On some open problems in Diophantine approximation

8/12/2019 On some open problems in Diophantine approximation

1/42

arXiv:1202

.4539v5

[math.NT

]22Dec2012

On some open problems in

Diophantine approximation

byNikolay Moshchevitin1

In memory ofVladimir Igorevich Arnold

(1937 2010)

Abstract. We discuss several open problems in Diophantine approximation. Among them there are

famous Littlewoods and Zarembas conjectures as well as some new and not so famous problems.

In the present paper I shall give a brief surwey on several problems in Diophantine aproximationwhich I was interested in and working on. Of course my briev survey is by no means complete. Themain purpose of this paper is to make the problems under consideration more popular among a widermathematical audience.

Basic facts concerning Diophantine approximation one can find in wonderful books [74, 29,133].As for the Geomerty of Numbers we refer to [30,56]. Some topics related to the problems discussedbelow are considered in recent surveys [104,151].

1 Littlewood conjecture and related problems

Every paper in Diohantine approximations should begin with the formulation of the Dirichlet theoremwhitch states that for real numbers 1,...,n, n 1there exist infinitely many positive integersqsuchthat

max1jn ||

qj||

1

q1/n

(here and in the sequel|| || stands for the distance to the nearest integer), or

lim infq+

q1/n max1jn

||qj|| 1.

The famous Littlewood conjecture in Diophantine approximatios supposes that for any two real num-bers 1, 2 one has

lim infq+

q||q1||||q2|| = 0. (1)The similar multidimensional problem is for given n 2 to prove that for any reals 1,...,n one has

liminfq+ q||q1| | | |qn|| = 0. (2)

This problem is not solved for anyn 2. Obviously the statement is false forn = 1: one can considera badly approximable number 1 such that

infqZ+

q||q1|| >0. (3)1 Research is supported by RFBR grant No.12-01-00681-a and by the grant of Russian Government, project 11.

G34.31.0053 and by the grant NSh-2519.2012.

1
http://arxiv.org/abs/1202.4539v5http://arxiv.org/abs/1202.4539v5http://arxiv.org/abs/1202.4539v5http://arxiv.org/abs/1202.4539v5http://arxiv.org/abs/1202.4539v5http://arxiv.org/abs/1202.4539v5http://arxiv.org/abs/1202.4539v5http://arxiv.org/abs/1202.4539v5http://arxiv.org/abs/1202.4539v5http://arxiv.org/abs/1202.4539v5http://arxiv.org/abs/1202.4539v5http://arxiv.org/abs/1202.4539v5http://arxiv.org/abs/1202.4539v5http://arxiv.org/abs/1202.4539v5http://arxiv.org/abs/1202.4539v5http://arxiv.org/abs/1202.4539v5http://arxiv.org/abs/1202.4539v5http://arxiv.org/abs/1202.4539v5http://arxiv.org/abs/1202.4539v5http://arxiv.org/abs/1202.4539v5http://arxiv.org/abs/1202.4539v5http://arxiv.org/abs/1202.4539v5http://arxiv.org/abs/1202.4539v5http://arxiv.org/abs/1202.4539v5http://arxiv.org/abs/1202.4539v5http://arxiv.org/abs/1202.4539v5http://arxiv.org/abs/1202.4539v5http://arxiv.org/abs/1202.4539v5http://arxiv.org/abs/1202.4539v5http://arxiv.org/abs/1202.4539v5http://arxiv.org/abs/1202.4539v5http://arxiv.org/abs/1202.4539v5http://arxiv.org/abs/1202.4539v5http://arxiv.org/abs/1202.4539v5http://arxiv.org/abs/1202.4539v5http://arxiv.org/abs/1202.4539v5


2/42

Any quadratic irrrationality satisfies (3); moreover as it was proved by V. Jarnk [63] the set of all 1satisfying (3) has zero Lebesgue measure but full Hausdorff dimension in R.

Of course Littlewood conjecture is true for almost all pairs (1, 2) R2. Moreover, from Gallaghers[51] theorem we know that for a positive-valued decreasing to zero function(q) the inequality

||q1||||q2|| (q)

for almost all pairs (1, 2) in the sense of Lebesgue measure has infinitely many solutions in integersq(respectively, finitely many solutions) if the series

q(q)log qdiverges (respectively, converges).

Thus for almost all (1, 2) we have

liminfq+

qlog2 q||q1||||q2|| = 0.

Einsiedler, Katok and Lindenstrauss[42] proved that the set of pairs(1, 2)for which (1) is not trueis a set of zero Hausdorff dimension (see also a paper by Venkatesh [150] devoted to this result). In myopinion Littlewood conjecture is one of the most exiting open problems in Diophantine approximations.Some argument for Littlewood conjecture to be true are given recently by Tao [145].

At the beginning of our discussion we would like to formulate Pecks theorem [120] concerningapproximations to algebraic numbers.

Theorem 1. Suppose thatn 2 and1, 1,...,n form a basis of a real algebraic field of degreen + 1.Then there exists a positive constant C = C(1,...,n) such that there exist infinitely many q Z+such that simultaneously

max1jn

||qj|| Cq1/n

and

max1jn1

||qj|| Cq1/n(log q)1/(n1)

.

Pecks theorem ia a quantitative generalization of a famous theorem by Cassels and Swinnwrton-Dyer [28]. We see that for a basis of a real algebraic field one has

liminfq+

qlog q||q1| | | |qn|| 0 (4)(see [29], Ch. V, 3).

A good inrtoduction to Littlewood conjecture one can find in[124]. Interesting discussion is in [18].

1.1 Lattices with positive minima

Suppose that 1, 1,...,n form a basis of a totally real algebraic field K = Q() of degree n+ 1. Thismeans that all algebraic conjugates = (1), (2),...,(n+1) to are real algebraic numbers. So there

2


3/42

exists a polynomial gj() with rational coefficients of degree n such that j = gj(). We considerconjugates

(i)j =gj(

(i)) and the matrix

=

1

(1)1 (1)n

1

(n)1 (n)n

.

LetGbe a diagonal matrix of dimension (n+1)(n+1)with non-zero diagonal elements. We considera lattice of the form

=G Zn+1.

Lattices of such a type are known as algebraic lattices.For an arbitrary lattice Rn+1 we consider its homogeneous minima

N() = inf z=(z0,z1,...,zn)\{0}

|z0z1 zn|.

One can easiily see that if is an algebraic lattice thenN()>0.Ifn = 1 and , are arbitrary badly appoximable numbers (that is satisfying (4)) then for

=

1 1

the lattice L= Z2 R2 will satisfy the propertyN(L) > 0. Of course one can take, in such away that L is not an algebraic lattice. So in the dimension n+ 1 = 2 there exists a lattice L R2which is not an algebraic one, butN(L)>0.

A famousOppenheim conjecturesupposes that in the casen 2any lattice satisfyingN(L)> 0is an algebraic lattice. This conjecture is still open. Cassels and Swinnerton-Dyer [28] proved thatfrom Oppenheim conjecture in dimension n= 2Littlewood conjecture (1) follows.

Oppenheim conjecture can be reformulated in terms of sailsof lattices (see [52, 53, 54]). A sailof a lattice is a very interesting geometric object whish generalizes Kleins geometric interpretationof the ordinary continued fractions algorithm. A connection between sails and Oppenheim conjecturewas found by Skubenko [143, 144]. (However the main result of the papers [143, 144] is incorrect:Skubenko claimed the solution of Littlewood conjecture, howewer he had a mistake in FundamentalLemma IV in [143].)

Oppenheim conjecture can be reformulated in terms of closure of orbits of lattices (see [143]) andin terms of behaviour of trajectories of certain dynamical systems. There is a lot of literature relatedto Littlewood-like problems in lattice theory and dynamical approach (see [42,72,43,89, 92,90,139,140, 150]). Some open problems in dynamics related to Diophantie approximations are discussed in[55].

1.2 W.M. Schmidts conjecture and Badziahin-Pollington-Velani theorem

For, [0, 1]under the condition += 1 and >0 we consider the sets

BAD(, ; ) =

= (1, 2) [0, 1]2 : inf

pNmax{p||p1||, p||p2||}

andBAD(, ) =

>0

BAD(, ; ).

3


4/42

In [136] Schmidt conjectured that for any 1, 2, 1, 2 [0, 1], 1+ 1 = 2+ 2 = 1the intersection

BAD(1, 1)

BAD(2, 2)

is not empty. Obviously if Schmidts conjecture be wrong then Littlewood conjecture be true. ButSchmidts conjecture was recently proved in a breakthrough paper by Badziahin, Pollington and Velani[4]. They proved a more general result:

Theorem 2. For any finite collection of pairs(j, j), 0 j , j 1, j+j = 1, 1 j r andfor any1 under the condition

infqZ+

q||q1|| >0 (5)the intersection

rj=1

{2 [0, 1] : (1, 2) BAD(j , j)} (6)

has full Hausdorff dimension.

Moreover one can take a certain infinite intersection in (6).

This result was obtained by an original method invented by Badziahin, Pollington and Velani.Authors preprint [107] is devoted to an exposition of the method in the simplest case. The onlypurpose of the paper [107] was to explain the mechanism of the method invented by Badziahin,Pollington and Velani. In this paper it is shown that for 0 < 21622 and1 such that

infqN

q2||q1|| ,

there exists 2 such that for all integers A, B with max(|A|, |B|)>0 one has||A1 B2|| max(A2, B2)

and henceinfqZ+

q1/2 max1j2

||qj|| >0.

This is a quantitative version of a corresponding results from [ 4].However the method works with two-dimensional sets only. Of course we can consider multidimen-

sional BAD-sets. For example for , , [0, 1]under the condition + + = 1and >0 one mayconsider the sets

BAD(, , ; ) =

= (1, 2, 3) [0, 1]3 : inf

pNmax{p||p1||, p||p2||, p||p3||}

and

BAD(, , ) =>0

BAD(, , ; ).

The question if for given (1, 1, 1) = (1, 1, 1) one has

BAD(1, 1, 1)

BAD(2, 2, 2) = remains open.

A positive answer is obtained in a very special cases (see, [1,82, 122]).Recently Badziahin [7] adopted the construction from [4]to Littlewood-like setting and proved the

following result.

