence of digits,replacing all occurrences of the digit p/q with u. Then

|x− x′| <∞∑i=1

∣∣∣∣u− p

q

∣∣∣∣ 3−i =3

2

∣∣∣∣u− p

q

∣∣∣∣ < 3−k

q.

Thus it takes at least (1/3)3k intervals of length 3−k/q to cover Su.Let Nu(ε) be the minimum number of intervals of length ε needed to cover

Su. Setting 3−k/qi = C/qαi , we estimate the Minkowski dimension of Su as

dimM(Su) = limi→∞

logNu(C/qαi )

α log qi + C≥ lim

i→∞

log(qα−1i ) + C

α log qi + C≥ 1− 1

α.

We again invoke the theorem of Falconer [2] to conclude the same boundon dim(Su).

50

When u is close to a single rational, the previous argument can be refinedto give a lower bound on dim(Su).

Let p/q be in lowest terms with p + q ≡ 0 mod 3 and 0 2 and let m = b q−1

2c. The graph G∂ has

2m vertices, labelled with nonzero integers from −m to m inclusive. From avertex labelled x, for every d1, d2 ∈ 0, p, q there is an edge labelled (d1, d2)pointing to vertex 3x+ d1 − d2 if this is a nonzero integer in [−m,m].

For each vertex v of G∂ there is a word of length < log3 q which does notlabel a path from v (to see this: if v > 0, take a word consisting entirelyof qs; otherwise, take a word consisting entirely of 0s). Concatenating thesewords appropriately, we construct a word γ of length c < q log3 q which doesnot label a path starting at any vertex of G∂.

Theorem 6. Let u ∈ R and suppose there are relatively prime integers p, q,

0 1− 1log 1/εq log q

− 1.

Sketch of proof. For each k > 0, let Wk ⊂ 0, p, qN be the set of sequencesa = ai such that for each j ≥ 0, r(σja) is at a distance at least q

23−k from

∂S, where σ is the left shift. An element of Wk is then a sequence such thatno substring of length k labels a path in G∂. When k is sufficiently large,we claim that the growth rate of Wk is close to 3 and approximates qSp/qfrom below. Consequently, we see that dim(Sp/q) is close to 1, and replacingoccurrences of p/q by u as in the previous proof will lead to a similar boundon dim(Su).

Let k > 2c. Then Wk contains all sequences of the form γ, w1, γ, w2, γ, . . .,where the wj are arbitrary words of length k−2c. Let W

(N)k denote the subset

of N -truncations of elements of Wk. If W(N)k contains l arbitrary words of

length k− 2c, then N ≈ l(c+ k− 2c). So it takes about 3l(k−2c) many words

51

of length l(k − c) to cover W(N)k ; thus, the growth rate of Wk is at least

ξ = 3k−2ck−c . Since c < q log3 q, we have log3 ξ > 1− q log3 q

k−q log3 q.

Now we let W ′k be the set of sequences obtained from Wk by replacing

all occurrences of digit p by the digit qu. By hypothesis, any correspondentsequences a ∈ Wk and a′ ∈ W ′

k lie within q2ε of each other, where k =

−blog3 3εc. By definition of Wk, if a′, b′ ∈ W ′k, then |r(a′) − r(b′)| ≥ q

63−k.

So for n > k, it takes at least #(W′(n)k ) ≈ ξn many intervals of length q

63−n

to cover W ′k. Thus, the Minkowski dimension of Su (and so the Hausdorff

dimension by [2]) is at least log3 ξ.

References

[1] Besicovitch, A. S., On the fundamental geometric properties of linearlymeasurable plane sets of points III, Mathematische Annalen 116 (1939),349–357.

[2] Falconer, K. J., Dimensions and measures of quasi self-similar sets,Proc. American Math. Soc. 106 (1989), 543–554.

[3] Kenyon, R., Projecting the one-dimensional Sierpinski gasket, IsraelJour. of Math. 97 (1997), 221–238.

[4] Marstrand, J., Some fundamental geometric properties of plane sets offractional dimension, Proc. London Math. Soc. 4 (1954), 257–302.

[5] Odlyzko, A., Nonnegative digit sets in positional number systems, Proc.London Math. Soc. 37 (1978) 213–229.

[6] Swiatek, G., Veerman, J. J. P., On a conjecture of Furstenberg, IsraelJour. of Math. 130 (2002), 145–155.

Edward Kroc, University of British Columbiaemail: [email protected]

52

8 Wiener’s ‘closure of translates’ problem and

Piatetski-Shapiro’s uniqueness phenomenon

after N. Lev and A. Olevskii [2]A summary written by Allison Lewko

Abstract

We present a counterexample to a conjecture of N. Wiener, showingthat the cyclic vectors in `p(Z) (and Lp(R)) cannot be characterizedin terms of the zero set of the Fourier transform for values of p in therange 1 < p < 2.

8.1 Introduction

In [4], Wiener characterized the cyclic vectors in `p(Z) and Lp(R) for p = 1, 2in terms of the zeros of the Fourier transform. We will focus on `p(Z) forsimplicity, as the situation in Lp(R) is similar. A vector c = cnn∈Z is calleda cyclic vector in `p(Z) (with respect to translations) if the linear span of itstranslates is dense in `p(Z). Wiener proved:

Theorem 1. [4] Let c = cnn∈Z.(i) c is a cyclic vector in `2(Z) if and only if the Fourier transform

c(t) :=∑n∈Z

cneint

is nonzero almost everywhere.(ii) c is cyclic in `1(Z) if and only if c(t) has no zeros.

One can interpret these results as being unified by the feature that c is acyclic vector if and only if the set

Zc := t ∈ T : c(t) = 0

containing the zeros of the Fourier transform is sufficiently “small,” where theappropriate notion of smallness depends on whether p = 1 or p = 2. Wienerexpected that the intermediate range of 1 < p < 2 would behave like someinterpolation of these two results, with cyclicity being characterized entirelyby Zc. It is quite natural to guess that one would need an intermediate

53

notion of “smallness” for Zc parameterized by p, as it is known for 1 < p < 2that requiring Zc = ∅ is too strong and requiring Zc have Lebesgue measurezero is too weak.

One positive result along these lines was proved by A. Beurling [1], whoshowed that when the Hausdorff dimension of Zc is sufficiently small withrespect to p, then c is a cyclic vector in `p(Z) for 1 < p < 2. Despite beingsharp, this criterion is not necessary, and hence provides only an implicationand not a characterization.

The main result of this work is that for any 1 < p < 2, one cannotcharacterize the cyclic vectors in `p(Z) by any criteria depending only on Zc.This is a corollary of the following theorem:

Theorem 2. [2] Let 1 0, and c vanishes on K, then c is not cyclic in `p(Z).

(b) There exists c ∈ `1(Z) such that c vanishes on K, and c is a cyclicvector in `p(Z).

This yields:

Corollary 3. [2] Given any p, 1 < p < 2, one can find two vectors in`1(Z) such that one is cyclic in `p(Z) and the other is not, but their Fouriertransforms have an identical set of zeros.

We thus have a definitive counterexample to Wiener’s conjecture for theseintermediate values of p.

The methods for obtaining the proof of the above theorem are inspiredby the Piatetski-Shapiro phenomenon. This phenomenon refers to compactsets in T representing a surprising balance of “smallness” and “largeness”properties. More precisely, Piatetski-Shapiro [3] constructed a compact setK supporting a nonzero distribution S with Fourier transform S(n) tendingto zero as |n| → ∞, but not supporting any such measure. This result wasunexpected, as it was previously believed that any compact set supportingsuch a distribution would also support such a measure.

The connection between the Piatetski-Shapiro phenomenon and the char-acterization of cyclic vectors is expressed by the following known facts. First,if c is a non-cyclic vector in `p(Z), then Zc supports a nonzero distributionwith Fourier coefficients in `q(Z), where q := p/(p− 1). The converse of this

54

statement turns out to be false, but two closely related results are known.If Zc supports a nonzero measure with Fourier coefficients in `q(Z), then cis a noncyclic vector in `p(Z). Also, if c is continuously differentiable andZc supports such a nonzero distribution, then c is noncyclic in `p(Z). Thus,the strategy for proving the main theorem is to find a compact set K that is“large enough” to support such a nonzero distribution but “small enough”so that it does not support such a measure.

8.2 Tools

We now give a high level outline of the proof. We first define an appropriatenotion of smallness for K which suffices to rule out the support of measureswith quickly vanishing Fourier coefficients while still allowing us to build asuitable distribution S supported on K. We employ the notion of a Helsonset:

Definition 4. A compact set K is called a Helson set if it satisfies any oneof the following equivalent conditions:

(i) Every continuous function on K admits extension to a function whoseFourier coefficients are in `1(Z).

(ii) There exists a constant δ1(K) > 0 such that, for every measure µsupported by K,

supn∈Z|µ(n)| ≥ δ1(K)

∫|dµ|.

(iii) There exists a constant δ2(K) > 0 such that, for every measure µsupported by K,

lim sup|n|→∞

|µ(n)| ≥ δ2(K)

∫|dµ|.

To establish that any Helson set is sufficiently “small” for our purposes,we must show every Helson set is contained in Zc for some cyclic c. For this,we prove that for any Helson set K, the set of vectors c with Zc ⊇ K thatare also cyclic in `p(Z) for every p > 1 is a countable intersection of open,dense sets. Hence, by the Baire category theorem, this set is nonempty.

In order to construct a Helson set supporting a suitable distribution, ithelps to identify some alternate properties that imply that a set K is a Helsonset. In particular, we consider totally disconnected compact sets K which,given any real-valued continuous function h on T with no zeros in K, allow

55

us to find a real trigonometric polynomial P which has absolute value > 1 onK, agrees with the sign of h everywhere on K, and has Fourier coefficientsbounded independently of h in terms of the `1-norm. We prove that any suchset K is also a Helson set.

This alternate set of criteria enables us to take an iterative approach toconstructing a Helson set K and the suitable distribution S it supports. Themain idea is to first construct a compact set K with much weaker properties.More precisely, it should support a reasonable approximation to the constant1 function (close in terms of the `q-norm on the Fourier coefficients) whilealso allowing one to find a trigonometric polynomial P with the propertiesdescribed above with respect to a specific, fixed h which is also a trigonomet-ric polynomial. Once we find such sets K1, K2, . . . for a dense sequence oftrigonometric polynomials, then we can take their intersection to arrive at afinal set K which is then a Helson set. We can also take the infinite productof the supported approximations to the constant 1 function as our nonzerosupported distribution with sufficiently vanishing Fourier coefficients. Thisis merely a birds-eye view of the outer structure of the proof - there are manytechnical challenges arising in the implementation, particularly in construct-ing the initial compact sets and 1-approximations to combine.

References

[1] Beurling, A., On a closure problem. Ark. Mat. 1 (1951), 301–303.

