Partial difference equations over compact Abelian

groups

Tim Austin

Courant Institute, NYUNew York, NY, USAtim@cims.nyu.edu

SETTING

Z a compact (metrizable) Abelian groupU1, . . . , Uk closed subgroups of Z

F(Z) {measurable functions Z −→ T = R/Z}(up to equality a.e.)∫

Z

· dz integral w.r.t. Haar probability measure

If w ∈ Z, f ∈ F(Z) then dwf(z) := f(z − w)− f(z)— discrete analog of directional derivative.

If f : Z −→ C, then ∇wf(z) := f(z − w)f(z)— multiplicative variant

PARTIAL DIFFERENCE EQUATION

Describe those f ∈ F(Z) for which

du1 · · · dukf(z) ≡ 0 ∀u1 ∈ U1, . . . , uk ∈ Uk.

ZERO-SUM PROBLEM

Describe those f1, . . . , fk ∈ F(Z) s.t.

duifi(z) ≡ 0 ∀ui ∈ Ui, i = 1,2, . . . , k

and

f1(z) + f2(z) + · · ·+ fk(z) ≡ 0.

Will focus on PDceEs in this talk.

MOTIVATION: SZEMERÉDI’S THEOREM

A k-AP in Z is a set of the form

{z, z + r, . . . , z + (k − 1)r} for some z, r ∈ Z.

It is non-degenerate if all k entries are distinct.

Theorem (Szemerédi ’75).∀δ > 0 ∀k ≥ 1 ∃N0 = N0(δ, k) ≥ 1 such that if N ≥ N0 and E ⊆ ZNwith |E| = δN , then

E ⊇ some non-degenerate k-AP.

MULTI-DIMENSIONAL ANALOG

Fix F = {v1, . . . , vk} ⊂ Zd, all vi distinct.

If z ∈ ZdN and r ∈ ZN , let

z + r · F = {z + rv1, . . . , z + rvk} mod N.Call this an F -constellation. It is non-degenerate if |z + r · F | = k.

Theorem (Furstenberg and Katznelson).∀δ > 0 ∀d, k ≥ 1 ∃N0 = N0(δ, k, d) ≥ 1 such that if N ≥ N0 andE ⊆ ZdN with |E| = δN

d then

E ⊇ some non-degenerate F -constn.

Szemerédi’s Theorem is special case

F = {0,1, . . . , k − 1} ⊂ Z.

Many proofs now known, using graph theory (Szemerédi), ergodic theory(Furstenberg), hypergraph theory (Nagle-Rödl-Schacht, Gowers, Tao) orharmonic analysis/additive combinatorics (Roth, Gowers).

Roth’s and Gowers’ harmonic-analysis proof gives much the best boundon N0(δ, k).

Some proofs generalize to higher dimensions. Roth-Gowers approach hasnot been generalized, and the known dependence of N0(δ, k, d) is gener-ally much worse when d ≥ 2.

SKETCH OF ROTH-GOWERS APPROACH

Will formulate this for multi-dimensional theorem as far as possible.

Important idea: estimate the fraction of all F -constellations that are con-tained in E. Let Z = ZdN .

Suppose φ1, . . . , φk : Z −→ C, and set

Λ(φ1, . . . , φk) :=∫ZN

∫ZdN

φ1(z + rv1)φ2(z + rv2) · · ·φk(z + rvk) dzdr.

Interpretation: if E ⊆ Z and φ1 = . . . = φk = 1E, then

Λ(1E, . . . ,1E) = P(random F -constn. lies in E

).

Idea is to show that if |E| = δN then

Λ(1E, . . . ,1E) ≥ const(δ, k) > 0.

Since

P(randomly-chosen F -constn. is degenerate

)−→ 0

as N −→∞, this proves the theorem.

Key tool for estimating Λ: directional Gowers uniformity norms.

Can formulate these for any compact U1, . . . , Uk ≤ Z:

If φ : Z −→ C, then

‖φ‖U(U1,...,Uk) :=( ∫U1

∫U2

· · ·∫Uk

∫Z

∇u1 · · ·∇ukφ(z) dzduk · · ·du1)2−k

.