4


5/42

Theorem 3. The set

{(1, 2) R2 : inf xZ,x3

x log x log log x ||1x||||2x|| >0}

has Hausdorff dimension equal to 2.Moreover if1 is a badly approximable number (that is (5) is valid) then the set

{2 R : inf xZ,x3 x log x log log x ||1x||||2x|| >0}has Hausdorff dimension equal to 1.

1.3 p-adic version of Littlewood conjecture

Let for a primep consider the p-adic norm | |p, that is ifn = pn1, (n1, p) = 1, Z+then |n|p = p.De Mathan and Teulie conjectured [93] that for any Rone has

liminfq+

q|q|p||q|| = 0.

This conjecture is known as p-adic or mixedLittlewood conjecture. Of course the conjecture is truefor almost all numbers. The conjecture can be reformulated as follows: to prove or to disprove thatfor irrational one has

infn,qZ+

q||pnq|| = 0.It worth noting that de Mathan and Teulie themselves proved[93] that their conjecture is valid forevery quadratic irrational.

As it is shown in [43] from Furstenbergs result discussed behind in 4), Subsection 1.4 it followsthat for distinctp1, p2 one has

infm,n,qZ+

q||pn1pm2q|| = 0

and so lim infq+

q|q|p1|q|p2||q|| = 0 (7)for all. Moreover from Bourgain-Lindenstrauss-Michel-Venkateshs result[9] it follows that for somepositiveone has

lim infq+

q(log log log q)|q|p1|q|p2||q|| = 0.We should note that the inequality (7) is not the main result of the paper [43] by Einsiedler andKleinbock. The main result from [43] establishes the zero Hausdorff dimension of the exceptional setin the problem under consideration.

Many interesting metric results and multidimensional conjectures are discussed in [22].Badziahin and Velani [5] generalized proved an analog of Theorem 2 for mixed Littlewood conjec-

ture:

Theorem 4. The set of reals satisfying

lim infq+

q log q log log q|q|p ||q|| >0

has Hausdorff dimension equal to one.

In this subsection we consider powers of primespn only. Instead of powers of a prime it is possibleto consider other sequences of integers. This leads to various generalizations. Many interesting resultsand conjectures of such a kind are discussed in [15,6]and [58].

5


6/42

1.4 Inhomogeneous problems

Shapira[139] proved recently two important theorems. We put them below.

Theorem 5. Almost all (in the sense of Lebesgue measure) pairs (1, 2) R2 satisfy the followingproperty: for every pair(1, 2) R2 one has

lim infq q||q1 1|| ||q2 2|| = 0.Theorem 6. The conclusion of Theorem 5 is true for numbers 1, 2 which form together with 1 abasis of a totally real algebraic field of degree3.

Here we should note that the third theorem from [139] follows from Khintchines result (see [70])immediately:

Theorem 7. Suppose that reals1 and2 are linearly dependent overZ together with1. Then thereexist reals1, 2 such that

infxZ+

x ||x1 1| | | |x2 2|| >0.

Theorem 7 is discussed in authors paper [110]. Moreover in this paper the author deduces fromKhintchines argument[70] the following

Theorem 8. Let(t) be a function increasing to infinity ast +. Suppose that for anyw 1 wehave the inequality

supx1

(wx)

(x) 0.

The following problem is an open one: is it possible that for a constant function(t) =0 > 0tthe conclusion of Theorem8remains true. If not, it means that a stronger inhomogeneous version ofLittlewood conjecture is valid.

We would like to formulate here one open problem in imhomogeneous approximations due to Harrap[57]. Harrap [57]proved that given , (0, 1), += 1 for a fixed vector

(1, 2) BAD(, ) (8)

the set

{(1, 2)

R2 : inf

q

max(q

||q1

1

||, q

||q2

2

||>0

} (9)

has full Hausdorff dimension. It is possible to prove that this set is an 1/2-winning set in R2 (wediscuss winning properties in Subsection 1.6 below). The following question formulated by Harrap[57]: to prove that the set (9) is a set of full Hausdorff dimension (and even a winning set) withoutthe condition (8). Of course in the case = = 1/2 the positive answer follows from Khintchinesapproach (see results from the paper [108] and the historical discussion there). However in the case(, ) = (1/2.1/2) Harraps question is still open 2.

2 A sketch of a proof for Harraps conjecture is given in a recent preprint [ 114].

6


7/42

1.5 Peres-Schlags method

In [121]Peres and Schlag proved the following result.

Theorem 9. Consider a sequencetn R, n= 1, 2, 3, .. Suppose that for someM 2 one hastj+1

tj

1 + 1

M, j Z+. (10)

Then with a certain absolute constant >0 for any sequence{tj} under the condition (10) there existsreal such that

||tj|| Mlog M

, j Z+. (11)

This result has an interesting history with starts from famous Khintchines paper [70]. We do notwant to go into details about this history and refer to pepers [104,111].

As it was noted by Dubickas [38], an inhomogeneous version of Theorem9is valid: with a certainabsolute constant >0 for any sequence{tj} under the condition (10) and for any sequence of realnumbers{j} there exists real such that

||tj j|| Mlog M

, j Z+. (12)

One can easily see that for any large integer Mthere exists an infinite sequence {tj} such that suchthat for any real there exists infinitely many j with

||tj|| 1M

(13)

(one may start this sequence {tj} with a finite part1, 2, 3,...,Mand then continue by1, M, 2M, 3M,...,MM, e.t.c.). Of cource constant 1 in the numerator of the right hand side may be improved. The open

question is to find the right order of approximation in the homogeneous version of this problem. Ingeneral Peres-Schlags method does not give optimal bounds. The conjecture is that Theorem9maybe improved on, and the optimal result should be stronger than the inequality (11).

There are several results and papers dealing with Peres-Schlags construction (see [ 23, 24,38,101,106, 110,111,121,125]). Here we refer to four such results.

The results in 1) and 2) below are taken from [ 110]. The paper [110] contains some other resultsrelated to Peres-Schlags method.

1) In Littlewood-like setting we got the following result.Letq, q= 1, 2, 3, .. be a sequence of reals. Given positive 214 and a badly approximable real

1 such that||1q|| 1

q q Z+, >1,

there existX0= X0(, ) and a real2 such that

infqX0

qln2 q ||q1| | | |q2 q|| .

If one consider the sequence q = 0the result behind was obtained in [23]. It is worse than BadziahinsTheorem3.

2) In Schmidt-like settind we proved the following statement.

7


8/42

Suppose that, > 0 satisfy+= 1. Letq, q= 1, 2, 3, .. be a sequence of reals. Let be anarbitrary real number. Let >0. Suppose that is small enough. Suppose that for a certain real1and forq X1 one has

||1q|| (ln q)

q1/ .

Then there existX0 = X0( , , X1) and a real2 such that

infqX0

max((qln q) ||q1 ||, (qln q) ||q2 q||) .

Note that if we take = 0, q 0 we get a result from[106] which is much worse that Badziahin-Pollilgton-Velanis Theorem2.

3) For a real we deal with the sequence ||q2||. Peres-Schlags argument gives the followingstatement (see[101]) which solves the simplest problem due to Schmidt [135].

Given a sequence{q} there exists such that for all positive integerqone has

||q2 q|| qlog q

(14)

(hereis a positive absolute constant).Zaharescu[153] proved that for any positive and irrational one has

lim infq+

q2/3||q2|| = 0. (15)

I do not know if this result may be generalized for the value

lim infq+

q2/3||q2 ||

with a real .Nevertheless even in the homogeneous case the lower bound (14) is the best known. So in the

homogeneous case we have a gap between (14) and (15).4) Furstenbergs sequence. Consider integers of the form2m3m, m , n Z+written in the increasing

order:s0= 1< s1 = 2< s2 = 3< s3 = 4< s5= 6< s6= 8< s7= 9< s8= 12< ...

Furstenberg [50](simple proof is given in[8]) proved that the sequence of fractional parts {sq}, q Z+is dense for any irrational . Hence for any one has

liminfq+

||sq || = 0.

Bourgain, Lindenstrauss, Michel and Venkatesh [9] proved a quantitative version of this result. In

particular they show that if saitisfies for some positive the conditioninfqZ+

q||q|| >0

then with some positive and for any one has

liminfq+

(log log log q)||sq || = 0. (16)

Of course Furstenberg had a more general result: instead of the sequence{sq} he considered anarbitrary non-lacunary multiplicative semigroup in Z+. The result (16) deals with this general settingalso.

8


9/42

Peres-Schlags method gives the following result: for an arbitrary sequenceq, q= 1, 2, 3,... thereexists irrational such that

infq2

qlog q||sq q|| >0.

One can see that the results from 1), 2) with q

0 are known to be not optimal. We do notknow if the original Theorem9and the results from 3), 4) with q 0are not optimal. However I amsure that in homogemeous setting the original Theorem9and the results from 3), 4) are not optimaland may be improved on. From the other hand it may happen that the order of approximation in thesetting with arbitrary sequence{q} is optimal for some (and even for all) results from 1), 2), 3), 4)and for lacunary sequences.

Of course Peres-Schlags method gives a thick set (a set of full Hausdorff dimension) ofs for whichthe discussed conclusions hold.

1.6 Winning sets

We give the definition of Schmidts (, )-games and winning sets. Consider, (0, 1), and a setS Rd. Whites an Blacks are playing the following game. Blacks take a closed ball B1 Rd withdiameterl(B1) = 2. Then Whites choose a ballW1 B1with diameterl(W1) =l(B1). Then Blackschoose a ball B2 W1 with diamater l(B2) = l(W1), and so on... In such a way we get a sequenceof nested balls B1 W1 B2 W2 with diameters l(Bi) = 2()i1andl(Wi) = 2()i1(i= 1, 2, . . . ). The set

i=1

Bi =i=1

Wi consists of just one point. We say that Whites win the game ifi=1

Bi S. A set Sis defined to be an (, )-winning set if Whites can win the game for any Blacksway of playing. A set Sis defined to be an-winning set if it is(, )-winning for every (0, 1).