[2] Lev, N. and Olevskii, A., Wiener’s ‘closure of translates’ problem andPiatetski-Shapiro’s uniqueness phenomonon. Ann. of Math. 174 (2011),519–541;

[3] Pyateckii-Sapiro, I. I., Supplement to the work “On the problem ofuniqueness of expansion of a function in a trigonometric series.”Moskov. Gos. Univ. Uc. Zap. Mat. 165(7) (1954), 79–97, English trans-lation in Selected Works of Ilya Piatetski-Shapiro, Amer. Math. Soc.Collected Works 15, 2000;

[4] Wiener, N., Tauberian theorems. Ann. of Math. 33 (1932), 1–100;

Allison Lewko, University of Texas at Austin and MicrosoftResearch New Englandemail: [email protected]

56

9 Bounded orthogonality systems and the Λ(p)-

set problem II

after Jean Bourgain [1]A summary written by Mark Lewko

9.1 Introduction

Here we outline the proof of the following theorem [1] (for motivation andbackground we refer the reader to Stefan’s summary):

Theorem 1. Let φn(x)Nn=1 be a bounded orthogonal system and 2 n2/p such that

||∑n∈S

anφn(x)||p p ||a||`2

Let ξn(ω) denote independent selectors taking the value 1 with probabilityδ = n2/p−1 and 0 otherwise. We will let A := anNn=1 denote a sequence ofreal numbers. The proof will proceed by estimating the quantity

E sup||a||`2≤1

||N∑n=1

anξnφn(x)||pp. (1)

Ultimately, one can show that this quantity is O(1). On the other hand,standard deviation inequalities show that the size of the selected set, S(ω) :=n ∈ [N ], ξn(ω) = 1), will be within a constant factor of n2/p with all butexponentially small probability. These two facts easily imply Theorem 1.Expanding the norm in (1), this may be rewritten as

E sup||a||`2

⟨N∑n=1

anξn(ω)φn(x),N∑n=1

anξn(ω)φn(x)

∣∣∣∣∣N∑n=1

anξn(ω)φn(x)

∣∣∣∣∣p−2⟩

(2)

We will now make a simplifying assumption: we will restrict attention tothe case when all of the coefficients are of the same scale (that is we restrict toa coefficient level set). In other words we consider B ⊂ [N ] such that an ∼ 2−i

for n ∈ B (here we use a ∼ b to denote b/2 ≤ a ≤ b). Clearly we have |B| =O(22i). A considerable amount of the technical difficulties of Bourgain’s proof

57

stem from handling all possible level sets simultaneously. Roughly speakingthe idea is to prove several (stronger) ‘restricted’ multilinear variants of thelevel sets estimates and then interpolate between them to recover the fullresult. (Some of these ideas are discussed in Stefan’s summary.) Of course,from dyadic pigeonholing one can use this single level set (or ‘restricted’)result to obtain the general result with a loss of log(N) in the Λ(p) constant.Let us now define the random variable

Kp(ω) := supB

sup||a||`2≤1

ai∼|B|−1/2

||N∑n=1

anξn(ω)φn(x)||pp.

Our goal is to show that EKp(ω) = O(1) . Expanding the norm (andomitting the sup over B for brevity) we have

E sup||a||`2

ai∼|B|−1/2

supB⊆[N ]

1√|B|

N∑n=1

1B(n)ξn(ω)

×

∣∣∣∣∣∣⟨φn(x),

N∑n=1

anξn(ω)φn(x)

∣∣∣∣∣N∑n=1

anξn(ω)φn(x)

∣∣∣∣∣p−2⟩∣∣∣∣∣∣ . (3)

This expression turns out to be difficult to work with because of joint de-pendencies on ω between the first and second coordinate of the inner product.The first key idea is that one may ‘decouple’ the dependencies using somefairly elementary probabilistic inequalities. The upshot is that the abovequantity can be essentially controlled by

Eω1Eω2 sup||a||`2

ai∼|B|−1/2

supB⊆[N ]

1√|B|

N∑n=1

1B(n)ξn(ω1)

×

∣∣∣∣∣∣⟨φn(x),

N∑n=1

anξn(ω2)φn(x)

∣∣∣∣∣N∑n=1

anξn(ω2)φn(x)

∣∣∣∣∣p−2⟩∣∣∣∣∣∣ . (4)

Here ω1 and ω2 are independent. Now, for fixed ω2, we may define

58

Q(ω2) ⊂ RN by

Q(ω2) =

⟨φm(x),

N∑n=1

anξn(ω)φn(x)

∣∣∣∣∣N∑n=1

anξn(ω)φn(x)

∣∣∣∣∣p−2⟩

1≤n≤N

:

a ∈ `2, suppa ≤ |B|

.

Thus our quantity can now be rewritten as

Eω2Eω1 supy∈Q(ω2)

∑i∈B

ξi(ω1)yi. (5)

If we replaced the random selectors ξi(ω1) with independent Gaussiansthen the above quantity is classically studied under the heading of ‘boud-edness of Gaussian processes’. A central result in this theory is Dudley’sentropy bound which states that the (Gaussian variant) of the above quan-tity is controlled by the entropy (or covering) numbers of the set Q(ω2).It turns out that Dudley’s method can be adapted to the case of selectorprocesses as well.

Let Q ⊂ Rn and let Nq(Q, t) denote the number of `q balls of radius tneeded to cover Q. We then have

Lemma 2. (Chaining Lemma) Let ξi denote independent selectors of meanδ. Then, denoting the diameter of Q as diam(Q), we have

E supy∈Q

∑i∈A|A|=m

ξi(ω)yi √δm+ log(δ−1)Diam(Q)

+ log−1/2(δ−1)

∫ ∞0

[logN2(Q, t)]1/2 dt.

In fact, slightly stronger bounds are available (and required in the argu-ments needed to remove the level set restriction) however we omit this forbrevity. We have now reduced matters to understanding the entropy functionN2(Q(ω2), t) and the quantity diam(Q(ω2)). Denote P|B| :=

∑i∈A aiφi(x) :

|A| ≤ |B|. Let f, g ∈ P|B| and vf , wg ∈ Q(ω2) the associated vectors in RN .We then have

59

|vf −wg| ≤

(N∑i=1

∣∣| ⟨φi, f |f |p−2⟩| − |

⟨φi, g|g|p−2

⟩|∣∣2) ≤ ||f |f |p−2− g|g|p−2||2

by Bessel’s inequality. We now make the further assumption that 2 < p < 3(the case 3 ≤ p < 4 can be handled by a similar argument. The resultfor larger p can subsequently be obtained by a more complicated iterativeversion of these arguments. We omit discussion of this here). Invoking thepointwise bound (using 2 < p < 3) f |f |p−2−g|g|p−2 |f−g|(|f |p−2 + |g|p−2)we may further bound the above by

≤ ||(f − g)|f |p−2||2 + ||(f − g)|g|p−2||2 ≤ ||f − g|| 23−p

where the last inequality follows from Holder’s inequality and the fact that||f ||2, ||g||2 ≤ 1. From this, we see that N2(Q(ω2), t) N p−2

4(P|B|, t/2). In

other words the quantity N2(Q(ω2), t) can be controlled by the entropy num-bers N p−2

4(P|B|, t/2). A similar calculation using Bessel’s inequality shows

that

diam(Q(ω2)) ≤ Kp(ω2)p/2|B|p−2

4 .

It now suffices to gain control of the entropy numbers N p−24

(P|B|, t/2).

This is the content of the following lemma.

Lemma 3. (Entropy lemma) For t < 1

logNq(P|B|, ct) |B|(

log

(1 +

N

|B|

)+ log(1/t)

)and for t ≥ 1

logNq(P|B|, ct) |B| log

(1 +

N

|B|

)t−ν

for some ν := ν(q) > 2.

This is proven using what has become known as the ‘support reductiontrick’ and Khinchin’s inequality. Collecting the estimates we have that (5)can be bounded by

δ1/2|B|p−2

4 Eω2Kp/2p (ω2) + |B|−1/2 log−1/2(δ−1)

60

×|B|1/2 log1/2(1 +N

|B|)

(∫ 1

0

log1/2(1/t)dt+

∫ ∞1

t−ν/2dt

)Using that |B| N2/p, δ = N2/p−1 and that the integrals are finite, we

arrive at the recursion

EωKpp(ω) ≤ C1EωKp/2

p (ω) + C2

which implies that EωKpp(ω) is bounded by a constant independent of N .

References

[1] Bourgain, J. Bounded orthogonal systems and the Λ(p)-set problem.Acta Math. 162 (1989), no. 3–4, 227–245.

Mark Lewko, UT-Austin / UCLAemail: [email protected]

61

10 Local estimates of exponential polynomi-

als and their applications to inequalities

of uncertainty principle type - part I

after F. L. Nazarov [3]A summary written by Christoph Marx

Abstract

A classical result by Pal Turan, estimates the global behavior of anexponential polynomial on an interval by its supremum on any arbi-trary subinterval. We discuss F. L. Nazarov’s extension of this “globalto local reduction” to arbitrary Borel sets of positive Lebesgue mea-sure. More recently, an observation by O. Friedland and Y. Yomdin,enlarges the class of sets to encompass even discrete, in particularfinite, sets of sufficient density.

10.1 Introduction

We consider an exponential polynomial, i.e. an expression of the form

p(t) =n∑k=1

ckeλkt , (1)

where both the coefficients ck and the frequencies λk are complex. Thenumber of non-vanishing coefficients defines its order. Following, dependingon the context, µ denotes the Haar measure on R or T := R/Z such that,respectively, µ(T) = 1 or µ([0, 1]) = 1.

A classical Lemma due to Pal Turan [4] estimates the global behaviorof (1) (order n) on an interval I ⊆ R by its supremum on any arbitrarysub-interval E ⊆ I:

supt∈I|p(t)| ≤ eµ(I)·max |Reλk| ·

(Aµ(I)

µ(E)

)n−1

supt∈E|p(t)| . (2)

Here, A > 0 is an absolute constant, independent of n.In particular, comparing (2) with an analogous result for algebraic poly-

nomials dating back to Chebyshev, implies that exponential polynomials oforder n behave like their algebraic counterparts of degree n− 1.

Following, we discuss the extension of Turan’s Lemma to arbitrary Borelsets E ⊆ I, achieved by F. L. Nazarov in [3], Chapter 1 therein:

62

Theorem 1 ([3]). Let p(t) be an exponential polynomial of order n of theform given in (1). Turan’s Lemma (2) holds for any Borel set E ⊆ I withµ(E) > 0.

Thus, the above mentioned analogy between exponential and algebraicpolynomials persists when considering arbitrary Borel sets, Theorem 1 thenbeing paralleled by Remez type estimates. For an extensive review of avail-able results for both algebraic and exponential polynomials, we refer to e.g.[2].