For example,

‖φ‖4U(U1,U2) =∫U1

∫U2

∫Z

φ(z − u1 − u2)φ(z − u1)φ(z − u2)φ(z) dzdu1du2.

Theorem (Gowers-Cauchy-Schwartz inequality).

|Λ(φ1, . . . , φk)| ≤ ‖φ1‖U(V12,...,V1k) · ‖φ2‖∞ · · · ‖φk‖∞,

where Vij := ZN · (vi − vj) ≤ ZdN for 1 ≤ i, j ≤ k.

Proved by repeated use of Cauchy-Schwartz inequality.

Corollary. If

‖1E − δ‖U(V12,...,V1k) ≈ 0,‖1E − δ‖U(V21,V23...,V2k) ≈ 0,

...

‖1E − δ‖U(Vk1,...,Vk,k−1) ≈ 0,

then Λ(1E, . . . ,1E) ≈ δk, =⇒ many F -constellations in E if N is large.

Intuition: in this case, E ‘behaves like a random set’, and therefore ‘con-tains many F -constellations by chance’.

Idea for full proof: if

‖1E − δ‖U(U1,...,Uk−1) 6≈ 0

for one of the lists of subgroups above, then deduce some special ‘struc-ture’ for E, which implies positive probability of F -constellations for a dif-ferent reason (not ‘by chance’). Proof is finished once have result for both‘random’ and ‘structured’ E.

In one-dimension, have: Inverse Theorem for Gowers norms.

In that case, Vij = ZN = Z for every i, j.

Roughly: If ‖φ‖∞ ≤ 1 and ‖φ‖U(Z,...,Z) ≥ η > 0, then ∃ a ‘locally polyno-mial phase function’ ψ : Z −→ S1 s.t.∣∣∣∣ ∫

Z

φ(z)ψ(z) dz∣∣∣∣ ≥ const(η) > 0.

Won’t define ‘locally polynomial phase functions’ here.

Related toy problem: suppose |φ(z)| ≤ 1 ∀z but∫Z

· · ·∫Z

∇u1 · · ·∇uk−1φ(z) dzdu1 · · ·duk−1 = 1.

This is possible only if φ : Z −→ S1 and

∇w1 · · ·∇wk−1φ(z) ≡ 1.

Exercise: ifZ = ZN ,N � k andN prime, this occurs iff φ = exp(2πif/N)for some polynomial f : ZN −→ ZN of degree ≤ k − 2.

There are more technical wrinkles for other Z, but still get some slightly-generalized notion of ‘polynomial of degree ≤ k − 2’.

In higher dimensions no good inverse theorem known for directional Gow-ers norms.

Simplest toy case: describe those φ for which |φ(z)| ≤ 1 and

‖φ‖U(U1,...,Uk) = 1.

As before, this is equivalent to φ : Z −→ S1 and

∇u1 · · ·∇ukφ ≡ 1 ∀u1 ∈ U1, . . . , uk ∈ Uk.

Writing φ = exp(2πif) with f : Z −→ T, this is exactly the partial differ-ence equation we started with.

EXAMPLES OF PDceEs AND SOLUTIONS

Let us write our partial difference equation as

dU1 · · · dUkf ≡ 0.

Example 1 If U1 = . . . = Uk = Z, then this is the case of ‘polynomialphase functions’ mentioned before.

Example 2 There are always some obvious solutions: can take

f = f1 + . . .+ fk,

where dUifi = 0 (equivalently, fi is lifted from Z/Ui −→ T).

Definition. If f solves

dU1 · · · dUkf ≡ 0,

then let’s say f is degenerate if can decompose it as

f = f1 + · · ·+ fk,

where

dU2dU3 · · · dUkf1 = dU1dU3 · · · dUkf2 = . . . = dU1 · · · dUk−1fk = 0.

– that is, each fi satisfies a simpler (and stronger) equation.

One is interested in classifying non-degenerate solutions, modulo degen-erate ones.

Example 3 On Z = T3, let f(θ1, θ2, θ3) = b{θ1}+ {θ2}c · θ3.

(For θ ∈ T, {θ} is its unique representative in [0,1), and b·c = integer part.)