Schmidt [131, 132, 133] proved that for any > 0 an -winning set is a set of full Hausdorff

dimension and that the intersection of a countable family of-winning sets is an-winning set also.For example the set BAD(1/2, 1/2) is an 1/2-winning set (more generally, form Schmidt [132] we

know that the set of badly approximable linear forms is a 1/2-winning set in any dimension). Given Rthe set

{ R : inf qZ+

q||q || >0}

is an1/2-winning set[108] (more generally, in [108] there is a result for systems of linear forms).In 1) - 4) in the previous subsection we discuss the existence of certain real numbers 2 and

. In all of these settings it is possible to show that the sets of corresponding 2 or have fullHausdorff Dimension. Badziahin-Pollington-Velanis Theorem 2, Badziahin-Velanis Theorem 4 formixed Littlewood setting and Badziahins Theorem3gives the sets of full Hausdorff dimension also.However neither in any result from 1) - 4) from the previous subsection, nor in Badziahin-Pollington-Velanis Theorem23 and Badziahins Theorem3we do not know if the sets constructed are winning.

3Very recently Jinpeng An in his wonderful paper[67]showed that under the condition

infqq1/||q1|| >0

the set{2 R : (1, 2) BAD(, )}

is 1/2-winning. So he proved the winning property in Theorem2. The construction from[67] of course gives a betterquanitiative version of the statement from [107]formulated in Section 1.2.

9


10/42

Moreover we do not know if the set BAD(, ) is a winning set in the case (, ) = (1/2, 1/2).The reason is that the property to be a winning set is a local property, but Peres-Schlags agrument

and Badziahin-Pollington-Velanis argument are non-local. The construction of badly approximablesets by Peres-Schlag and Badziahin-Pollington-Velani suppose that at a certain level we have a col-lection of subsegments of a given small segment and we must choose some good subsegments fromthe collection. The methods do not enable one to say something about the location of good segments

form the collection under consideration. All the methods give lower bound for the number of goodsubsegments in the collection. This is enough to establidh the full Hausdorff dimension, but does notenough to prove the winning property.

I would be very interesting to study winning properties of the sets arising from Peres-Schlagsmethod and Badziahin-Pollington-Velanis method.

I do not know any natural example of a set of badly approximable numbers in a natural Dio-phantine problem which has full Haudorff dimension but which is not a winning set in the sence ofSchmidt games.

At the end of this subsection I would like to fomulate a result by Badziahin, Levesley and Velani[6]:

Theorem 10. For, (0, 1), += 1 and primep the set{ R : inf

qZ+qmax(|q|1/p , ||q||1/)> 0} (17)

is1/4-winning set.

The main result from [6] is more general than Theorem 10: it deals not with p-adic norm| |ponly but with a norm associated with an arbitrary bounded sequence of integersD. It is interestingto undestand if the set (17) is1/2-winning.4

2 Best approximations

For positive integers m, nwe consider a real matrix

=

11 m1

1n mn

. (18)

Suppose thatx Zn, x Zm \ {0}. (19)

We consider a norm| |n in Rn and a norm| |m in Rm We are interested mostly in the sup-norm

||ksup= max1jk |j|

or in the Euclidean norm

||k2 = k

j=1

|j|2

In his next paper [68] Jinpeng An proved that the two-dimensional set BAD(, ) is (32

2)1-winning set. Theconstruction due to Jinpeng An seems to be very elegant and important. Probably it can give a solution to the multi-dimensionsl problem. As for the constant (32

2)1, it seems to me that it may be improved to 1/2.

4recently Yaqiao Li [88]proved that the set from Theorem10 is 1/2-winning indeed.

10


11/42

for a vector = (1,...,k) Rk where k is equal to n orm.Let

z= (x,y) Zm+n, x Zm, y Zn, = 1, 2, 3,... (20)be the infinite sequence of best approximation vectors with respect to the norms| |m, | |n . We usethe notation

M= |x|m, = |x y|n .Recall that the definition of the best approximation vector can be formulated as follows: in the set

{z= (x,y) Rm+n : |x|m M,|x y|n }there is no integer points z= (x,y) different from 0, z.

In this section we formulate some open problems related to the sequence of the best approximations.

2.1 Exponents of growth forM

We are interested in the valueG() = lim inf

M1/ . (21)

First of all we recall well-known general lower bounds. In [14] it is shown that for sup-norms inRm and Rn for any matrix under the condition (19) one has

G() 21

3m+n1 .

This is a generalization of Lagarias bound [84]for m = 1.In fact Lagarias result deals with an arbitrary norm: for any norm | |n on Rn and a vector Rn

that has at least one irrational coordinate, the inequality

M+2n+1 2M+1+M

is true for all 1. SoG() 1,n where 1,n is the maximal root of the equation t2n+1 = 2t + 1.

There is another well known statement which is true in any norm. Given a norm ||n in Rn, considerthe contact number K= K(| |

n) This number is defined as the maximal number of unit balls with

respect to the norm| |n without interior common points that can touch another unit ball. Considera vector Rn that has at least one irrational coordinate. Then the inequality

M+K M+1+M

holds for all 1. SoG() 2,n where 2,n is the maximal root of the equation tK =t + 1.

The general problem is to find optimal bounds for the value

inf

G() (22)

for fixed dimensions m, n and for fixed norms| |m, | |n (the infimum here is taken over matrices satisfying (19)). This problem seems to be difficult. The only case where we know the answer is the

case m = n = 1 (of course in this case there is no dependence on the norms). For m = n = 1 thetheory of continued fractions gives

inf

G() =G

1 +

5

2

=

1 +

5

2 .

Here we consider the case m= 1, n= 2 when better bounds are known. In Subsections 2.1.1 and2.1.2 we formulate the best known results for sup-norm and Euclidean norm. Any improvement ofthese bounds may be of interest. Of course any generalizations to larger values of n are of interesttoo. We write g1,2; for the infimum (22) in the case of sup-norm and m= 1, n= 2 and g1,2;2 for theinfimum (22) in the case of the Euclidean norm and m = 1, n= 2

11


12/42

2.1.1 Case m= 1, n= 2, sup-norm

In [97]Moshchevitin improved on Lagarias result form [84] by proving

g1,2;

8 + 133133

111

= 1.28040+, 3=

1 +

5

2 .

2.1.2 Case m= 1, n= 2, the Euclidean norm

Improving on Romanovs result from[126], Ermakov [44]proved that

g1,2;2 1.228043.

Here we should note that this result involves numerical computer calculations.

2.1.3 Brentjes example related to cubic irrationalities

Brentjes[13]considers the following example. Let 4= 1.324+ be the unique real root of the equation

t3 =t + 1.

Consider the lattice consisting of all points of the form

() =

Re

Im

,

where is an algebraic integer from the field Q(4) and is one of its algebraic conjugates. The

triple(1). (4), (

24)

form a basis of the lattice . Brentjes consideres the sequence of the best approximations w , =1, 2, 3, ... But his definition differs from our definition behind. A vector w= (w0, w1, w2) is a bestapproximation (in Brentjes sense) if the only points of the lattice belonging to the cylinder

{= (0, 1, 2) R3 : |0| |w0|,|1|2 + |2|2 |w1|2 + |w2|2}are the points 0, w. Brentjes shows that these best approximations form a periodic sequence andthat

lim

|w|1/ =4 = 1, 324+

(here | | stands for the Euclidean norm in R3). This Brentjes result can be easily obtained by meansof the Dirichlet theorem on algebraic units.

Cusick [33] studied the best approximations for linear form

24x1+ (24 4)x2 y

in the case m = 2, n= 1 and in sup-norm.

In my opinion the following question remains open: for the vector =

424

R2 (we consider

the casem= 1, n= 2) find the value ofG() defined in (21). Is it equal to 4 or not? Probably thesolution should be easy.

Lagarias [84] conjectured that in the case m = 1, n= 2for the value G()defined in (18) we have

infover all norms on R2

inf

G() =4.

12


13/42

2.2 Degeneracy of dimension: m= n = 2

The simplest facts concerning the degeneracy of dimension of subspaces generated by the best approx-imation vectors are discussed in [104].

If

det

11

21

12 22

= 0

and (19) is satisfied then for any0 the set of integer vectors {z, 0} span the whole spaceR4. Sodim span{z, 0} = 4.

For a matrix under the condition (19) the equality

dim span{z, 0} = 3never holds. These facts are proven by Moshchevitin (see Section 2.1 from [104]).

The following question is an opened one. Does there exist a matrix (probably, with zero deter-minant) and satisfying (19) such that for all0 (large enough) one has

dim span{z, 0} = 2 ?

3 Jarnks Diophantine exponents

For a real matrix (18) satisfying (19) we consider function

(t) = minx=(x1,...,xm)Zm:0


14/42

3.1 Jarnks theorems

In [65]Jarnk proved the following theorem.

Theorem 11. Suppose that satisfies (19).(i) Suppose thatm= 1, n 2and the column-matrix consist at least of two lineqrly independent

overZ together with 1 numbers1j , Then

2

1 . (23)

(ii) Suppose thatm= 2. Then ( 1). (24)

(iii) Suppose thatm 3 and (5m2)m1 then

mm1 3. (25)

From the other hand Jarnk proved

Theorem 12. (i) Letm 2. Take realT >2. Then there exists satisfying (19) such that

() =Tm, () =Tm1.

(ii) Letm= 1, n 2. Take realT >2 satisfying

Tn1 > Tn2 +n2k=0

Tk, Tn >1 + 2

n1k=1

Tk.

Then there exists satisfying (19) such that

() = 1 1T 1

T2 ... 1

Tn1.

() =T Tn1 Tn2 ... 1Tn1 +Tn2 +...+ 1

We see that from (i) of Theorem12it follows that for >2m1 there exists such that

() =, () = (()) mm1 .

From (ii) of Theorem12it follows that for m = 1and arbitraryn for any


15/42

Theorem 13. The following statemens are valid for the exponents of two-dimensional Diophantineapproximations.

(i) For a vector-row = (1, 2) R2 such that1, 1 and 1 are linearly independent overZ forthe values

w= (), w= (), v= (), v= () (26)

the following statements are valid:

2 w +, w= 11 w,

v(w 1)v+w

v v w+ 1

w . (27)

(ii) Given four real numbers(w, w, v , v), satisfying (27) there exists a vector-row = (1, 2) R2, such that (26) holds.

This theorem is known as four exponent theorem. It combines together Khintchines thansferenceinequalities [70] for ordinary exponents , and Jarnks equality [64]for uniform exponents , aswell as some new results [19].