More recently, Theorem 1 was extended further to certain discrete, inparticular finite, sets of sufficient density [1]. To this end, Friedland andYomdin introduce the metric span of a set E ⊆ I for a given pair (P, I):

Definition 2 ([1]). Let p(t) be an exponential polynomial of order n of

the form given in (1) and I an interval. Set m := n(n+1)2

+ 1, C(n) :=

m(2m + 1)2m22m2and λ := max |Imλk|. Letting d := C(n)µ(I)λ, introduce

the “frequency bound” M(p, I) := bd2c + 1. The metric span of a set E ⊆ R

is defined byω(p,I)(E) := sup

ε>0ε[M(ε, E)−M(p, I)] , (3)

where M(ε, E) is the ε-covering number of E 1.

Remark 3. (i) Clearly, for any measurable E, ω(p,I)(E) ≥ µ(E).

(ii) ω(p,I)(E) > 0 if [M(ε, E)−M(p, I)] > 0, for some ε > 0. In particular,ω(p,I)(E) > 0, for discrete sets of sufficient density.

(iii) The number M(P, I) characterizes the complexity of sub-level sets, t ∈I : |p(t)| ≤ δ (see also Lemma 8, below).

Theorem 4 ([1]). Replacing µ(E) by ω(p,I)(E), the statement of Theorem 1holds for any E ⊆ R with ω(p,I)(E) > 0.

We mention, that Theorem 4 was preceded by an analogous result foralgebraic polynomials [5].

1We recall, that given a metric space (M,d) and ε > 0, one defines the ε-coveringnumber of a subset X ⊆M as the minimal number of ε-balls needed to cover X.

63

10.2 Nazarov’s theorem

Following, we present the main ideas in the proof of Theorem 1. We will focuson the case when p(t) has purely imaginary frequencies, p(t) =

∑nk=1 cke

iλkt,λ1 < · · · < λn. Without loss of generality, we take I = [−1

2, 1

2].

Theorem 1 is based on two crucial Lemmas, one quantifying the numberof zeros in a vertical strip (see Lemma 7, below), the other allowing reductionof order by a weak-type estimate of the logarithmic derivative (see Lemma6, below).

10.2.1 Bernstein-type estimates and order reduction

The strategy of order reduction is most transparent when p(t) is a trigono-metric polynomial, i.e. λk = mk ∈ Z. Substituting z = e2πit, considerp(z) =

∑nk=1 ckz

mk as a Laurent polynomial on the unit circle. We shallshow:

Theorem 5 (see Theorem 1.4 in [3]). Given E ⊂ T, µ(E) > 0, one has

‖p‖W :=n∑k=1

|ck| ≤

16e

π

1

µ(E)

n−1

supz∈E|p(z)| . (4)

To prove Theorem 5, one inductively reduces the order of p(z) by con-structing a sequence of Laurent polynomials p = pn, pn−1, . . . , p1 satisfying

(Ind1) ordpk = k

(Ind2) ‖pk‖W ≥ π16‖pk−1‖W

such that

µ

z ∈ T :

∣∣∣∣pk−1(z)

pk(z)

∣∣∣∣ > t

≤ 1

t, (5)

for 2 ≤ k ≤ n.pk−1 is obtained from pk =:

∑ks=1 dsz

rs , r1 < . . . rk ∈ Z, choosing one ofthe following Laurent polynomials of order k − 1

q(z) :=d

dz(z−r1pk(z)) or q(z) :=

d

dz(z−rkpk(z)) , (6)

which guarantees a lower bound of ‖.‖W as indicated in (Ind2).

64

Thus, one can reduce the order of p from n to 1,( π16

)n−1

‖p‖W ≤ ‖p1‖ = |p1(z0)| =∣∣∣∣p1(z0)

p(z0)

∣∣∣∣ · |p(z0)| , (7)

arriving at (4), provided there exists some z0 ∈ E satisfying∣∣∣∣p1(z0)

p(z0)

∣∣∣∣ ≤ e

µ(E)

n−1

. (8)

Existence of such z0 follows from a measure estimate of the exceptional setwhere (8) is violated. Noticing that |pk−1(z)

pk(z)| can be realized as a logarithmic

derivative of an algebraic polynomial, such estimate is accomplished by thefollowing Bernstein-type Lemma:

Lemma 6 (see Lemma 1.2 in [3]). Let g(z) be an algebraic polynomial ofdegree n. Then,

µ

z ∈ T :

∣∣∣∣g′(z)

g(z)

∣∣∣∣ > y

≤ 8n

πy. (9)

10.2.2 The role of “zero counting”

Quantifying the distribution of zeros of exponential polynomials is a crucialingredient for both the Theorems 1 and 4. Nazarov’s argument is based onthe Langer lemma:

Lemma 7 (see Lemma 1.3 in [3]). Let p(z) =∑n

k=1 ckeiλkz, 0 = λ1 < λ2 <

· · · < λn =: λ, be an exponential polynomial not vanishing identically. Then,the number of complex zeros of p(z) in an open vertical strip x0 < Rez <x0 + ∆ of width ∆ > 0 does not exceed (n− 1) + λ∆

2π.

In particular, based on Lemma 7, one concludes that complex zeros zjof the given exponential polynomial p(z) are sufficiently separated: Orderingzj according to increasing |Rezj|, the inequality

|Rezj| ≥ πj − (n− 1)

(n− 1), (10)

holds.

65

We employ the Hadamard factorization theorem,

p(z) = ceazκ∏j=1

(z − zj)∏j>κ

(1− z

zj

)ez/zj =: ceazQ(z)R(z) , (11)

with κ chosen such that Q(z) contains all zeros of p(z) on [−1/2, 1/2].By (10), κ can be estimated depending on the relation between λ and

n− 1,

κ =

2(n− 1) , if λ ≤ n− 1 ,

2λ , if λ > n− 1 .(12)

Using order reduction similar to Sec. 10.2.1, it suffices to consider λ ≤ n−1.The argument principle allows to quantify the contribution of the zero-

free part of p(z),

maxz∈[−1/2,1/2]

|ceazR(z)| ≤ 3n−1 minz∈[−1/2,1/2]

|ceazR(z)| . (13)

Finally, the polynomial Q(z) is dealt with using a Cartan-like estimate:Given 0 < h < 1/8, it is shown that for z outside an exceptional subsetΩh ⊆ [−1/2, 1/2] with µ(Ωh) ≤ 8h < 1 = µ(I), one has

|Q(z)|max|Q(t)| : t ∈ [−1/2, 1/2]

≥

8h

32 3√

4

n−1

. (14)

Thus, letting h = µ(E)/8, we may combine all the pieces to arrive at

supz∈I|p(t)| ≤ 3n−1 inf

z∈I|ceazR(z)| · sup

z∈I|Q(z)| (15)

≤

96 3√

4

µ(E)

n−1

infz∈I|ceazR(z)| · inf

z∈E∩Ω|Q(z)| (16)

≤

96 3√

4

µ(E)

n−1

supz∈E|p(z)| . (17)

10.3 Extensions

We conclude by briefly commenting on Theorem 4. For δ := supt∈E |p(t)|,consider the sublevel set Vδ := t ∈ I : |p(t)| ≤ δ. Clearly, one has E ⊆

66

Vδ. The main idea in [1] is to reduce Theorem 4 to Theorem 1 by showingω(p,I)(E) ≤ µ(Vδ) (see Lemma 2.3 in [1]).

Again, “counting zeros” provides the key ingredient and also explains thedefinition of M(p,I):

Lemma 8 (see Lemma 2.2 in [1]). For p(t) as in Theorem 4, given η > 0the number of non-degenerate solutions of the equation |p(t)|2 = η in theinterval I does not exceed d = C(n)µ(I)λ. Here, C(n) and λ are defined asin Definition 2.

Lemma 8 allows to estimate from above the ε-covering number of Vδ,which in turn yields ω(p,I)(E) ≤ µ(Vδ), as claimed.

References

[1] O. Friedland, Y. Yomodon, An observation on the Turan-Nazarov in-equality., preprint (2012); available on arXiv:1107.0039v2 [math.FA].

[2] M. I. Ganzburg Polynomial inequalities on measurable sets and theirapplications., Constr. Approx. 17 (2001), 275 - 306.

[3] F. L. Nazarov, Local estimates of exponential polynomials and theirapplications to inequalities of uncertainty principle type., Algebra iAnaliz 5 5 (1993), no. 4, 3 - 66 (Russian); translation in St. Pe-tersburg Math. J. 5 (1994), no. 4, 663 - 717; also available underhttp://www.math.msu.edu/fedja/pubpap.html.

[4] P. Turan, Eine neue Methode in der Analyses und deren Anwendungen,Acad. Kiado, Budapest (1953).

[5] Y. Yomdin, Remez-type inequality for discrete sets, Israel Journal ofMathematics 186 (2011), no. 1, 45 - 60.

Christoph Marx, UCI / Caltechemail: [email protected]

67

11 Salem sets and restriction properties of

Fourier transforms

after G. Mockenhaupt [2]A summary written by Eyvindur Ari Palsson

Abstract

We give an analogue on the real line of the restriction phenomenaof the Fourier transform which was first discovered in the late sixtiesby E.M. Stein for higher dimensions.

11.1 Classical restriction

Restriction problems involve restricting the Fourier transform of a functionto a subset of Rn and showing that the restriction is well defined and can becontrolled by some norm of the original function. A classical result of thisflavor is the Stein-Tomas theorem.

Theorem 1. If f ∈ Lp(Rn) where 1 ≤ p ≤ 2n+2n+3

and n > 1 then

‖f‖L2(dµ) ≤ C‖f‖Lp(Rn)

where µ is the uniform measure on the unit sphere Sn−1 in Rn.

There have been many generalizations of this result, mainly for situa-tions where the unit sphere is replaced by some smooth submanifold of Rn

satisfying suitable curvature conditions. Prior to Mockenhaupt’s result [2]one could have believed that restriction was genuinely a higher dimensionalphenomena because all the generalizations of the Stein-Tomas result had incommon that their setting was in higher dimensions.

11.2 Two notions of dimension

In order to do restriction on the real line then we need to find suitable subsetsthat we can restrict to. With that goal in mind we first discuss two notionsof dimension.

Fix α > 0 and let E ⊂ Rn. For ε > 0 define

Hεα(E) = inf

(∞∑j=1

rαj

)

68

where the infimum is taken over all countable coverings of E by discsD(xj, rj)with rj < ε. Then let Hα(E) = lim

ε→0Hεα(E). One can show that there is a

unique number a0, called the Hausdorff dimension of E or dimH(E), suchthat Hα(E) =∞ if α < α0 and Hα(E) = 0 if α > α0. For integer dimensionsthen the Hausdorff dimension coincides with the regular notion of dimension,however it allows us to extend the notion of dimension to non-integers. Asan example then one can for example show that the Cantor middle third sethas Hausdorff dimension log(2)/ log(3).