Then

f(θ1, θ2, θ3)− f(θ1, θ2, θ3 + θ4) + f(θ1, θ2 + θ3, θ4)− f(θ1 + θ2, θ3, θ4) + f(θ2, θ3, θ4) = 0

Think of these as five different functions of (θ1, θ2, θ3, θ4) ∈ T4.

By differencing, we can annihilate the last four terms, leaving

dU1dU2dU3dU4f = 0

for suitable U1, . . . , U4 ≤ T4.

f(θ1, θ2, θ3)− f(θ1, θ2, θ3 + θ4) + f(θ1, θ2 + θ3, θ4)− f(θ1 + θ2, θ3, θ4) + f(θ2, θ3, θ4) = 0

Disclosure: this equation is not arbitrary. It is the equation for a 3-cocyclein group cohomology H3m(T,T).

One can prove that if f were degenerate, then it would be a coboundary:i.e., represent the zero class in H3m(T,T).

But H3m(T,T) ∼= Z, and I chose f above to be a generator. Hence non-degenerate.

More generally, Hpm(T,T) ∼= Z for all odd p. For each odd p, choosinga generator gives a non-degenerate solution to a PDceE with p + 1 sub-groups.

This leads to many new ‘cohomological’ examples.

Example 4 Let Z = T2 × T2, and define σ, c : Z −→ T by

σ(s, x) = {s1}{x2} − b{x2}+ {s2}c{x1 + s1} mod 1and

c(s, t) = {s1}{t2} − {t1}{s2} mod 1

Then

σ(s, x+ t)− σ(s, x) = σ(t, x+ s)− σ(t, x) + c(s, t).Interpret this as a zero-sum problem on T2 × T2 × T2.

As for Example 3, repeated differencing annihilates all but one term, leav-ing a new PDceE for the subgroups

U1 = {(s, t, x) | s = x+ t = 0}, U2 = {(s, t, x) | s = x = 0},

U3 = {(s, t, x) | x+ s = t = 0}, U4 = {(s, t, x) | s = t = 0}.

σ(s, x+ t)− σ(s, x) = σ(t, x+ s)− σ(t, x) + c(s, t).

The reveal:

Written multiplicatively (i.e., for maps Z −→ S1), this equation becomes

σ(s, x+ t)

σ(s, x)= c(s, t)

σ(t, x+ s)

σ(t, x).

This is the Conze-Lesigne equation, important in the study of two-stepnilrotations.

Now a more complicated argument shows that σ is a non-degenerate solu-tion to the PDceE. Also: not obtained from a cocycle in group cohomology.

SOME POSITIVE RESULTS

Assume U1 + . . . + Uk = Z. (Otherwise, just work on each coset ofU1 + . . .+ Uk separately.)

Let M ⊆ F(Z) be the subgroup of solutions to the PDceE associated toU1, . . . , Uk. It is globally invariant under rotations of Z.

Let M0 ⊆ M be the further subgroup of degenerate solutions. It is alsoglobally Z-invariant.

Lastly, let | · | : T −→ [0,1/2] be distance from 0 in T.

Theorem (Small solutions are always degenerate).∀k ≥ 1 ∃η > 0 such that

f ∈M and∫Z|f | < η =⇒ f ∈M0.

Corollary. The quotient M/M0 is a countable discrete group.

Theorem (Stability for approximate solutions).∀k ≥ 1 ∀ε > 0 ∃δ > 0 such that if

f ∈ F(Z) and∫U1· · ·∫Uk

∫Z|du1 · · · dukf(z)| < δ

then

∃g ∈M s.t.∫Z|f − g| < ε.

Lastly, suppose Z = Td. A function f : Td −→ T is step-polynomial if ∃a partition

Z = P1 ∪ · · · ∪ Pm

such that

• each Pi is defined by linear inequalities in {θi} for (θ1, . . . , θd) ∈ Td,and

• each restriction f |Pi is some polynomial function of ({θi})di=1, evalu-ated mod 1.

Theorem (Basic solns are step-polynomial, mod M0). If f ∈M , then

f = f1 + g

for some step-polynomial f1 ∈ M and some g ∈ M0. Also, have boundson ‘complexity’ of f1 that depend only on ‘quantitative smoothness’ of f ,not on choice of Z: this gives a nontrivial result even for Z finite.