From Theorem13it follows that in the case m= 1, n= 2 the inequality (23) is the best possible

and cannot be improved. Also in the case m= 2, n= 1 the inequality (24) is the best possible. Thecases m = 1, n= 2 and m= 2, n= 1 are the only cases when the optimal bounds for in terms of are known. In the next two subsections we will formulate the best known improvements of JarnksTheorem11. However all these improvements are far from optimal. The only possible exception isTheorem14below. The bound of Theorem14may happen to be the optimal one, however I am notsure.

To find optimal bounds for in terms of (even for specific values of dimensions m, n) is aninteresting open problem.

3.3 Case m+n= 4

Moshchevitin proved the following results In the case m= 1, n= 3 he get[112]

Theorem 14. Suppose thatm= 1, n= 3 and the vector = (1, 2, 3) consists of numbers linearlyindependent, together with 1, overZ. Then

2

1 +

1 2

+ 4

1

. (28)

.

The inequality (28) is better than Jarnks inequality (23) for all values of().

In [102,104]the following two theorems are proved (a proof of Theorem16was just sketched).

Theorem 15. Suppose thatm= 3, n= 1 and the matrix = (1, 2, 3) consists of numbers linearlyindependent overZ together with1. Then

+ 1

27

4+

1

1

2

. (29)

The inequality (29) is better than Jarnks inequality (25) for all values of().

15


16/42

Theorem 16. Consider four real numbersij , i , j = 1, 2 linearly independent overZ together with 1.Letm= n = 2 and consider the matrix

=

11

21

12 22

satisfying (19). Then

1 +

(1 )2 + 4(22 2+ 1)2

. (30)

The inequality (30) improves on the inequality (24) for ()

1,1+

5

2

2.5

It may happen that Theorem 14 gives the optimal bound.6 I am sure that the inequality fromTheorem16may be improved.

3.4 A result by W.M. Schmidt and L. Summerer (2011)

Very recently Schmidt and Summerer [138] improved on Jarnks Theorem11in the cases m= 1and

n= 1.Form = 1 and arbitrary n 2they obtained the bound

2 + (n 2)(n 1)(1 ) . (31)

As for the dual setting with n = 1 andm 2 they proved that

(m 1) 2

1 + (m 2) . (32)

No analogous inequalities are known in the case when both n andmare greater than one.The result by Schmidt and Summerer deals with successive minima for one-parametric families of

lattices and relies on their earlier research [137] and Mahlers theory of compound and pseudocompoundbodies [91]. We suppose to write a separate paper concerning Schmidt-Summerers result and itspossibble extensions, jointly with O. German.

Here we should note that in the cases m = 1, n = 3 and m = 3, n = 1 the inequalities (28) and(29) are better than (31) and (32), correspondingly.

3.5 Special matrices

Here we would like to formulate one open problem which seems to be not too difficult. Consider

a special set Wof matrices . A matrix belongs to W if (19) holds and moreover there existsinfinitely many(m+n)-tuples ofconsecutivebest approximation vectors

z, z+1, z+2,...,z+m+n1

consisting oflinearly independent vectors in Rm+n. The definitoion ofzj Zm+n (see (20)) is givenin the very beginning of Section 2.

5Recently the author[115] improved Theorem16. Now the result is better than (24) for all admissible values of6Very recently the author proved that Theorem14 gives the optimal bound.

16


17/42

I think that it is not dificult to improve on all the inequalities from Jarnks Theorem 11in thecase W.7

Moreower I think that in this case it is possible to get optimal inequalities and to prove theoptimality of these inequalities by constructing special matrices W.

4 Positive integers

In this section we consider collections of real numbers = (1,...,m), m 2. (The indexn = 1 isomitted here.) We are interested in small values of the linear form

||1x1+... +nxn||

in positive integers x1,...,xn. Put

+(t) =+;(t) = minx1,...,xmZ+, 00;2. ||1x1(i) +2x2(i)|| (max{x1(i), x2(i)}) 0 as i +.

7In fact, it can be easily seen that for Wone has() G (),

whereG is the largest root of the equation

n1j=1

xj + 1 +

m1j=1

xj = 0.

For m + 1, n= 1the value ofG = 1 coinsides with the analogous value from (23) from Jarnks Theorem11. In the

casem = 1, n= 3 the value ofG = 12

1 +

1

2+ 4

1

coinsides with those from authors Theorem14.

17


18/42


19/42

4.2 A counterexample to W.M. Schmidts conjecture

In the paper [134] W.M. Schmidt wrote that he did not know if the exponent in Theorem17maybe replaced by a lagrer constant. At that time he was not able even to rule a possibility that thereexists an infinite sequence (x1(i), x2(i)) Z2 with condition 1. and such that

||1x1(i) +2x2(i)|| (max{x1(i), x2(i)})2 c() (40)with some large positive c(). Later in [136] he conjectured that the exponent may be replaced byany exponent of the form 2 , > 0 and wrote that probably such a result should be obtained byanalytical tools. It happened that this conjecture is not true. In [109]the author proved the followingresult.

Theorem 18. Let = 1.94696+ be the largest real root of the equationx4 2x2 4x+ 1 = 0. Thereexist real numbers1, 2 such that they are linearly independent overZ together with 1 and for everyinteger vector(x1, x2) Z2 withx1, x2 0 andmax(x1, x2) 2200 one has

||1x1+2x2|| 12300(max(x1, x2))

.

Theorem18shows that W.M. Schmidts conjectue discussed in previous subsection turned out tobe false.Here we should note that for the numbers constucted in Theorem 18one has

=(+ 1)2(2 1)

4 = 3.1103+, =

(+ 1)2

2 = 2.2302+.

So (, ) A2 and the inequality (38) gives

+ + 2

2 1= 1.413+.

However from the proof of Theorem18(see [109]) it is clear that for the numbers constructed one has+= = 1.94696

+.

4.3 m= 2: large domains

The original paper [134] contained 5 remarks related to Theorem 17. One of these remarks wasas follows. The condition 1. in Theorem17may be replaced by a condition|1,1,x1(i) +1,2x2(i)| 1, 0 and1< t r 2 satisfy the condition

(1 )((t2r tr t r 1) +t2) + (1 )(t2 1) 0. (42)Then there exist infinitely many integer points(x1, x2) such that

(x1, x2) (, 0) and ||1x1+2x2|| c1(max(|x1|, |x2|))r,or

(x1, x2) (1, ) and ||1x1+2x2|| c2(max(|x1|, |x2|))t.Herec1,2 are positive constants depending on and, .

19


20/42

Thurnheer [147] considered three special cases of his Theorem 19:1. by putting

= 7/4, = 0 (43)

and t = r = 2one can see that there exist infinitely many integer vectors (x1, x2)such that

(x1, x2) (7/4, 0) and ||1x1+2x2|| c3(max(|x1|, |x2|))2 (44)

(with a certain value ofc3 > 0);2. by putting

1< 7/4, =7 4

4 (45)

and t = r = 2one can see that there exist infinitely many integer vectors (x1, x2)such that

(x1, x2) (, ) and ||1x1+2x2|| c4(max(|x1|, |x2|))2 (46)

(with a certain value ofc4 > 0);3. for any

(1, 7/4]and = 0 one can consider the largest root s() of the equation

x3 2( 1)x2 2x 1 = 0. (47)

Then one can see that there exist infinitely many integer vectors (x1, x2) such that

(x1, x2) (, 0) and ||1x1+2x2|| c5(max(|x1|, |x2|))s() (48)

(with a certain value ofc5 > 0); note thats(1) = = 1+

5

2 , and this gives Schmidts bound (34).

4.4 Large dimension (m >2)

4.4.1 A remark related to Davenport-Schmidts result

Another remark to Theorem 1 from[134]tells us that no great improvement is affected by allowinga large number of variables. W.M. Schmidt showed the following result to be true.

Theorem 20. There exists a vector = (1, . . . , m), m 3 such that:1, 1,...,m are linearly independent overZ; for any positive there exists a positivec() such that

1x1+ +mxm > c()

max1im

|xi|2

for all integersx1, . . . , xk under the conditionxi > 0, i= 1, . . . , k.To prove Theorem 20 one should use a result by H. Davenport and W.M. Schmidt from the

paper [34]. This result is based on existence of very singular vectors. Theorem 20 shows that form 3 for linearly independent collection it may happen that

+() 2. (49)

20


21/42

4.4.2 General Thurnheers lower bounds

Here we formulate three general results by Thurnheer from [148]. Its particular case (inequality (37))was discussed above. Thurnheer used the Euclidean norm to formulate his result. Of course it is notof importance and we may use sup-norm.

Given > 0 consider the domain

= =

(x1,...,xm) Rm : |xm| max1jm1 |xj |

.

Theorem 21. Suppose that1, 1,...,m are linearly independent overZ. Put

v(m) =1

2

m 1 + m2 + 2m 3

> m 1

m.

Then there exists infinitely many integer vectors(x1,..,xm) such that

||1x1+...+mxm|| ( max1jm1

|xj|)v(m)

(here is an arbitrary small fixed positive number).

Another Thurnheers result deals with aproximations from a larger domain. Forw >0 put

(w) =

(x1,...,xm) Rm : |xm| (1 +)

m1j=1

|xj|2w/2

(x1,...,xm) Rm :m1j=1

|xj|2 1

, >0.

Theorem 22. Put

w= w(m) = 1 + 1

m+

1

m2.

Then for any real and and for any positivethere exists infinitely many integer vectors(x1, . . ,xm) (w) such that

||1x1+...+mxm|| (1 +)( max1jm1

|xj |)m.

Another result deals with a lower bound in terms of.

Theorem 23. Suppose that1, 1,...,m are linearly independent overZ. Suppose that

1m

< = () 1m 1 . (50)

Put

u0(m, ) =

1

2m

(m 1)2 + 1 +

((m 1)2 + 1)2 + 4m2(m 1)()2

Then for anyu < u0(m,

) there exists infinitely many integer vectors(x1,..,xm) such that

||1x1+... +mxm|| ( max1jm1

|xj |)u.

21


22/42

4.4.3 From Thurnheer to Bugeaud and Kristensen

Bugeaud and Kristensen [17]considered the following Diophantine exponents. Let1 l m. Considerthe set

(m, l) = (m, l) =

(x1,...,xm Rm : max

l+1jm|xj| max

1jl|xj|

.

Diophantine exponent m,l = m,l() is defined as the supremum over all such that the inequality

||1x1+... +mxm|| ( max1jm

|xj|)

has infinitely many solutions in(x1,...,xm) (m, l) Zm.