A theorem of Frostman shows that if E ⊂ Rn is a compact set withdimH(E) = α then there is a probability measure µ supported on E satisfy-ing µ(Br(x)) ≤ Crα, where Br(x) denotes a ball of radius r centered at x.Therefore the β-energy of µ defined as

Iβ(µ) =

∫∫dµ(y)dµ(x)

|x− y|β

is finite as long as β < α.Frostman’s theorem also shows that if Iα(µ) < ∞ for some probability

measure µ supported on a compact set E, then dimH(E) ≥ α. Using the factthat on the Fourier side we can write

Iα(µ) = c

∫Rn

|dµ(ξ)|2

|ξ|n−αdy

we conclude that Iα(µ) <∞ provides some information on the size of dµ.We define the Fourier dimension of a compact set E ⊂ Rn, denoted by

dimF(E), as the supremum of β ≥ 0 such that for some probability measuredµ supported on E

|dµ(x)| ≤ C|x|−β/2.As an example then the unit sphere in Rn has Fourier dimension n − 1.Observe that the condition implies that Iα(µ) < ∞ for α < β so we alwayshave dimF(E) ≤ dimH(E).

11.3 Main result

Theorem 2. Let µ be a compactly supported positive measure on Rn whichsatisfies the following properties.

(i) There is β > 0 such that |du(x)| ≤ C|x|−β/2.

69

(ii) There is α > 0 such that µ(Br(x)) ≤ Crα for every ball Br(x) of radiusr centered at x.

Then for 1 ≤ p < 2(2n−2α+β)4(n−α)+β

, we have

‖f‖L2(dµ) ≤ C‖f‖Lp(Rn).

Observe that if µ is the uniform measure on the unit sphere Sn−1 in Rn

then dimH(E) = dimF(E) = n− 1 and the range of p’s becomes

1 ≤ p <2n+ 2

n+ 3

which up to the right endpoint recovers the Stein-Tomas theorem.Note that trivially ‖f‖L2(dµ) ≤ C‖f‖L1(Rn) so we are mainly interested in

cases where we can have non-trivial bounds. Observe that

2(2n− 2α + β)

4(n− α) + β= 1 +

β

4(n− α) + β

so we always get some non-trivial bounds. Further note that for fixed αand n then this function is increasing in β so it is of interest to push β ≤dimF(E) ≤ dimH(E) as high as possible, where E = supp µ, in order to geta maximal range of non-trivial results.

11.4 Salem sets

Compact sets E ⊂ Rn that fulfill dimF(E) = dimH(E) are called Salem sets,named after R. Salem. We have already seen one such example which isthe uniform measure on the unit sphere Sn−1 in Rn. The existence of suchsubsets on the real line was first shown by R. Salem [3].

Another example of a Salem set, due to R. Kaufman [1], is the set Et,t > 0, of those real numbers x ∈ [0, 1] such that

‖qx‖ ≤ q−1−t

has solutions for arbitrarily large integers q. Here ‖x‖ denotes the distanceto the nearest integer. R. Kaufman showed that dimF(E) = dimH(E) = 2

2+t.

J. P. Kahane has further provided a rich class of Salem sets by showingthat images of compact sets of a given Hausdorff dimension under Brownianmotion are almost surely Salem sets.

70

Going through R. Salem’s construction, which was based on a generaliza-tion of the classical Cantor type construction, then one obtains both a set Ewith dimH(E) = α and a measure dF that fulfills

|dF (x)| ≤ Cε|x|−α2

+ε.

Now for this particular set and measure we can state a real line analog of theStein-Tomas result.

Corollary 3. Let 1 ≤ p < 2(2−α)4−3α

and dF be as in Salem’s construction fora Salem set on the real line. Then there is a constant C such that

‖f‖L2(dF ) ≤ C‖f‖Lp(Rn)

11.5 Sketch of proof of theorem

First observe‖f‖L2(dµ) ≤ ‖f ∗ dµ‖p′‖f‖p

where 1p

+ 1p′

= 1. It is thus sufficient to show that the convolution operator

T (f) := dµ ∗ f is bounded from Lp → Lp′

for p′ > 2(2n − 2α + β)/β.Let (φk)

∞k=0 be a Littlewood-Paley decomposition, that is,

∑k≥0 φk = 1 and

supp φk ⊂ 2k−1 ≤ |x| ≤ 2k. Then decompose T (f) =∑

k≥0 Tk(f) where

Tk(f) = (φkdµ) ∗ f.

By condition (i) in the theorem then

‖Tk‖L1→L∞ ≤ ‖φkdµ‖∞ ≤ C2−kβ2 .

Using Plancherel’s theorem and condition (ii) from the theorem one can show

‖Tk‖L2→L2 ≤ C2k(n−α).

Interpolating the two bounds above one obtains

‖Tk‖Lp→Lp′ ≤ C2k( 2n−2α+β

p′ −β2

).

Thus T =∑Tk is bounded for p′ > 2(2n− 2α + β)/β.

71

References

[1] Kaufman, R., On the theorem of Jarnik and Besicovitch. Acta Arith-metica 39 (1981), 265–267;

[2] Mockenhaupt, G., Salem sets and restriction properties of Fourier trans-forms. GAFA 10 (2000), 1579–1587;

[3] Salem, R., On singular monotonic functions whose spectrum has a givenHausdorff dimension. Ark. Math. 1 (1950), 353–365;

[4] Wolff, T., Lectures on harmonic analysis With a foreword by CharlesFefferman and preface by Izabella Laba. Edited by Laba and CarolShubin. University Lecture Series, 29. American Mathematical Society,Providence, RI, (2003).

Eyvindur Ari Palsson, University of Rochesteremail: [email protected]

72

12 Maximal operators and differentiation

theorems for sparse sets: Part II

after I. Laba and M. Pramanik [2]A summary written by Alex Rice

Abstract

We state the main results on maximal operators for sets yielded bya random Cantor-type construction. Further, we discuss the deductionof these results from the transverse correlation condition in the mainresult of Part I.

12.1 Introduction

Operators defined by maximal averages have been of central concern in har-monic analysis for many years. Most classically, the Lebesgue differentiationtheorem follows from estimates on the Hardy-Littlewood maximal operator,M , defined by

Mf(x) = supr>0

1

|Br(x)|

∫Br(x)

|f(y)|dy,

where Br(x) denotes the ball of radius r centered at x and | · | denotesLebesgue measure. More generally, one can consider maximal averages, anddeduce differentiation theorems, over sets besides balls. The following resultconsiders the case of spheres, and the proof in dimension d = 2 inspired muchof our subsequent discussion.

Theorem 1 (Stein [3]: d > 2, Bourgain [1]: d = 2). Consider the sphericalmaximal operator on Rd defined by

MSd−1f(x) = supr>0

∫Sd−1

|f(x+ ry)|dσ(y),

where σ is the normalized Lebesgue measure on the unit sphere Sd−1. Ifd ≥ 2, then MSd−1 is bounded on Lp(Rd) for p > d

d−1, and this range of p is

optimal.

Many results of this type are known for other sets under varying smooth-ness and curvature conditions, but no similar theory has developed so far in

73

one dimension. When making connections between harmonic analysis andadditive combinatorics, we often make an analogy between curvature and“randomness”, for example with regard to decay estimates on Fourier trans-forms. With this analogy in mind, we proceed in one dimension with therole of a “curved surface” played by the randomized Cantor-type construc-tion from Part I.

12.2 Main results

For the remainder of our discussion, we let Skk∈N be a decreasing sequenceof subsets of [1, 2] yielded by the random Cantor-type construction from PartI, so the Hausdorff dimension of S = ∩∞k=1Sk is 1 − ε with 0 ≤ ε < 1/3. Inparticular, we have that |Sk| 0 as k →∞, and the densities φk = 1Sk/|Sk|converge weakly to a probability measure µ supported on S. We also assumethat Sk satisfies the transverse correlation condition in the conclusion ofthe main result of Part I, which holds with positive probability.

Theorem 2. The restricted maximal operators M and M, defined by

Mf(x) = sup1<r<2, k≥1

1

|Sk|

∫Sk

|f(x+ ry)|dy

and

Mf(x) = sup1<r<2

∫R|f(x+ ry)|dµ(y),

are bounded from Lp[0, 1] to Lq(R) for any p, q such that

1 + ε

1− ε< p <∞ and 1 < q <

1− ε2ε

p (∞ if ε = 0), (1)

and from Lp(R) to Lq(R) whenever 1 0, k≥1

ra

|Sk|

∫Sk

|f(x+ ry)|dy

and

Maf(x) = supr>0

ra∫R|f(x+ ry)|dµ(y),

are bounded from Lp(R) to Lq(R). In particular, M := M0 and M := M0

are bounded on Lp(R) for all p > 1+ε1−ε .

74

12.3 Motivation

In his proof of Theorem 1 in dimension d = 2, Bourgain relies on observationsconcerning the size of intersections of thin annuli. Roughly, he observes thatlarge intersections between two affine copies of a thin annulus, such as thoseresulting from “internally tangent” annuli in a clamshell configuration, areextremely rare, and the remaining generic, or transverse, intersections arevery small. However, one can see that when considering “square annuli”,large intersections are much more common due to the lack of curvature. Onemight expect that an adaptation of Bourgain’s method is possible as longas some analog of these observations on intersections holds. Motivated bythe analogy between curvature and randomness, we turn to the transversecorrelation condition, analogous to Gowers’ higher order uniformity fromadditive combinatorics, to adapt Bourgain’s method to the random Cantor-type construction.

12.4 Linearization and discretization

Recall from the construction that each set Sk is a union of Pk intervals oflength δk. While we don’t discuss the details here, one can establish themaximal estimates in Theorem 2 by examining the linearized and discretizedauxiliary operators Φk defined by

Φkf(x) =

∫Rf(z)Vk,x(z)dz,

where Vk,x(z) = σk

(z−c(x)r(x)

), σk = φk − φk+1, and c(x) and r(x) are functions

taking values in discrete, equally spaced sets C ⊆ [−4, 0] and R ⊆ [1, 2],respectively. Further, Φkf is supported in [−4, 0] and c(x) satisfies

|x− c(x)| ≤ δk ≤ P−1k for all x ∈ [−4, 0]. (2)

Lp[0, 1]→ Lq[−4, 0] operator norm bounds on Φk can be deduced from esti-mates on

supΩ⊆[−4,0]

‖Φ∗k1Ω‖n|Ω|n−1

n

, (3)

where Φ∗k is the “adjoint” operator defined by

Φ∗kg(z) =

∫Rg(x)Vk,x(z)dx.

75

As discussed in Part I, one must allow n in (3) to be arbitrarily large in orderto obtain the full results of Theorem 2, but here we restrict to n = 2 for asimplified exposition.

12.5 Transverse correlations

Recall that for a fixed k and a pair of affine transformations g1, g2, we considerthe collection of internal tangencies, Fint, which roughly corresponds to pairsof intervals I1, I2 which are close together and satisfy g1(I1)∩ g2(I2) 6= ∅. Ofcourse, when dealing with affine transformations we can equivalently considertranslation-dilation pairs, where (c, r) ∈ R2 corresponds to g(x) = rx + c,and here we consider the collection of pairs of transformations

Uk =(

(c1, r1), (c2, r2))

: c1, c2 ∈ C, r1, r2 ∈ R,

where C and R are as in section 12.4. We partition this collection intothe pairs of transformations which have many internal tangencies, and thosewhich do not. Specifically, we let

U intk = A ∈ Uk : #(Fint) ≥

√Pk

andU trk = Uk \ U int

k .

Recall that for A ∈ Uk and functions f1, f2 on R, we define the correlationof f1, f2 according to A by

Λ(A; f1, f2) =

∫Rf1

(z − c1

r1

)f2

(z − c2

r2

)dz,

and we write Λ(A; f) for Λ(A; f, f). The main component we wish to illumi-nate is the following, which states that control on correlations for transversepairs according to σk yields the desired type of estimate on Φ∗k.

Proposition 3. IfsupA∈Utr

k

|Λ(A;σk)| ≤ C0(k), (4)

then

supΩ⊆[−4,0]

‖Φ∗k1Ω‖2

|Ω|1/2≤ C

(max

P−1/2k

|Sk+1|, C0(k)

)1/2

for an absolute constant C > 0.

76

The main result of Part I can be used to show that (4) holds with C0(k)decaying exponentially in k, which is sufficient for the purposes of Theorem2. When encountering pairs in U int

k , we use the following two facts, the firstof which allows us to conclude that such pairs are rare, and the second ofwhich provides a trivial upper bound on the relevant correlation.

Lemma 4. If A = (c1, r1), (c2, r2) ∈ Uk and #(Fint) ≥ L, then

|c1 − c2| ≤ min4, 160/L.

While we omit the details of Lemma 4, we do provide a proof of the fol-lowing bound which ignores any potential cancellation in the integral definingΛ(A;σk).

Lemma 5. For all k ≥ 1 and A ∈ Uk, we have

|Λ(A;σk)| ≤8

|Sk+1|.

Proof. Recalling that σk = φk+1−φk, we see that Λ(A;σk) expands as a sumof four terms of the form ±Λ(A;φk+λ1 , φk+λ2) with λ1, λ2 ∈ 0, 1, and weestimate each in absolute value. Using that the sequence |Sk| is decreasingand the scaling parameters ri ≤ 2, we have

|Λ(A;φk+λ1 , φk+λ2)| = 1

|Sk+λ1||Sk+λ2|

∫R

1Sk+λ1

(z − c1

r1

)1Sk+λ2

(z − c2

r2

)dz

≤ 1

|Sk+1||Sk+λ2|

∫R

1Sk+λ2

(z − c2

r2

)dz ≤ 2

|Sk+1|,

and the lemma follows.

Proof of Proposition 3. Fixing Ω ⊆ [−4, 0], we see

‖Φ∗k1Ω‖22 =

∥∥∥∫Ω

Vk,x(·)dx∥∥∥2

2

=

∫R

(∫Ω

Vk,x1(z)dx1

)(∫Ω

Vk,x2(z)dx2

)dz

=

∫Ω2

(∫RVk,x1(z)Vk,x2(z)dz

)dx1dx2

=

∫Ω2

Λ(A(x1, x2);σk)dx1dx2

=

∫Θ1tΘ2

Λ(A(x1, x2);σk)dx1dx2,

77

where A(x1, x2) =(

(c(x1), r(x1)), (c(x2), r(x2)))∈ Uk,

Θ1 = (x1, x2) ∈ Ω2 : A(x1, x2) ∈ U intk ,

andΘ2 = (x1, x2) ∈ Ω2 : A(x1, x2) ∈ U tr

k .First we estimate the integral over Θ1, on which we can only trivially boundthe integrand, but we see that the integration is restricted to a small set.Specifically, we have by Lemma 4 that

Θ1 ⊆

(x1, x2) ∈ Ω2 : |c(x1)− c(x2)| ≤ 160P−1/2k

⊆

(x1, x2) ∈ Ω2 : |x1 − x2| ≤ 320P−1/2k

,

where the last inclusion uses (2). In particular we know that

|Θ1| ≤ 640P−1/2k |Ω|,

which combined with Lemma 5 yields∫Θ1

|Λ(A(x1, x2);σk)|dx1dx2 ≤5120P

−1/2k

|Sk+1||Ω|.

Further, the correlation condition (4) immediately yields∫Θ2

|Λ(A(x1, x2);σk)|dx1dx2 ≤ C0(k)|Ω|2 ≤ C0(k)|Ω|,

and the proposition follows.

References

[1] Bourgain, J. Averages in the plane over convex curves and maximaloperators. J. Analyse Math. 47 (1986), 69-85.

[2] Laba, I. and Pramanik, M. Maximal operators and differentiation the-orems for sparse sets. Duke Math. J. 158 (2011), 347-411.

[3] Stein, E. M. Maximal functions: Spherical means. Proc. Nat. Acad. Sci.U.S.A. 73 (1976), 2174-2175.

Alex Rice, UGAemail: [email protected]

78

13 Maximal operators and differentiation the-

orems for sparse sets, part I

after I. Laba and M. Pramanik [1]A summary written by Pablo Shmerkin

Abstract

The proof of Lp bounds for maximal operators associated to asequence Sk of sparse sets depends critically on a multiscale higher-order uniformity condition, given in terms of a bound on certain cor-relation integrals. I sketch the proof of these correlation bounds fora sequence Sk constructed via a random Cantor iteration. In con-junction with the second part, this establishes the existence of sparsesets for which the corresponding maximal operators are bounded fromLp to Lq for appropriate choices of (p, q).

13.1 Introduction

Let Sk : k ≥ 1 be a decreasing sequence of compact subsets of R. Associ-ated to this sequence is the maximal operator

Mf(x) = supr>0,k≥1

1

|Sk|

∫Sk

|f(x+ ry)|dy.

(|Sk| is the Lebesgue measure of Sk.) In order to obtain Lp bounds forthis and related operators, a critical step is to obtain good estimates forcorrelations of the form

Λ(g1, . . . , gn;σk) =

∫ n∏`=1

σk(g−1` (z))dz, (1)

where

σk =1Sk+1

|Sk+1|− 1Sk|Sk|

measures the “oscillation” of the sequence Sk, and g1, . . . , gn are affinefunctions. The idea is that for “transversal” tuples of affine functions thereshould be a large amount of cancelation in the integral, provided the sets Skare sufficiently “random”. At the other extreme, if the g` are “tangential” (i.e.some of them are nearly equal) then one cannot do better than the trivial

79

estimate for the integral, but on the positive side there will be “relativelyfew” such bad tuples. In this note we define a random construction of Cantoriteration type, and prove efficient bounds for the correlations Λ(g1, . . . , gn;σk)under appropriate transversality conditions on the affine maps.

13.2 Construction of the sets Sk

We start by describing the construction of the sets Sk. The specifics of therandom choices differ slightly from those in [1], and help avoid a number oftechnical steps in the proofs and reduce the number of parameters involved.The construction will depend on an integer N ≥ 2 and a sequence εk ofnumbers in (0, 1). For simplicity, we also assume that N1−εk ∈ N for all k.

Informally, the sets Sk will be constructed as follows: we subdivide [1, 2]into N equal subintervals, and pick N1−ε1 of them at random; let S1 bethe union of these intervals. We next subdivide each of the intervals thatmake up S1 into N subintervals of equal length, and pick N1−ε2 of them atrandom, with all the choices independent. We let S2 be the union of thepicked intervals of this level, and so on. We now give a formal definition thatwill also allow us to define useful notation. Let

I = Ik = i = (i1, i2, . . . , ik) : 1 ≤ ir ≤ N, 1 ≤ r ≤ k.

Elements of Ik will index N -adic intervals of step k in the construction; theleft point of the interval associated to i ∈ Ik is

α(i) = 1 +i1 − 1

N+ . . .+

ik − 1

Nk,

and the length of the interval is δk = N−k. The interval in question is thenIk(i) = [α(i), α(i) + δk].

Next we define, for each k, a collection of indicator random variablesXi : i ∈ Ik taking values in 0, 1 that will describe the collection of“chosen” intervals. To begin, let X1 be a random subset of 1, . . . , N withN1−ε1 elements (the choice is uniform over all such subsets), and let X1 bethe indicator function of X1.

Now suppose the family Xk(i) : i ∈ Ik has been defined. Let Yi :i ∈ Ik be independent random subsets of 1, . . . , N with N1−εk+1 elements,write Yi for the indicator function of Yi, and set Xk+1(ia) = Xk(i)Yi(a).

80

(Here and throughout, ia denotes the juxtaposition (i1, . . . , ik, a).) Finally,we set Sk =

⋃i∈Ik:Xi=1 Ik(i). Let

Pk =k∏i=1

N1−εi .

Then Sk is made up of Pk intervals of length δk. The following facts areeither immediate from the definitions, or follow from standard arguments:

1. The sequence Sk is decreasing; let S =⋂∞k=1 Sk.

2. The densities 1Sk/|Sk| converge weakly to a measure µ supported on S.

3. The Hausdorff dimension of S is dimH S = lim infk→∞1k

∑ki=1(1− εi).

13.3 Internal tangencies and transversal intersections

In order to obtain the most general results, it is important to allow n in (1)to be arbitrarily large. However, the case n = 2 is enough to obtain L2+δ

estimates and already requires the main ideas needed to tackle the generalcase, while being notationally much simpler. We therefore restrict ourselvesto n = 2 in the sequel.

The pair of affine functions in (1) will ultimately be drawn from certaindiscrete family, but most of the analysis will be for a fixed pair g1, g2. Wealways assume that gi(x) = rix + ci, where ri ∈ [1, 2] and ci ∈ [−4, 0]. Weindex the pairs of intervals which have nonempty intersection after beingmapped by g1, g2:

F =

(i1, i2) ∈ I2k : g1(Ik(i1)) ∩ g2(Ik(i2)) 6= ∅

.

It will be critical to distinguish between two kinds of intersections: those inwhich the intervals Ik(i1), Ik(i2) are close to each other (at a distance boundedby a constant multiple of their size), and the rest. The precise definition isas follows. Let

Fint = (i, j) ∈ F : ii = ji for i ≤ k − 1, |ik − jk| ≤ 4 .This collection is referred to as the class of internal tangencies. Its com-plement Ftr = F \Fint will be called the class of transversal intersections.

It can be checked that, in a precise way, a large number of internal tan-gencies forces the maps g1 and g2 to be close, and viceversa. We say thatthe pair (g1, g2) is transversal if the number of internal tangencies is smallin the following quantitative sense: #Fint <

√Pk.

81

13.4 Main result

We can now state the main result.