So Diophantine exponent m,1 corresponds just to +. Thurnheers Theorems 21 and 23 have thefollowing interpretation in terms ofm,m1.

Suppose that1, 1,...,m are linearly independent over Z. Then

m,m1 v(m). (51)

If in addition (50) holds thenm,m1 u0(m, ). (52)

Bugeaud and Kristensen formulate the following result.

Theorem 24. Suppose that1, 1,...,m are linearly independent overZ. Then

m,l l

m +land

m,m1

1 +

.

In fact linearly indenendency condition here is necessary.Of course from this theorem the bound (51) follows immediatelly.Here we should note that the main results of the paper[17] deal with metric prorerties of exponents

m,l. Also in [17]several interesting problems are formulated.

4.5 Open questions

1. What is the optimal exponent inf1,2 independent +() in the problem for a linear form in twopositive variables? Is it or or something else between and?8

2. What are the best possible lower bounds for + and + in terms of , and ? Anyimprovement of any of the lower bounds (35, 37,39) will be of interest, in my opinion. Of course anyimprovement of lower bounds for m.l given in (51,52) as well as of the bounds from Theorem 24 willbe of interest.

3. As it was shown behind for 1,...,m, m 3 linearly independent over Ztogether with 1 it mayhappen (49). But in view of Theorem18I may conjecture that for m = 3(or even for an arbitrary m)there exist a collection of linearly independent numbers 1, 1,...,m such that+()< 2.

4. What are optimal exponents in the Thurnheers setting for large domains, ? In particular,are the values of parameters , from (43) and (45) optimal to get (44) and (46) or not? What is

8 Recently Damien Roy[127] anounced that the exponent = 1+5

2 = 1.618+ from Schmidts Theorem17is optimal.

22


23/42

the optimal values of s() to conclude that (48) has infinitely many solutions in integers (x1, x2)?Any improvements of the discussed results is of interest. Similar questions may be formulated formulti-dimensional results.

In view of our Theorem18I think that it is possible to solve some of the problems formulated inthis subsection.

5 Zaremba conjecture

For an irreducible rational fraction aq Q we consider its continued fraction expansiona

q = [b0; b1, . . . , bs] =

=b0+ 1

b1+ 1

b2+ 1

b3+ + 1

bs

, bj =bj(a) Z+, j 1. (53)

The famous Zarembas conjecture [154] supposes that there exists an absolute constant k with thefollowing property: for any positive integerq there exists a coprime to qsuch that in the continuedfraction expansion (53) all partial quotients are bounded:

bj(a) k, 1 j s= s(a).

In fact Zaremba conjectured that k = 5. Probably for large prime qeven k = 2sould be enough, as itwas conjectured by Hensley .

5.1 What happens for almost alla (mod q)?N.M. Korobov [80] showed that for prime qthere exists a,(a, q) = 1 such that

max

b(a) log q.

Such a result is true for composite q also. Moreover Rukavishnikova[128] proved

Theorem 25.

1

(q)#a Z : 1 a q, (a, q) = 1, max1js(a) bj(a) T

log q

T .

Here we would like to note that the main results of Rukavishnikovas papers [128,129] deal withthe typical values of the sum of partial quotients of fractions with a given denominator: she proves ananalog of the law of large numbers.

5.2 Exploring folding lemma

Niederreiter [117] proved that Zarembas conjecture is true for q= 2, 3, Z+ with k = 4, and forq= 5 with k = 5. His main argument was as follows. If the conjecture is true for qthen it is true for

23


24/42

Bq2 with bounded integer B. The construction is very simple. Consider continued fraction (53) withb0(a) = 0 and its denominator written as a continuant:

q= b1, b2, . . . , bs.

Definea byaa

1 (mod q)

(the signshould be chosen here with respect to the parity ofs). Thena

q = [0; bs(a), . . . , b2(a), b1(a)] =

b1, . . . , bs1b1, . . . , bs

andq= b1, . . . , bs1 = c1, c2, . . . , cs, cj =bsj .

At the same time ifc1 2then

q a

q

= [0; 1, c1

1, . . . , cs] = c1 1, c2, . . . , cs1, c1 1, c2, . . . , cs

So we see thatb1, . . . , bs1, bs, X, 1, c1 1, c2, . . . , cs =

= b1, . . . , bs1bsX, 1, c1 1, c2, . . . , cs + b1, . . . , bs11, c1 1, c2 . . . , cs =

= b1. . . . , bs1, c1 1, c2, . . . , cs

X+ c1 1, c2, . . . , cs1, c1 1, c2, . . . , cs +

b1, . . . , bs1b1, . . . , bs

=

= b1. . . . , bs1, c1 1, c2 . . . , cs (X+ 1) .This procedure is known as folding lemma.

By means of folding lemma Yodphotong and Laohakosol showed [152] that Zarembas conjectureis true for q= 6 and k = 6. Komatsu[75]proved that Zarembas conjecture is true for q= 7r2r, r =

1, 3, 5, 7, 9, 11and k = 4. Kan and Krotkova [69] obtained different lower bounds for the number

f= #{a (mod pm) : a/pm = [0; b1, . . . . bs], bj pn}

of fractions with bounded partial quotients and the denominator of the form pn. In particular theyproved a bound of the form

f C(n)m, C(n), >0.

Another applications of folding lemma one can find for example in [16,32,77,78] and in the papersrefered there. I think that A.N. Korobov proved Niederreiters result concernind powers of 2 and 3

independently in his PhD thesis [77].

5.3 A result by J. Bourgain and A. Kontorovich (2011)

Recently J. Bourgain and A. Kontorovich[10,11] achieved essential progress in Zarembas conjecture.Consider the set

Zk(N) := {q N : a such that (a, q) = 1, a/q = [0; b1,...,bs], bj k}

( so Zarembas conjecture means thatZk(N) = {1, 2,...,N}). In a wonderful paper[10] they proved

24


25/42

Theorem 26. Fork large enough there exists positivec= c(k) such that forN lagre enough one has

#Zk(N) =N O(N1c/ log logN).

For example it follows from Theorem26that fork large enough the set nZk(N)contains infinitelymany prime numbers.

Another result from [10]is as follows.

Theorem 27. Fork= 50 the setNZ50(N) has positive proportion inZ+, that is

#Z50(N) N.

These wondeful results follow from right order upper bound for the integral

IN=

10

|SN()|2d, (54)

where

SN() =qNN(q)e

2iq

,

and N(q) = Nk(q) is the number of integers a, (a, q) = 1 such that all the partial quotients in thecontinued fraction expansion for aq are bounded by k

For example positive proportion result follows from the bound

IN SN(0)2

N

by the Cauchy-Schwarz inequality.The procedure of estimating of the integral comes from Vinorgadovs method on estimating of

exponential sums with polynomials (Weyl sums). The main ingredient of the proof (Lemma 7.1 from[10]) needs spectral theory of automorphic forms and follow from a result by Bourgain, Kontorovichand Sarnak from [12].

I think that it is possible to simplify the proof given by Bourgain and Kontorovich and to avoid theapplication of a difficult result from [12]. Probably for a certain positive proportion result A. Weilsestimates on Kloosterman sums should be enough.9

5.4 Real numbers with bounded partial quotients

In this subsection we formulate some well-known results concerning real numbers with bounded partialquotients. We deal with Cantor type sets

Fk = { [0, 1] : = [0; b1, b2, . . . ], bj k}.

For the Hausdorff dimension dim Fk Hensley [61]proved

rk = dim Fk = 1 62

1

k72

4log k

k2 +O

1

k2

, k

9It was actually done recently by Igor Kan and Dmitrii Frolenkov [47,48,49]. Moreover Kan and Frolenkov improvedon some results by Bourgain and Kontorovich on Zaremba conjecture.

25


26/42

Exlicit estimates for dim Fk for certain values of k one can find in[66]. Another result by Hensley[59,60] is as follows. For the sums of the values

Nk(q) = #{a (mod q) : a/q= [0; b1, . . . . bs], bj k}considered in the previous subsection Hensley proved

qQ

Nk(q) constant Q2r

k , Q +. (55)

We need a corollary to (55). Consider the set

B(k, T) = {a (mod q) : a/q= [0; b1, . . . . b, . . . , bs],b1,...,b T = bj k}.Then forT qone has

#B(k, T) k qrk .

5.5 Zarembas conjecture and points on modular hyperbola

When we are speaking about modular hyperbola we are interested in the distribution of the pointsfrom the set

{(x1, x2) Z2q : x1x2 (modq)}.For a wonderful survey we would like to refer to Shparlinski [141].

Proposition. Suppose that forT C

q there existx1, x2 (modq) such that

x1, x2 B(k, T),and

x1x2 1 (modq).Then Zarembas conjecture is is true with a certaink depending onk andC.

From A. Weils bound for complete Kloosterman sums we know that the points on modular hypebolaare uniformily distributed in boxes of the form

I1 I2, I= [X, X+Y]provided

Y1 Y2 q3/2+.So we have the following

Corollary. Suppose that

1+2 0. (66)

That is why we have

2nj=1

j,n j

2n

2= 2n

10

(?(x) x)2dx +Rn, |Rn| 4. (67)

The question is as follows. Is it true that for the remainder in (67) one hasRn 0 asn ?This question is motivated by the famous Franel theorem. Instead of Stern-Brocot sequence Fn

consider Farey series FQwhich consist of all rational numbersp/q [0, 1], (p, q) = 1with denominators Q. Suppose thatFQ form an increasing sequence

1 =ro,Q < r1,Q < ... < rj,Q < rj+1,Q < ... < r(Q),Q = 1, (Q) =qQ

(q)

(here()is the Euler totient function). Then for the limit distribution function one has

limQ

#{r FQ: r x}(Q) + 1

=x

and the integral similar to (66) is equal to zero. Franels theorem (see [85]) states that the asymptoticformula

(Q)

j=1rj,Q

j

(Q)2

=O(Q1+), Q .

for all positive is equivalent to Riemann Hypothesis. In fact the well-known asymptotic equalitynQ

(n) = o(Q), Q

leads to(Q)j=1

rj,Q j

(Q)

2=o(1), Q .

A analogous formula forRn from (67) is unknown, probably.