Theorem 1. For each k, let U trk be a family of transversal pairs (g1, g2) with

at most A := 4δ−2Lk+1 elements for some constant L ≥ 1 (these families and

the constant L will be defined in the second part). There exists C > 0 suchthat with positive probability (on the choices of X1, Yi), the following holdsfor all k:

sup(g1,g2)∈U tr

k

Λ(g1, g2;σk) ≤ C√Lk logN N2εk+1 N−

∑kj=1(1+3εj)/2. (2)

We make some remarks:

1. Given ε ∈ [0, 1/3), with appropriate choices of the parameters we canensure that dimH S = 1 − ε, |S| = 0, and the right-hand side of (2)decays exponentially in k.

2. There is a corresponding version for transversal n-tuples (g1, . . . , gn)that we omit, but is needed to prove the most general Lp → Lq esti-mates.

3. The bounds obtained can be seen as a multiscale analog to second orderuniformity conditions in additive combinatorics (for larger values of none would obtain higher order uniformity-type estimates).

In the rest of the note we outline the proof of Theorem 1, trying toemphasize the main ideas and skipping all calculations. The proof goes byconditioning on Sk (or, more precisely, on Xi : i ∈ Ik), and proving that(2) holds with large probability for k + 1. A little algebra shows that

σk(z) =1

Pk+1δk+1

∑i∈Ik

Xk(i)N∑a=1

(Yi(a)−N−εk+1)1Ik+1(ia)(z).

From here it is easy to deduce that

Λ(g1, g2;σk) =1

(Pk+1δk+1)2

∑(i,j)∈F:Xk(i)Xk(j)=1

Γ(i, j), (3)

82

Γ(i, j) =N∑

a,b=1

(Yi(a)−N−εk+1)(Yj(b)−N−εk+1)|g1(I(ia)) ∩ g2(I(jb))|.

Because we are conditioning on Sk, the randomness comes in the Γ(i, j). Wesplit the sum (3) as Ξint

k +Ξtrk , where the first corresponds to pairs (i, j) ∈ Fint

and the second in Ftr. Under the assumption that (g1, g2) is transversal (i.e.

#Fint < P1/2k ), simple geometric arguments yield a deterministic bound

Ξintk ≤

4δkP1/2k

(Pk+1δk+1)2.

13.5 A martingale argument, and conclusion of theproof

The crux of the proof is to estimate Ξtrk . Since Yi(a) has mean N−εk+1 for all

i ∈ Ik and a ∈ 1, . . . , N, the random variables Γ(i, j), (i, j) ∈ Ftr all havezero mean (recall that i 6= j for (i, j) ∈ Ftr). If they were also independent,the desired estimates would follow from standard exponential concentrationbounds, but this is not the case: Γ(i1, j1) and Γ(i2, j2) are correlated wheneveri1, j1 ∩ i2, j2 6= ∅. The main idea to handle this issue is to decompose

Ftr into a bounded number of subsets F(i)tr , such that each sum∑

Γ(i, j) : XiXj = 1, (i, j) ∈ F(i)tr

can be reordered as a martingale (recall that a sequence of random variablesMr∞r=0 is a martingale if E(Mr+1|Br) = Mr for all r, where Br is anincreasing filtration of σ-algebras). Martingales satisfy concentration boundsessentially as good as sums of iid random variables. A classical example is:

Theorem 2 (Azuma’s Inequality). Let Mr be a martingale, and suppose|Mr+1−Mr| ≤ cr a.s. for some positive numbers cr. Then for all T ∈ N andall λ ∈ R,

P(|MT −M0| ≥ λ) ≤ 2 exp

(− λ2

2∑T

r=1 c2r

).

The families F(i)tr are obtained as follows. An easy consequence of the

definition of Ftr is that the projections (i, j) → i, (i, j) → j are at most 4-to-1 on Ftr, so we can decompose Ftr into 16 classes, on each of which the

83

projections are both injective. We further subdivide each of these classes intotwo subsets, so that on each class αk(i)− αk(j) is either everywhere positiveor everywhere negative (by transversality it cannot be 0).

Let F(i)tr be one of the classes constructed in this way, and without loss

of generality suppose αk(i) < αk(j) for all (i, j) ∈ F(i)tr . We enumerate F(i)

tr =(i1, j1), . . . , (iT , jT ), where αk(jt) is strictly increasing in t (there can be norepetitions thanks to the injectivity of the projection). Further, let Br bethe σ-algebra generated by Zi : α(i) < α(jr). One can then check thatthe sequence M0 = 0, Mr =

∑rt=1 Γ(it, jt) for 1 ≤ r ≤ T , is a martingale.

The essential reason for this is that at most one set it, jt (1 ≤ t < r) canintersect ir, jr.

Applying Azuma’s inequality with λ = 4δk√

2Pk log(200k2A) we get, af-ter some easy estimates,

P(∑

Γ(i, j) : XiXj = 1, (i, j) ∈ F(i)tr > λ

)≤ 1

100k2A.

So far the analysis has been for a fixed transversal pair (g1, g2). Now, since

by assumption #U trk ≤ A, and there are at most 32 classes F(i)

tr , we concludethat

P(

Ξtrk >

32λ

(δk+1Pk+1)2

)≤ 1

3k2.

As∑

k1

3k2 < 1, the proof is finished after a little more algebra to combine thebounds for Ξtr

k and Ξintk into the form given in the statement of the theorem

(we note that with the bound√Pk in the definition of transversal pair (g1, g2),

the estimates for Ξtrk and Ξint

k are essentially equal).

References

[1] Laba, I. and Pramanik, M., Maximal operators and differentiation the-orems for sparse sets. Duke Math. J. 158 (2011), no. 3, 347–411;

Pablo Shmerkin, University of Surreyemail: [email protected]

84

14 Bounded orthogonality systems and the

Λ(p)-set problem I

after Jean Bourgain [1]A summary written by Stefan Steinerberger

Abstract

We describe a pivotal result of Jean Bourgain in the theory ofΛ(p)−sets; it states that the property of being Λ(p) is - in a certainsense - a generic property and highlights the efficiency of interfacingelements from analysis and probability theory.

14.1 Introduction

We consider the one-dimensional torus T and the standard Lp−norm.

Definition 1. Let p > 0. A set E ⊂ Z is called a Λ(p) set if there is a0 < q < p such that for all functions f : T→ C with

supp f ⊂ E

the inequality‖f‖Lp . ‖f‖Lq

holds for an absolute implicit constant depending only on E.

If this property holds for some 0 < q < p, then it is known to holds for all0 < q < p (with the implicit constant depending also on q) thereby justifyingthe name Λ(p)-set. The origin of this definition can be traced back to an oldconjecture, which states that the squares k2 : k ∈ N are a Λ(p) set for allp < 4, or, alternatively,∥∥∥∥∥

N∑k=1

ake2πik2x

∥∥∥∥∥Lp(T)

.p

∥∥∥∥∥N∑k=1

ake2πik2x

∥∥∥∥∥L2(T)

(1)

Early results are due to Rudin, who gave explicit constructions based onthe following theorem.

Theorem 2 (Rudin, 1960). Let 1 < s ∈ N and E ⊂ N. If there is a constantC < ∞ such that any integer n ∈ N has at most C representations as thesum of s elements of E, then E is a Λ(2s) set.

85

Based on results of this type exploiting the combinatorial properties ofthe Lp norm for p ∈ 2N, it was shown that there exist Λ(2n)-sets which arenot Λ(2n+ε)-sets for any ε > 0. We know from results of Bachelis-Ebenstein,Rosenthal and Hare that this is wrong for any Λ(p)−set with 0 < p < 2. Themain problem has therefore been the construction of sets showing that forp ≥ 2 the inclusion of Λ(p+ ε) ⊆ Λ(p) is strict.

This problem was resolved by Jean Bourgain as a consequence of a muchmore powerful theorem with many implications on the structure of Λ(p)−sets.His theorem holds in the more general context of uniformly bounded orthog-onal systems - a simplified version can be stated as follows.

Theorem 3 (Bourgain, 1989, simplified). Let 2 n2/p such that E is a Λ(p)−set. In fact, thisproperty holds for generic sets E of size n2/p.

Remarks.

1. It is easy to see that n2/p is the maximal size of a Λ(p) set containedin 1, 2, . . . , n.

2. Standard results from harmonic analysis allow to extend the result tobuild infinite Λ(p)-sets which, by density arguments alone, are Λ(p) butnot Λ(p+ ε).

3. The theorem implies that being a Λ(p)-set is less special a propertythan previously thought - however, it has no implications for specificsets (i.e. squares).

Λ(p)−sets have applications in the study of Lp−improving measures onT whose convolution with a Lp function yields a smoother Lp+ε function -this property is known to be equivalent to ’large’ Fourier coefficients of themeasure having a Λ(p)−type structure.

Conversely, arguments in the early style of Rudin have matured into state-ments linking the multiplicative structure and the Λ(p)−constant.

Theorem 4 (Bourgain & Chang, 2003). Given ε > 0 and p > 2, there existsδ(ε, p) > 0 such that for any set A ⊂ Z the growth bound |A2| ≤ |A|1+ε

impliesλp(A) ≤ |A|δ,

where δ → 0 as ε→ 0.

86

14.2 Sketch of the general proof

Very roughly summarized, the proof can be said to use a decompositioninto big and small terms to get a bootstrap argument on the size of theΛ(p)−constant; for the bootstrap to close one needs to estimate complicatedexpressions. The core element of the proof is to exploit the fact that byrandomizing over sets the structure of these expressions can be dealt with inan averaged sense.

We demonstrate the way the bootstrap is set up for 2 < p ≤ 3 and definefor any set S ⊂ 1, 2, . . . , n

KS = sup‖a‖2≤1

‖∑i∈S

aie2πix‖Lp(T).

Fix a number 0 < γ < 1 satisfying

(1− γ2)(p−2)/2 + γp < 1

and partition 1, 2, . . . , n (up to 1 point) into sets I, J such that

mini∈I|ai| ≥ max

j∈J|aj|∑

i∈I

a2i < γ2 and

∑j∈J

a2j < 1− γ2.

Having our partition, we define

x(u) =∑i∈I

aiφi(u) y(u) =∑j∈J

ajφj(u)

and wish to estimate ∫|x(u) + y(u)|pdu

using (valid for 2 < p ≤ 3)

|x+ y|p ≤ |x+ y|2|y|p−2 + (1 + |x|)p + 2x(1 + |x|)p−2y + (1 + |x|)p−2y2.

We arrive at∫|x(u) + y(u)|pdu ≤ ‖x+ y‖2

p + ‖1 + |x|‖pp + 2∣∣⟨y, x(1 + |x|)p−2

⟩∣∣+∣∣⟨y, y(1 + |x|)p−2

⟩∣∣≤ K2

SKp−2S

(∑J

a2j

)(p−2)/2

+KpS

(∑I

a2i

)p/2

+ CKp−1S + 2

∣∣⟨y, x(1 + |x|)p−2⟩∣∣+

∣∣⟨y, y(1 + |x|)p−2⟩∣∣ .

87

Now the partition becomes valuable:

(1− γ2)(p−2)/2 + γp < 1,

which allows a rearrangement of the type

KpS ≤ CKp−1

S + other terms,

where the other terms are precisely of the scalar-product-type.

The next steps consist of introducing a dummy variable z(u) (inheritingproperties given to x(u), y(u) by virtue of the partition procedure) and usingdecoupling inequalities. This leads ultimately to terms that can be dyadicallydecomposed into expression where the combinatorial nature is well embod-ied in the entropy-type structure - the above lemma then reduces things tostudying certain geometric properties of the function space when regardedas embedded in a high dimensional Euclidean space (i.e. its diameter anddistance relations); these properties can be studied using Bessel’s inequalityand certain algebraic inequalities.

References

[1] Bourgain, J. Bounded orthogonal systems and the λ(p)-set problem.Acta Math. 162 (1989), no. 3–4, 227–245.

[2] Bourgain, Jean; Chang, Mei-Chu. On the size of k−fold sum and productsets of integers. J. Amer. Math. Soc. 17 (2004), no. 2, 473–497

[3] Rudin, Walter. Trigonometric series with gaps. J. Math. Mech. 9 (1960),203–227.

Stefan Steinerbegrer, University Bonnemail: [email protected]

88

15 On a problem of Erdos on sequences and

measurable sets, &

Infinite patterns that can be avoided by

measure.

after K. Falconer [3] and M. Kolountzakis [4] (respectively)A summary written by Krystal Taylor

Abstract

Erdos conjectured that given an infinite set A of real numbers,there always exists a measurable set of positive measure which containsno affine copy of A. Some partial progress has been made on solvingthis problem. In particular, K. Falconer proves this conjecture in thecase that A is given by a not-too-rapidly-decreasing sequence of realnumbers. M. Kolountzakis shows that for every infinite set A, thereis a set E of positive measure such that x + tA ⊂ E fails for almostall (Lebesgue) pairs (x, t).

15.1 Introduction: known classes of non-universal sets

Definition 1. A set A of real numbers is called universal (in measure) if forevery measurable E ⊂ R with positive Lebesgue measure there are x, t ∈ Rsuch that

x+ tA := x+ ta : a ∈ A ⊂ E.

One can verify that all finite sets of reals are universal. Erdos [2] askedwhether there exist any infinite universal sets. While no universal sets areknown, there are some classes of infinite sets which have been shown to benot universal.Komjath [5] proves that for every infinite set A ⊂ [0, 1], there is anothersubset of [0, 1], of measure arbitrarily close to 1 that does not contain anytranslate of A (no dilations allowed). Kolountzakis [4] gives an alternativeproof of Komjath’s result.Bourgain [1] explores a 3-dimensional version of the problem for sets withCartesian product structure. He shows that any given set in R3 of the typeS×S×S, where S ⊂ R is infinite, is not universal, defining universality in 3dimensions. His method allows for non-isotropic scaling along the three axis.

89

Furthermore, he shows that for sets of real numbers of the type S1 +S2 +S3,where the Sj are infinite, are not universal. He points out that a variant ofhis method shows that sums such as

2−n+ 2−n (1)

are not universal. Bourgain uses a probabilistic construction.Falconer proves that any sequence xn of positive reals decreasing to 0 andsatisfying

limn→∞

xn+1

xn= 1 (2)

is not universal. He constructs a probabilistic Cantor-type set with positivemeasure which avoids all affine copies of a given sequence xn satisfying (2).His result is stated below in Theorem 2.The problem in the case of geometrically decreasing sequences is open. Thatis, no non-universal sequence xn ↓ 0 is known which satisfies xn+1 ≤ ρxn, forsome fixed ρ < 1. Moreover, it is not even known whether all uncountablesets are not universal [4].Kolountzakis [4] proves that almost all copies of a given set of real numberscan be avoided by some set of positive measure. His results are stated belowin Theorems 3 and 4. As mentioned above, Komjath’s result follows as asimple consequence [4].

15.2 Results

15.2.1 Sequences with ’slow decay’ are universal

In this section, we present a result of Falconer which classifies all sequenceswith slow decay as non-universal.

Theorem 2. Let xn∞n=1 be a decreasing sequence of real numbers convergentto 0 such that

limn→∞

xn+1

xn= 1.

Then there exists a closed set E with µ(E) > 0 such that for any numbers b,c, with c 6= 0, cxn − b /∈ E for infinitely many n.

90

15.2.2 Infinite sets are ’almost everywhere’ universal

Here we present two results due to Kolountzakis which say that almost allcopies of an infinite set of real numbers can be avoided by some set of positivemeasure. Theorem 4 below is a strengthening of Theorem 3, but the proofsare different.

Theorem 3. Let A ⊂ R. There is a set E ⊂ [0, 1] of measure arbitrarilyclose to 1 such that the set of pairs

(x, t) : (x+ tA) ⊂ E

has measure 0 (Lebesgue measure in R2).

Theorem 4. Let A ⊂ R. There is a set E ⊂ [0, 1] of measure arbitrarilyclose to 1 such that

µt : ∃ x such that (x+ tA) ⊂ E = 0.

Here and throughout, µ denotes the Lebesgue measure in R.

15.3 Sketch of proofs

15.3.1 Construction of the set E in Theorem 2

The purpose of this section is to present some of the ideas in the proof ofFalconer’s Argument. For a given sequence satisfying the conditions of thetheorem, Falconer constructs a set E and shows that, for any integer m, thereexists an integer n(m) so that

∞⋂n=n(m)

1

xn(E + b) ⊂ 0 (3)

for all real numbers b. As a consequence, for any c 6= 0, there exists n ≥ n(m)so that cxn /∈ E+b. Iterating this process generates infinitely many elementsof the sequence for which cxn /∈ E + b.

To construct E, choose numbers λk (1 ≤ k < ∞) such that 0 < λk < 1and

∑∞k=1 λk <

12. Let lk be a rapidly decreasing sequence of lengths to be

determined. LetEk = ∪∞r=−∞ [rlk, rlk + lk(1− λk)]

91

and let E =⋂∞k=1Ek. Provided that lk < 1, for 1 ≤ k <∞, it is immediate

that µ(E) > 0. To see this, observe that [0, 1]\Ek contains at most ( 1lk

+ 1)intervals of length lk, and so

µ([0, 1]\E) ≤∞∑k=1

µ([0, 1]\Ek) ≤∞∑k=1

λklk(1

lk+ 1) < 1.

By observing the overlapping of certain intervals, Falconer shows that

∞⋂n=n(k)

1

xn(Ek + b) ⊂

[−2lk/xn(k), 2lk/xn(k)

], (4)

whenever 0 ≤ b ≤ lk. Then by the periodicity of Ek, (4) holds for all real b.One can observe that it is in this aforementioned overlapping that the decaycondition is used and that the argument fails for geometric sequences.

Proving (4) reduces to showing that for each k and for each 0 ≤ b ≤ lk,(lk(1− λk) + b

xn(k)

,∞)⊂

∞⋃n=n(k)

(1

xn(Ek + b)

)c, (5)

where we recall that 0 < λk < 1. Here and throughout Ac denotes the com-pliment of the set A in RThe idea now is to fix k ∈ N, fix 0 ≤ b ≤ lk, and write the interval on theleft-hand-side of (5) as a union of overlapping intervals, each of which lies

in some(

1xn

(Ek + b))c

for some n ≥ n(k). It is at this point that the decay

condition on the sequence xn plays a role in the argument.

15.3.2 Construction of the set E in Theorem 3

The purpose of this section is to present some of the ideas in the proof ofKolountzakis’s Argument.

First, we observe that is is sufficient to prove the theorem in the case thatA is a sequence of positive reals decreasing to 0. Next, fix an interval [α, β],

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where 0 < α < β <∞, for the scaling parameter. The random Cantor-typeset, E, is defined as an intersection

E =∞⋂j=1

Fj,

where Fj is defined by dividing the unit interval into mj equal subintervalsand keeping each of them independently and with equal probability pj. Theprobabilities pj are taken such that

∞∏j=1

pj = q and∞∏j=1

pjj = 0,

something which is possible for every q ∈ [0, 1]. The integers mj are definedto be large enough that

ta1, · · · , taj (6)

are in separate intervals of Fj for t ∈ [α, β]. In particular, 1mj

is smaller than

half of the minimum gap (the largest interval containing no points) of thenumbers αa1, · · · , αaj.Now, for a fixed x ∈ [0, 1], it holds that x ∈ E if x is in exactly one of theintervals making up Fj for each value of j. Therefore

Pr(x ∈ E) =∞∏j=1

pj = q.

Therefore, Eµ(E) = q.

The next step is to show that

Eµ(x, t) : x+ tA ⊂ E = 0, (7)

for any x ∈ [0, 1] and t ∈ [α, β]. ( Here, we are abusing notation by using µ todenote either the Lebesgue measure on R or R2 depending on the context.)It would follow from (7) that

µ(x, t) : x+ tA ⊂ E = 0 (8)

almost everywhere, and we could conclude then that there exists a set E,with µ(E) ≥ q which satisfies (8), thus concluding the proof of the theorem.

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Let φ(x, t) be the indicator function of the set of pairs (x, t) so that x+ tA ⊂E. Observe that

Eµ(x, t) : x+ tA ⊂ E = E

∫ 1

0

∫ β

α

φ(x, t)dtdx

=

∫ 1

0

∫ β

α

Eφ(x, t)dtdx

= 0.

To prove (7), it suffices to show then that Eφ(x, t) = 0. For the set x + tAto be contained in E, it is neccessary that all intervals of stage j whichcontain a point in (6) must be kept. Since there are exactly j such intervals,the probability of this happening is exactly pjj, and the probability of this

happening for all stages is∞∏j=1

pjj = 0.

References

[1] Bourgain, J. Construction of sets of positive measure not containing anaffine image of a given infinite structure. Israel J. Math. 60 (1987) 333-344.

[2] Erdos, P. My Scottish book ’problems’. The Scottish Book (ed. R.D.Mauldin, Birkhauser), Boston, 1981, 35-43.

[3] Falconer, K. On a problem of Erdos on sequences and measurable sets.Proc. Amer. Math. Soc. 90 (1984), 77-78.

[4] Kolountzakis, M. Infinite patterns that can be avoided by measure. Bull.London Math. Soc. 29 (1997), 415-424.

[5] Komjath, P. Large sets not containing images of a given sequence.Canad. Math. Bull. 26 (1983), 41-43.

Krystal Taylor, Technionemail: [email protected]

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16 Averages in the plane over convex curves

and maximal operators

after J. Bourgain [2]A summary written by Joshua Zahl

Abstract

We prove certain Lp → Lp bounds on the Bourgain circular maxi-mal function.