31


32/42

6.4 Values of derivative

It is a well-known fact that if for x [0, 1] the derivative ?(x) exists then ?(x) = 0 or ?(x) = +.For a real irrational xrepresented as a condinued fraction expansionx= [a0; a1,...,an, ..]we consider

the sum of its first partial quotients Sx(t) =a1+... +at. Define

1 =2log 1+

5

2

log2 = 1.388+, 2=

4L5 5L4L5 L4

= 4.401+, Lj = logj+j

2 + 4

2 j

log2

2 .

Improving on results by Paradis, Viader and Bibiloni [119], Kan, Dushistova and Moshchevitin [40]proved the following four theorems.

Theorem 31. (i) Assume for an irrational numberx there exists such a constantCthat for all naturalt one has

Sx(t) 1t+ log t

log2+C.

Then?(x) exists and?(x) = +.(ii) Let(t) be an increasing function such that limt+ (t) = +. Then there exists such an

irrational numberx

(0, 1) that?(x) does not exist and for anyt one has

Sx(t) 1t +log t

log2+ (t).

Theorem 32. Let for an irrational numberx (0, 1) the derivative?(x) exists and?(x) = 0. Thenfor any real function= (t) under conditions

(t) 0, (t) = o

log log t

log t

, t

there existsT depending on such that for allt T one has

maxut

(Sx(u)

1u)

2log 1 log2

log2 t log t

(1

t(t)).

(ii) There exists such an irrationalx (0, 1) that?(x) = 0 and for all large enought

Sx(t) 1t

16log 1 8log2log2

t log t

1 + 25

log log t

log t

.

Theorem 33. (i) Assume for an irrational numberx there exists such a constantCthat for all naturalt one has

Sx(t) 2t C.Then?(x) exists and?(x) = 0.

(ii) Let(t) be an increasing function such that limt+ (t) = +. Then there exists such anirrational numberx (0, 1) that?(x) does not exist and for anyt we have

Sx(t) 2t (t).Theorem 34. (i) Assume for an irrational numberx(0, 1) the derivative?(x) exists and?(x) =+. Then for any large enought one has

maxut

(2u Sx(u))

t

108.

(ii) There exists such an irrationalx (0, 1) that?(x) = + and for large enought we have2t Sx(t) 200

t.

32


33/42

A weaker result is due to Dushistova and Moshchevitin [39]. All these results are related to deepanalysis of sets of values of continuants. Theorems31 and33 are optimal, but Theorms32 and34 arenot optimal. It should be interesting to prove optimal bounds related to this two last theorems.

Here we should note that the paper [40]contains some other results related to sets of real numberswith bounded partial quotients. Probably some of these results may be improved.

It is possible to prove that for any from the interval

1 2

there exist irrationals x, y,z [0, 1]such that

limt

Sx(t)

t = lim

tSy(t)

t = lim

tSz(t)

t =

and ?(x) = 0, ?(y) = +, but?(z) does not exist.Theorems31- 34 show that

1= sup

R : lim sup

t

St(x)

t < = ?(x) = +

, (68)

2= inf

R : lim inf

tSt(x)

t > = ?(x) = 0

. (69)

6.5 Denjoy-Tichy-Uitz family of functions

There are various generalizations of the Minkowski question mark function ?(x). One of them wasconsidered by Denjoy [35] and rediscovered by Tichy and Uitz[146].

For (0, 1) we define for x [0, 1]a functiong(x) in the following way. Putg(0) = 0, g(1) = 1.

Then ifg is defined for two neighbooring Farey fractions ab < cd , we put

g

a +c

b +d

= (1 )g

ab

+g

cd

.

So we define g(x) for all rational x [0, 1]. For irrationalx we define g(x) by continuouty.The family{g} constructed consist of sungular functions. One can easily see thatg1/2(x) =?(x),

that is the Minkowski question mark function is a member of this family. Hence g1/2 has a cleararithmetic nature: it is the limit distribution function for Stern-Brocot sequencesFn. Recall that

Fn = {x= [0; a1,...,at] : a1+... +at = n+ 1}.Zhabitskaya [155] find out that for = 352 the function g(x) is the limit distribution function

for the sequencesn= {x= [[1; b1,...,bl]] : b1+...+bl = n+ 1}

associated with semiregular continued fractions

[[1; b1, b2, b3,...,bl]] = 1 1b1 1

b2 1b3 1

bl

, bj Z, bj 2.

33


34/42

It happens that disrtibution functions of some other sequences associated with special continuedfractions do not belong to the family{g} (see[156, 157]).

It is interesting to find other values of for which the function g(x)is associated with an explicitand natural object.

Another open problem related to the family{g} is as follows. Analogously to (68,69) we define

1() = sup

R : lim suptSt(x)

t < = g(x) = + ,2() = inf

R : lim inf

tSt(x)

t > = g(x) = 0

.

The study of the functionsj(), 0<


35/42

[12] J. Bourgain, A. Kontorovich, P. Sarnak, Sector estimates for hyperbolic isometries. GAFA,20(5):1175 - 1200, (2010).

[13] A.J. Brentjes, Mulltidimensional continued fraction algorithhms, Mathmatical Centre Tracts,145, Amsterdam, 1981.

[14] Y. Bugeaud, M. Laurent, On exponents of homogeneous and inhomogeneous Diophantine ap-

proximation, Moscow Math. J., vol. 5, no. 4 (2005), 747766.

[15] Y. Bugeaud, M. Drmota, B. de Mathan, On a mixed Littlewood conjecture in Diophantineapproximation. Acta Arith. 128 (2007), 107-124.

[16] Y. Bugeaud, Diophantine approximation and Cantor sets. Math. Ann. 341 (2008), 677-684.

[17] Y. Bugeaud, S. Kristensen, Diophantine exponents for mildly restricted approximation, Arkiv. f.Mat. 47 (2009), 243 - 266.

[18] Y. Bugeaud, Multiplicative Diophantine approximation, Semminaires & Congres, 20, 2009, p.107 - 127

[19] Y. Bugeaud, M. Laurent, On transferrence inequalities in Diophantine approximations, II. Math-ematische Zeitschrift 265:2 (2010) 249-262.

[20] Y. Bugeaud, R. Broderick, L. Fishman, D. Kleinbock, B.Weiss, Schmidt games, fractals, andnumbers normal to no base, Math. Res. Lett. 17 (2010), 307321.

[21] Y. Bugeaud, S. Harrap, S. Kristensen and S. Velani, On shrinking targets for Z actions on tori,Mathematika 56 (2010), 193-202.

[22] Y. Bugeaud, A. Haynes, S, Velani, Metric consideration conserning the mixed Littlewood con-jecture, preprint available atarXiv:0909.3923v2 (2011).

[23] Y. Bugeaud, N. Moshchevitin, Badly approximable numbers and Littlewood-type problems.Math. Proc. Cambridge Phil. Soc. 150 (2011), 215 - 226.

[24] Y. Bugeaud, N. Moshchevitin, On fractional parts of powers of real numbers close to 1, Mathe-matische Zeitschrift, DOI: 10.1007/s00209-011-0881-z, (to appear).

[25] V. A. Bykovskii, On the error of number-theoretic quadrature formulas, Dokl. Math. 67, No. 2,175-176 (2003).

[26] V. A. Bykovskii, The discrepancy of the Korobov lattice points. Izvestiya. RAN. Ser. Matem. (toappear, in Russian).

[27] V.A. Bykovskii, The discrepancy of the Korobov lattice points. Program and Abstrct Book p.10-11. 27th Journees Arithmetiques conference.

[28] J. W. S. Cassels and H. P. F. Swinnerton-Dyer, On the product of three homogeneous linearforms and indefinite ternary quadratic forms, Philos. Trans. Roy. Soc. London, Ser. A, 248 (1955),7396.

[29] J. W. S. Cassels, An introduction to Diophantine approximation, Cambridge University Press,1957.

35
http://arxiv.org/abs/0909.3923http://arxiv.org/abs/0909.3923


36/42

[30] J. W. S. Cassels, An introduction to the Geometry of Numbers, Springer-Verlag, 1959.

[31] M.-C. Chang, Partial quotients and equidistribution, Comptes Rendus Mathematique. 349:13-14(2011), 713 - 718.

[32] H. Cohn, Symmetry and specializability in continued fractions, Acta Arithmetica, 57:4 (1996)297 - 320.

[33] T.W. Cusick, Best Diophantine approximations for ternary linear forms, J. Reine Angew. Math.,315 (1980), 40 - 52.

[34] H. Davenport and W. M. Schmidt, A theorem on linear forms, Acta Arith. 14 (1968), 209 - 223.

[35] A. Denjoy, Sur une fonction reele de Minkowski, J. Math. Pures Appl. 17 (1938), pp. 105-151.

[36] M. Drmota, R. F. Tichy, Sequnces, discrepancies and applications, Lecture notes in mathematics,1651, 1997.

[37] A. Dubickas On the fractional parts of lacunary sequences, Mathematica Scand. 99 (2006),

136-146.

[38] A. Dubickas An approximation by lacunary sequence of vectors, Probability and Computing,17.3 (2008), 339 - 345.

[39] A.A. Dushistova, N.G. Moshchevitin, On the derivativ of Minkowsli function ?(x), Fundamen-talnaya i prikladnaya matematika, 16:6 (2010), 33 - 44 (in Russian).

[40] A. A. Dushistova, I. D. Kan and N. G. Moshchevitin, Differentiability of the Minkovski questionmark function. To appear in Journal of Mathematical Analysis ansd Applications, preprint availableatarXiv:0903.5537v1 (2009).

[41] M. Einsiedler, J. Tseng, Badly approximable systems of affine forms, fractals, and Schmidt games,Journal fur die Reine und Angewandte Mathematik, 660 (2011) 83 - 97.

[42] M. Einsiedler, A. Katok, and E. Lindenstrauss, Invariant measures and the set of exceptions tothe Littlewood conjecture, Ann. of Math. 164 (2006), 513 - 560.

[43] M. Einsiedler and D. Kleinbock, Measure rigidity and p-adic Littlewood-type problems, Com-positio Math. 143 (2007), 689702.

[44] E. Ermakov, Simultaneous two-dimensional best Diophantine approximations in the Euclideannorm, preprint avaliable atarXiv:1002.2713v1

[45] L. Fishman, Schmitd games, badly approximable matrices and fractals, Journal Number theory129, 9 (2009), 2133-3153.