16.1 Introduction

16.1.1 Background

In [1], Bourgain defined the “circular maximal function”: for a boundedmeasurable function f : R2 → R, the circular maximal function of f is givenby

Mf(x) = supt>0

1

2πt

∫C(x,t)

f(y)dy, (1)

where C(x, t) is the circle centered at x of radius t, and dy is the 1–dimensionalarclength measure on the circle C. Bourgain established the bound

‖Mf‖p ≤ Cp ‖f‖p , 2 < p <∞. (2)

This result has several consequences:

1. Differentiation theorems: Define

Atf(x) =1

2πt

∫C(x,t)

f(y)dy. (3)

Then if f is a bounded measurable function,

f = limt→0

Atf a. e . (4)

2. “circle sets”: Let K ⊂ R2 be a compact set such that for each x ∈[0, 1]2, K contains a circle centered at x. Then K has positive Lebesguemeasure.

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3. Estimates on the wave equation in 2+1 dimensions: Let u : R+×R2 → R be a solution to the wave equation u = 0, u(0, x) =0, ut(0, x) = f(x). Then we obtain certain estimates on u of theform

‖u(t, x)‖L∞t (R)Lp1x (R2) . ‖f‖Lp2x (R2) . (5)

See i.e. [4], Section 3.2 for more details.

16.1.2 New Results

In this paper [2], Bourgain proves an analogue of (2) for a more generalclass of curves. Let Γ be the boundary of a compact, convex, centrallysymmetric set in R2. Assume furthermore that Γ is a smooth curve and hasnon-vanishing curvature. Let σ be the arclength measure on Γ.

For 0 < t <∞, and for f a bounded measurable function on R2, define

AΓ,tf(x) =

∫f(x+ ty)σ(dy). (6)

Thus if Γ is the unit circle, then AΓ,tf(x) = Atf(x), where the latter operatoris as defined in (3). Define the maximal operator

MΓf(x) = supt>0

AΓ,tf(x). (7)

The main result of [2] is the following theorem.

Theorem 1. Let f be a bounded measurable function on R2. Then

‖MΓf‖p ≤ CΓ,p ‖f‖p , 2 < p <∞. (8)

When Γ is the unit circle, then the operator MΓ is not bounded in L2.Thus the Lp bounds from Theorem 1 are best possible.

Rather than giving a sketch of Theorem 1 in its full form, we will onlyconsider the case where Γ is the unit circle. The proof of this special casecaptures all of the main ideas of the general proof, but it avoids many of themessy plane geometry details that need to be considered in the full theorem.

96

16.2 Proof the Theorem 1

As noted in the introduction, we will only consider the case where Γ is theunit circle. Thus we will use the operators M and At rather than MΓ andAΓ,t.

16.2.1 Reduction to a geometric problem

In this section, we use a variety of (by now standard) techniques from har-monic analysis to reduce the problem of bounding ‖Mf‖p to a quantitativeform of the following geometric question:

Question 2. Given a collection of circles C, such that for each point x ∈[0, 1]2 there is a circle C ∈ C centered at x, how many pairs of circles can betangent to each other?

We will make this question precise and show how certain quantitativebounds on circle tangencies give us the desired bounds on Mf .

First, we shall decompose f into a collection of diadic pieces using a Haarwavelet basis. We will write

∆kf = E[f |Dk]− E[f |Dk+1], (9)

where Dk is the σ–algebra generated by diadic squares of length 2−k. Thusthe function E[f |Dk] is constant on each diadic square of side length 2−k,and the value of E[f |Dk] on each square is equal to the average of f on thatsquare.

We will similarly decompose the measure σ:

σ = σ0 +∞∑k=1

2k−1σk, (10)

where

σ0 = χ1|y|≤2,

σk = χ1≤|y|≤1+2−k − χ1≤|y|≤1+2−k+1 .(11)

In order to control ‖Mf‖p, it suffices to get sufficiently good control onAt(∆kf) for each k. Once this control has been obtained, we can replacethe maximum suptAtf by a sum

∑k suptAt(∆kf), and this supremum can

in turn be replaced by a sum over an appropriately chosen sequences of t’s.

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Then, by summing carefully we use the theory of square functions (see i.e.[5]) to obtain the desired Lp bounds on Mf . In order to make this work, wewill need to establish the bound∥∥∥∥ sup

t∼2−(k+s)

At(∆kf)

∥∥∥∥p

. 2−α(p)s ‖∆kf‖p . (12)

For all choices of k, s ≥ 0. We will prove (12) in the case where k = 0 and fis supported in the unit square. The general case can be recovered by scalingarguments. Equation (12) would follow if for some α > 0 we could establishthe bound ∥∥∥∥sup

t∼1

∣∣∣ ∫ ∆0f(x+ ty)σk(y)dy∣∣∣∥∥∥∥p

. 2−k(1+α) ‖f‖p . (13)

We will control the RHS of (13) through a certain dualization process.The operator f 7→ supt∼1

∣∣ ∫ ∆0f(x+ ty)σk(y)dy∣∣ is not linear, and this will

make it difficult to find a dual operator. To fix this problem, we shall definea radius function t(x), and we shall consider the expression∥∥∥∥∫ ∆0f(x+ t(x)y)σk(y)dy

∥∥∥∥p

. (14)

If we can show that (14) . 2−k(1+α) ‖f‖p , where the implicit constant doesn’tdepend on the choice of function t(x), then we will have established (13).

Let Vx be the measure with Radon-Nikodym derivative

Vx(y)

dµ= χt(x)<|x−y|<t(x)+n−1 − χt(x)+n−1<|x−y|<t(x)+2n−1 , (15)

where µ is Lebesgue 2–measure, so∫f(x+ t(x)y)dσk(y) =

∫f(y)Vx(y)dy. (16)

By duality arguments and Marcinewitz interpolation, it suffices to establishthe bound ∥∥∥∥∫

Ω

Vx(y)dx

∥∥∥∥Lqy

. n−1−α|Ω|1/q (17)

for some α > 0, and for all measurable sets Ω and all 1 < q < 2.

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First, we have the following two bounds:∥∥∥∥∫Ω

Vx(y)dx

∥∥∥∥L1y

. n−1|Ω|, (18)∥∥∥∥∫Ω

Vx(y)dx

∥∥∥∥L2y

. n−1 log n|Ω|1/2. (19)

Equation (18) follows from Fubini’s theorem, while (19) follows from somegeneral results from [3], which we will not describe here. Interpolating be-tween (18) and (19) establishes (17) for α = 0, but we need to obtain (17)for some α > 0. In order to do this, we will need to improve either (18) or(19). Neither of these bounds can be improved directly, but instead we havethe following lemma:

Lemma 3. Let Vx(y) and Ω be as defined above. There exists an absoluteconstant ε > 0 and a decomposition Ω = Ω1 t Ω2 such that∥∥∥∥∫

Ω1

Vx(y)dx

∥∥∥∥L1y

. n−1−ε|Ω1|, (20)∥∥∥∥∫Ω2

Vx(y)dx

∥∥∥∥L2y

. n−1−ε|Ω2|1/2. (21)

Lemma 3 gives us (17), which in turn establishes Theorem 1 (in the casewhere Γ is the unit circle).

Proof of Lemma 3. Lemma 3 contains some intricate geometric arguments,so we shall only give a general sketch of the main ideas involved.

For each x ∈ Ω, let Ex be the annulus of thickness n−1 centered at x ofradius t(x). Let

Ωx = y ∈ Ω: Ey is (n−1+ε)–tangent to Ex. (22)

Fix some value of δ that we will choose later. Using the greedy algorithm,select a sequence of points x1, . . . , xJ ∈ Ω such that

|Ωx1| > n−1+δ,

|Ωx2\Ωx1| > n−1+δ,

...

|ΩxJ\⋃j<J

Ωxj | > n−1+δ.

(23)

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Let

Ω1 =⋃

Ωxj ,

Ω2 = Ω\Ω1.(24)

We will first establish (21). Let

D = (x, y) ∈ Ω2 × Ω2 : y ∈ Ωx. (25)

D is the set of points x, y where Ex and Ey are almost tangent. Ω2 has beenconstructed so that D is small. We write∫ (∫

Ω2

Vxdx)2

dy

=

∫(x1,x2)∈Ω2×Ω2

〈Vx1 , Vx2〉dx1dx2

=

∫D〈Vx1 , Vx2〉dx1dx2 +

∫Ω2×Ω2\D

〈Vx1 , Vx2〉dx1dx2 (26)

(27)

The first term of (26) is small because the set D is small, while the secondterm is small because of the following heuristic:

Heuristic 4. If two annuli Ex1 and Ex2 are far from being tangent, then〈Vx1 , Vx2〉 is very small:

The difference between the areas of the regions F++ ∪F−− and F+− ∪F−+ issmall.

We will now establish (20). Define

Ω′1 = Ωx1 ,

Ω′2 = Ωx1\Ωx1 ,

Ω′3 = Ωx3\(Ωx1 ∪ Ωx2),

...

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It suffices to show that ∥∥∥∥∥∫

Ω′j

Vx(y)dx

∥∥∥∥∥L1y

. n−1−ε|Ω′j| (28)

for each index j. After a translation and scaling, we can assume that xj = 0and t(xj) = 1, i.e. Exj is the n−1/2 neighborhood of the unit circle. Afterdiadic pigeonholing (and a corresponding refinement of Ω′j by a factor oflog n), we can also assume that there exists a diameter d such that |x| ∼ dfor all x ∈ Ω′j. After a suitable modification of the functions Vx : x ∈ Ω′j,we establish (28) using the Cauchy-Schwartz inequality and a suitable boundon the L2–norm of

∫Ω′jVx(y)dx. Again, we need to show that not too many

tangencies can occur amongst the annuli from

Ex : x ∈ Ω′j. (29)

In general, two circles (or annuli) can be tangent to each other even iftheir centers are far apart from each other. However, the annuli in (29) areall almost tangent to the unit circle, and thus we can apply the followingheuristic:

Heuristic 5. Let C(x, r), C(x′, r′) be circles, both of which contain a com-mon point z, which lies slightly outside the unit circle. Suppose C(x, r) andC(x′, r′) are each nearly tangent to the unit circle. Then the extent to whichC(x, r) and C(x′, r′) are tangent to each other is controlled by |x− x′|.

Using this heuristic, we can control the number of almost tangencies in(29), and this in turn allows us to establish (28).

References

[1] J. Bourgain. On the spherical maximal function in the plane. Preprint.IHES, 1985.

[2] J. Bourgain. Averages in the plane over convex curves and maximaloperators. J. Anal. Math. 47(1):69–85, 1986.

[3] J. Bourgain. On high dimensional maximal functions associated to con-vex bodies. Am. J. Math. 108(6):1467–1476, 1986.

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[4] W. Schlag. Lp → Lq estimates for the circular maximal function. Ph.D.Thesis. California Institute of Technology, 1996.

[5] E. Stein. Harmonic analysis: real-variable methods, orthogonality, andoscillatory integrals. Princeton University Press, Princeton NJ, 1993.

Joshua Zahl, UCLAemail: [email protected]

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