[46] D. Frolenkov, A numerically explicit version of the Polya-Vinogradov inequality, Moscow Journalof Combinatorics and Number Theory, 1:3 (2011), 234250.

[47] D. Frolenkov, I. Kan, A reinforcement of the Bourgain-Kontorovichs theorem by elementarymethods, preprint availabe atarXiv:1207.4546(2012).

[48] D. Frolenkov, I. Kan, A reinforcement of the Bourgain-Kontorovichs theorem, preprint availabeatarXiv:1207.5168(2012).

36
http://arxiv.org/abs/0903.5537http://arxiv.org/abs/1002.2713http://arxiv.org/abs/1207.4546http://arxiv.org/abs/1207.5168http://arxiv.org/abs/1207.5168http://arxiv.org/abs/1207.4546http://arxiv.org/abs/1002.2713http://arxiv.org/abs/0903.5537


37/42

[49] D. Frolenkov, I. Kan, A note on the reinforcement of the Bourgain-Kontorovichs theorem,preprint availabe atarXiv:1210.4204(2012).

[50] H. Furstenberg, Disjointness in ergodic theory, minimal sets, and a problem in Diophantineapproximation, Math. Systems Theory 1 (1967), 1 49.

[51] P. Gallagher, Metric simultaneous Diophantine aproximations, J. London Math. Soc. 37 (1962),

387 390.

[52] O.N. German, Sails and norm minima of lattices, Sbornik Mathematics, 196:3 (2005), 337 - 365.

[53] O.N. German, Klein polyhedra and norm minima of lattices, Doklady Mathematics, 406:3 (2006),38 - 41.

[54] O.N. German, Klein polyhedra and lattices with positive norm minima, J. Number TheoryBordeaux 19 (2007), 175 - 190.

[55] A. Gorodnik, Open problems in dynamics and related fields, Journal of Modern Dynamics, 1(2007), no. 1, 1 - 35.

[56] P.M. Gruber, C.G. Lekkerkerker, Geometry of Numbers, North-Holand Mathematical Library,V. 37, 1987.

[57] S. Harrap, Twisted inhomogeneous Diophantine approximation and badly approximable sets,Acta Arithmetica, 151 (2012), 55 - 82.

[58] S. Harrap, A. Haynes, The mixed Littlewood conjecture for pseudo-absolute values, preprintavailable atarXiv:1012.0191v2 (2012).

[59] D. Hensley, The distribution of badly approximable numbers and continuants with boundeddigits, Theorie des nombres (Quebec, PQ, 1987), 371 385, de Gruyter, Berlin, 1989.

[60] D. Hensley, The distribution of badly approximable rationals and continuants with boundeddigits. II, J. Number Theory 34:3 (1990), 293 334.

[61] D. Hensley, Continued fraction Cantor sets, Hausdorff dimension and functional analysis, J.Number Theory 40:3 (1992), 336 358.

[62] E. Hlawka, Zur angen herten Berechnung mehrfacher Integrale, Monatshefte fur Math., 66:2,(1962), 140 - 151.

[63] V. Jarnk, Zur metrischen Theorie der diophantischen Approximationen. Prace Mat.-Fiz. 36(1928/29), 91 - 106.

[64] V. Jarnk, Zum Khintcineschen "Ubertragungssats, Travaux de lInstitut Mathematique deTbilissi, 3, 193 - 216 (1938).

[65] V. Jarnk, Contribution a la theorie des approximations diophantiennes lineaires et homogenes,Czechoslovak Math. J. 4 (1954), 330 - 353 (in Russian, French summary).

[66] O. Jenkinson, On the density of Hausdorff dimensions of bounded type continued fraction sets:the Texan conjecture, Stochastics and Dynamics, 4 (2004), 63-76.

37
http://arxiv.org/abs/1210.4204http://arxiv.org/abs/1012.0191http://arxiv.org/abs/1012.0191http://arxiv.org/abs/1210.4204


38/42

[67] Jinpeng An, Badziahin-Pollington-Velanis theorem and Schmidts game, preprint available atarXiv:1203.2998v1.

[68] Jinpeng An, Two-dimensional badly approximable vectors and Schmidts game, preprint availableatarXiv:1204.3610v1 (2012).

[69] I.D. Kan, N.A. Krotkova, Quantitative generalizations of Niederreiters result concerning con-

tinuants, Chebyshevskii Sbornik 12: 1(37), (2011), 105 - 124 (in Russian); preprint available atarXiv:1109.1633v1 (2011,in Russian, summary in English).

[70] A.Y. Khinchine, Uber eine klasse linear Diophantine Approximationen, Rendiconti Circ. Math.Palermo, 1926, 50, p.170 - 195.

[71] J.R. Kinney, Note on a singular function of Minkowski, Proc. Amer. Math. Soc. 11 (1960), p.788 - 789.

[72] D.Y. Kleinbock and G.A. Margulis, Flows on homogeneous spaces and Diophantine approxima-tion on manifolds, Ann. of Math. (2) 148:1 (1998), 339 - 360,

[73] D. Kleinbock, B. Weiss, Modified Schmidt games and a conjecture of Margulis, preprint availableat arXiv:arXiv:1001.5017v2 (2010)

[74] J. F. Koksma, Diophantische Approximationen, Ergeb. Math. Grenzgeb., vol. 4, Springer, Berlin1936,

[75] T. Komatsu, On a Zarembas Conjecture For Powers, Saraevo Journal of Mathematics, Vol.1(13).2005, 9-13.

[76] S. Konyagin, Arithmetic properties of integers with missing digits: distribution in residue classes,Periodica Mathematica Hungaria., 2001, v. 42 (1 - 2), p. 145 - 162.

[77] A.N. Korobov, PhD thesis, Moscow Lomonosov State University, 1990 (in Russian).

[78] A.N. Korobov, Continued fractions of certain normal numbers, Mathematical Notes, 47:2 (1990)128 - 132,

[79] N.M. Korobov, Approximate evolution of repeated integrals. Dokl. Akad. Nauk SSSR 124, 1207- 1210 (1959, in Russian).

[80] N.M. Korobov, Number-theoretical methods in numerical analysis, Moscow, 1963 (in Russian).

[81] N.M. Korobov, Some problems in the theory of Diophantine approximation, Russian Mathemat-ical Surveys, 22:3 (1967) 80 - 118.

[82] S. Kristensen, R. Thorn, S. Velani, Diophantine approximation and badly approximable sets,Advances in Math. 203 (2006) p. 132 - 169

[83] L. Kuipers, H. Niederreiter, Uniform distribution of sequences, John Wiley & sons, 1974.

[84] J.S. Lagarias, Best simultaneous Diophantine approximation I. Growth rates of bes approximationdenominators, Trans. Amer. Math. Soc., 1982, V. 272, No 2, p. 545 - 554.

[85] E. Landau, Vorlesungen uber Zahlentheorie, Vol. 2, New York, 1969.

38
http://arxiv.org/abs/1203.2998http://arxiv.org/abs/1204.3610http://arxiv.org/abs/1109.1633http://arxiv.org/abs/1001.5017http://arxiv.org/abs/1001.5017http://arxiv.org/abs/1109.1633http://arxiv.org/abs/1204.3610http://arxiv.org/abs/1203.2998


39/42

[86] G. Larcher, On the Distribution of Sequences Connected with Good Lattice Points. Monatsheftefur Mathematics 101 (1986), pp. 135 - 150.

[87] M. Laurent, Exponents of Diophantine approximations in dimension two, Canad.J.Math. 61, 1(2009),165 - 189.

[88] Yaqiao Li, The winning property of mixed badly approximable numbers, manuscript (2012).

[89] E. Lindenstrauss, B. Weiss, On sets invariant under the action of the diagonal group, Ergod. Th.& Dynam. Sys. (2001), 21, 1481 - 1500.

[90] E. Lindenstrauss, U. Shapira, Homogeneous orbit closures and applications, preprint available atarXiv:1101.3945(2011).

[91] K. Mahler, On compound convex bodies I, II . Proc. London Math. Soc. (3) 5 (1955), p. 358 -384

[92] G. Margulis, Diophantine approximation, lattices and flows on homogeneous spaces, in Apanorama of number theory or the view from Bakers garden (Zurich, 1999), Cambridge Univ.

Press, 2002, p. 280 - 310.

[93] B. de Mathan and O. Teulie. Problemes Diophantiens simultanes, Monatsh. Math. 143 (2004),229 - 245.

[94] H. Minkowski, Gesammelte Abhandlungen vol.2 (1911).

[95] N.G. Moshchevitin, On integers with missing digits, Dokl. Math. 65, No. 3, 350-352 (2002).

[96] N.G. Moshchevitin, On numbers with missing digits: an elementary proof of a result of S. V.Konyagin, Chebyshevski Sb. 3, No. 2(4), 93-99 (2002, in Russian)

[97] N.G. Moshchevitin, On the best two-dimensional joint Diophantine approximations in the sup-norm, Moscow Univ. Math. Bull. 60 (2005), no. 6, 29 - 32.

[98] N.G. Moshchevitin, On numbers with missing digits: solvability of the congruence x1x2 (modp), Dokl. Math. 74, No. 2, 744 - 747 (2006)

[99] N.G. Moshchevitin, I.D. Shkredov, On the multiplicative properties modulo m of numbers withmissing digits, Mathematical Notes, Mathematical Notes, 81:3 (2007) 338 - 355.

[100] N.G. Moshchevitin, Sets of the form A+Band finite continued fractions, Sbornik Mathematics,198 (2007) no. 3-4, 537 - 557.

[101] N.G.Moshchevitin, On small fractional parts of polynomials, J. Number Theory 129 (2009), 349- 357.

[102] N.G. Moshchevitin, Contribution to Vojtech Jarnk, preprint available at arXiv:0912.2442v3(2009).

[103] N.G. Moshchevitin, Diophantine approximations with positive integers: a remark to W.M.Schmidts theorem, preprint available atarXiv:0904.1906(2009).

[104] N.G. Moshchevitin, Khintchines singular Diophantine systems and their applications., RussianMathematical Surveys. 65:3 43 - 126 (2010).

39
http://arxiv.org/abs/1101.3945http://arxiv.org/abs/0912.2442http://arxiv.org/abs/0904.1906http://arxiv.org/abs/0904.1906http://arxiv.org/abs/0912.2442http://arxiv.org/abs/1101.3945


40/42

[105] N. Moshchevitin, D. Ushanov, On Larchers theorem concerning good lattice points and multi-plicative subgroups modulo p, Uniform Distribution Theory 5 (2010), no. 1, 45 - 52.

[106] N. Moshchevitin, On simultaneously badly approximable numbers, Bull. London Math. Soc., v.42, 1 (2010), p. 149 - 154.

[107] N.G. Moshchevitin, Schmidts conjecture and Badziahin-Pollington-Velanis theorem, preprint

available atarXiv:1004.4269v1 (2010).

[108] N.G. Moshchevitin, A note on badly approximable affine forms and winning sets, Moscow Math-ematical Journal 11:1 (2011), 129 - 137

[109] N.G. Moshchevitin, Positive integers: counterexample to W.M. Schmidts conjecture, preprintavailable atarXiv:1108.4435(2011)

[110] N.G. Moshchevitin, On certain Littlewood-like and Schmidt-like problems in inhomogeneousDiophantine approximations, preprint available at arXiv:1101.2032v2 (2011).

[111] N.G. Moshchevitin, Density modulo 1 of lacunary and sublacunary sequences: application of

Peres-Schlags construction, Journal of Mathematical Sciences, Vol. 180, No. 5 (2012), 610 - 625.

[112] N.G. Moshchevitin, Exponents for three-dimensional simultaneous Diophantine approximations,Czechoslovak Mathematical Journal, 62 (137), (2012), 127 - 137.

[113] N.G. Moshchevitin, Diophantine approximations with positive integers: some remarks, preprintavailable atarXiv:1108.4435(2012).

[114] N.G. Moshchevitin, On Harraps conjecture in Diophantine approximations, preprint availableatarXiv:1204.2561(2012).

[115] N.G. Moshchevitin, Diophantine exponents for systems of linear forms in two variables, preprintavailable atarXiv:1209.1697(2012).

[116] H. Niederreiter, Existance of good lattice points in the sense of Hlawka, Monatshefte fur Math.,86:3, (1978/1979), 203 - 219.

[117] H. Niedderreiter, Dyadic fractions with small partial quontiens, Monatsh. Math., V.101 (1986),p.309 - 315.

[118] J. Paradis, P. Viader, L. Bibiloni, A new light on Minkowskis?(x)function, J. Number Theory.,73 (1998), 212 - 227.

[119] J. Paradis, P. Viader, L. Bibiloni, The derivative of Minkowskis ?(x) function, J. Math. Anal.Appl., 253. (2001), 107 - 125.

[120] L. G. Peck, Simultaneous rational approximations to algebraic numbers, Bull. Amer. Math.Soc. 67 (1961), 197 - 201.

[121] Y. Peres, W. Schlag, Two Erdos problems on lacunary sequences: chromatic numbers andDiophantine approximations, Bull. London Math. Soc., v. 42, 2 (2010), p. 295 - 300.

[122] A.D. Pollington, S. L. Velani, On simultaneously badly approximable pairs, J. London Math.Soc., 66 (2002), p. 29 - 40.

40
http://arxiv.org/abs/1004.4269http://arxiv.org/abs/1108.4435http://arxiv.org/abs/1101.2032http://arxiv.org/abs/1108.4435http://arxiv.org/abs/1204.2561http://arxiv.org/abs/1209.1697http://arxiv.org/abs/1209.1697http://arxiv.org/abs/1204.2561http://arxiv.org/abs/1108.4435http://arxiv.org/abs/1101.2032http://arxiv.org/abs/1108.4435http://arxiv.org/abs/1004.4269


41/42

[123] A. van der Poorten, Symmetry and folding of continued fractions Journal de Theorie des Nom-bres de Bordeaux, 14: 2 (2002) 603 - 611.

[124] M. Queffelec, An introduction to Littlewoods conjecture, Seminaires & Congres 20, (2009), p.129 - 152.

[125] I. P. Rochev, On distribution of fractional parts of linear forms, Fundamentalnaya i prik-

ladnaya matematika, vol. 16 (2010), no. 6, pp. 123 - 137 (in Russian), preprint available atarXiv:0811.1547v1

[126] M.V. Romanov, Simultaneous two-dimensional best Diophantine approximations in the Eu-clidean norm., Moscow Univ. Math. Bull. 61 (2006), no. 2, 34

[127] D. Roy, Diophantine approximation with sign constraints, Lec-ture at Fields Institute Number Theoretical Seminar, 22 Oct. 2012,http://www.fields.utoronto.ca/programs/scientific/12-13/numtheoryseminar/index.html

[128] M.G. Rukavishnikova, Probabilistic bound for the sum of partial quotients of fractions with afixed denominator, Chebyshevskii sbornik, 7:4 (2006), 113 - 121 (in Russian).

[129] M.G. Rukavishnikova, The law of large numbers for the sum of the partial quotients of a rationalnumber with fixed denominator, Mathematical Notes, 2011, 90:3, 418 - 430.

[130] R. Salem, On some singular monotonic functions which are strictly increasing, Trans. Amer.Math. Soc. 53 (3) (1943), 427 - 439.

[131] W.M. Schmidt, On badly approximable numbers and certain games, Trans. Amer. Math. Soc.,623 (1966), p. 178 - 199.

[132] W.M. Schmidt, Badly approximable systems of linear forms, J. Number Theory, 1 (1969) 139 -154.

[133] W.M. Schmidt, Diophantine Approximations, Lect. Not. Math., 785 (1980).

[134] W.M. Schmidt, Two questions in Diophantine approximations, Monatshefte fur Mathematik82, 237 - 245 (1976).

[135] W.M. Schmidt, Small fractional parts of polynomials, Regional Conference Series in Mathe-matics, No. 32 (1977) AMS, Providence.

[136] W.M. Schmidt, Open problems in Diophantine approximations, in "Approximations Diophanti-ennes et nombres transcendants Luminy, 1982, Progress in Mathematics, Birkhauser, p.271 - 289(1983).

[137] W. M. Schmidt, L. Summerer, Parametric Geometry of Numbers and applications, Acta Arith-metica 140 No.1, p. 67 - 91 (2009).

[138] W. M. Schmidt, L. Summerer, Diophantine approximations and parametric Geometry of Num-bers, Monatshefte fur Mathematik (to appear).

[139] U. Shapira, A solution to a problem of Cassels and Diophantine properties of cubic numbers,Annals of Mathematics, 173 (2011), 543 - 567.

[140] U. Shapira, Grids with dense values, preprint available atarXiv:1101.3941(2011).

41
http://arxiv.org/abs/0811.1547http://www.fields.utoronto.ca/programs/scientific/12-13/numtheoryseminar/index.htmlhttp://arxiv.org/abs/1101.3941http://arxiv.org/abs/1101.3941http://www.fields.utoronto.ca/programs/scientific/12-13/numtheoryseminar/index.htmlhttp://arxiv.org/abs/0811.1547


42/42

[141] I. Shparlinski, Modular Hyperbolas, preprint available atarXiv:1103.2879v2 (2011).

[142] M.M. Skriganov, Constructions of uniform distributions in terms of geometry of numbers, Al-gebra and Analysis, 6:3, (1994), 200 - 230.

[143] B.F. Skubenko, Minima of a decomposable cubic form of three variables, Zapiski nauchnyh semLomi 168 (1988), 125 - 139 (in Russian).

[144] B.F. Skubenko, Minima of a decomposable cubic form of degreenofn variables, Zapiski nauch-nyh sem Lomi 183 (1990), 142 - 154 (in Russian).

[145] T. Tao, Continued fractions, Bohr sets, and the Littlewood conjecture,http://terrytao.wordpress.com/2012/01/03/continued-fractions-bohr-sets-and-the-littlewood-conject

[146] R. F. Tichy, J. Uitz, An extension of Minkowskis singular function, Appl. Math. Lett., 8 (1995),39 - 46.

[147] P. Thurneer, Zur diophantischen Approximationen von zwei reelen Zahlen, Acta Arithmetica,44 (1984), 201 - 206.

[148] P. Thurneer, On Dirichlets theorem concerning diophantine approximations, Acta Arithmetica54 (1990), 241 - 250.

[149] D.M. Ushanov, Bykovskiis theorem and generalized Larchers theorem, Mathematical Notes(2012, to appear, in Russian), preprint available at arXiv:1202.6025v2.

[150] A. Venkatesh, The work of Einsiedler, Katok and Lindenstrauss on the Littlewood conjecture,Bull. Amer. Math. Soc. (N.S.) 45 (2008), no. 1, p. 117 - 134.

[151] M. Waldschmidt, Report on some recent advances in Diophantine approximation, preprint avail-able atarXiv:0908.3973v1.

[152] M. Yodphotong, and V. Laohakosol, Proofs on Zarembas Conjecture for Powers of 6, Proceed-ings of the International Conference on Algebra and Its Applications 2002, 278-282.

[153] A. Zaharescu, Small values of{n2}, Invent. Math. (1995), 121, p. 379 - 388.[154] S.K. Zaremba, La methode des bons treillis pour le calcul des integrales multiples./ Appl. of

Number Theory to Numerical Analysis., ed. S. K. Zaremba, N. Y.,1972, p. 39119

[155] E.N. Zhabitskaya, On arithmetical nature of Tichy-Uitzs function, Functiones et Approximatio,43:1(2010), 15 - 22.

[156] E. Zhabitskaya, Continued fractions with minimal remainders, Uniform Distribution Theory 5(2010), no.2, 55 - 78.

[157] E. Zhabitskaya, Continued fractions with odd partial quotients, accepted in International Jour-nal of Number Theory, preprint available at arXiv:1110.5270v1 (2011).
http://arxiv.org/abs/1103.2879http://terrytao.wordpress.com/2012/01/03/continued-fractions-bohr-sets-and-the-littlewood-conjecture/#more-5605http://arxiv.org/abs/1202.6025http://arxiv.org/abs/0908.3973http://arxiv.org/abs/1110.5270http://arxiv.org/abs/1110.5270http://arxiv.org/abs/0908.3973http://arxiv.org/abs/1202.6025http://terrytao.wordpress.com/2012/01/03/continued-fractions-bohr-sets-and-the-littlewood-conjecture/#more-5605http://arxiv.org/abs/1103.2879

On some open problems in Diophantine approximation

Documents

Transcript of On some open problems in Diophantine approximation