Issues

Journal & Issues

Volume 17 (2022): Issue 1 (December 2022)

Volume 16 (2021): Issue 2 (December 2021)

Volume 16 (2021): Issue 1 (June 2021)

Volume 15 (2020): Issue 2 (December 2020)

Volume 15 (2020): Issue 1 (June 2020)

Volume 14 (2019): Issue 2 (December 2019)

Volume 14 (2019): Issue 1 (June 2019)
The sixth International Conference on Uniform Distribution Theory (UDT 2018) CIRM, Luminy, Marseilles, France, October 1–5, 2018

Volume 13 (2018): Issue 2 (December 2018)

Volume 13 (2018): Issue 1 (June 2018)

Volume 12 (2017): Issue 2 (December 2017)

Volume 12 (2017): Issue 1 (June 2017)

Volume 11 (2016): Issue 2 (December 2016)

Volume 11 (2016): Issue 1 (June 2016)

Journal Details
Format
Journal
eISSN
2309-5377
First Published
30 Dec 2013
Publication timeframe
2 times per year
Languages
English

Search

Volume 17 (2022): Issue 1 (December 2022)

Journal Details
Format
Journal
eISSN
2309-5377
First Published
30 Dec 2013
Publication timeframe
2 times per year
Languages
English

Search

8 Articles
Open Access

The Seventh International Conference on Uniform Distribution Theory (UDT 2021)

Published Online: 31 May 2022
Page range: i - ii

Abstract

Open Access

Bounds on the size of Progression-Free Sets in ℤmn

Published Online: 31 May 2022
Page range: 1 - 10

Abstract

Abstract

In this note we give an overview of the currently known best lower and upper bounds on the size of a subset of ℤnm avoiding k-term arithmetic progression. We will focus on the case when the length of the forbidden progression is 3. We also formulate some open questions.

Keywords

  • progression-free sets
  • cap set problem
  • polynomial method

MSC 2010

  • 11B25
  • 05D99
Open Access

Products of Integers with Few Nonzero Digits

Published Online: 31 May 2022
Page range: 11 - 28

Abstract

Abstract

Let s(n) be the number of nonzero bits in the binary digital expansion of the integer n. We study, for fixed k, ℓ, m, the Diophantine system

s(ab)= k, s(a)= ℓ, and s(b)= m

in odd integer variables a, b.When k =2 or k = 3, we establish a bound on ab in terms of and m. While such a bound does not exist in the case of k =4, we give an upper bound for min{a, b} in terms of and m.

Keywords

  • sum of digits
  • digital expansion
  • factors

MSC 2010

  • Primary: 11A63
  • Secondary: 11B83
Open Access

On a Class of Lacunary Almost Newman Polynomials Modulo P and Density Theorems

Published Online: 31 May 2022
Page range: 29 - 54

Abstract

Abstract

The reduction modulo p of a family of lacunary integer polynomials, associated with the dynamical zeta function ζβ(z)of the β-shift, for β> 1 close to one, is investigated. We briefly recall how this family is correlated to the problem of Lehmer. A variety of questions is raised about their numbers of zeroes in 𝔽p and their factorizations, via Kronecker’s Average Value Theorem (viewed as an analog of classical Theorems of Uniform Distribution Theory). These questions are partially answered using results of Schinzel, revisited by Sawin, Shusterman and Stoll, and density theorems (Frobenius, Chebotarev, Serre, Rosen). These questions arise from the search for the existence of integer polynomials of Mahler measure > 1 less than the smallest Salem number 1.176280. Explicit connection with modular forms (or modular representations) of the numbers of zeroes of these polynomials in 𝔽p is obtained in a few cases. In general it is expected since it must exist according to the Langlands program.

Keywords

  • lacunary integer polynomial
  • zeroes
  • factorization
  • Lehmer’s problem
  • Chebotarev density theorem
  • Frobenius density theorem
  • number of zeroes modulo

MSC 2010

  • 11C08
  • 11R09
  • 11R45
  • 12E05
  • 13P05
Open Access

Insertion in Constructed Normal Numbers

Published Online: 31 May 2022
Page range: 55 - 76

Abstract

Abstract

Defined by Borel, a real number is normal to an integer base b ≥ 2 if in its base-b expansion every block of digits occurs with the same limiting frequency as every other block of the same length. We consider the problem of insertion in constructed base-b normal expansions to obtain normality to base (b + 1).

Keywords

  • normal numbers
  • de Bruijn sequences
  • combinatorics on words

MSC 2010

  • 11K16
  • 05C45
  • 68R15
Open Access

A Typical Number is Extremely Non-Normal

Published Online: 31 May 2022
Page range: 77 - 88

Abstract

Abstract

Fix a positive integer N ≥ 2. For a real number x ∈ [0, 1] and a digit i ∈ {0, 1,..., N − 1}, let Πi(x, n) denote the frequency of the digit i among the first nN-adic digits of x. It is well-known that for a typical (in the sense of Baire) x ∈ [0, 1], the sequence of digit frequencies diverges as n →∞. In this paper we show that for any regular linear transformation T there exists a residual set of points x ∈ [0,1] such that the T -averaged version of the sequence (Πi(x, n))n also diverges significantly.

Keywords

  • normal numbers
  • digit frequencies
  • regular linear transformations

MSC 2010

  • 11K16
Open Access

On Some Properties of Irrational Subspaces

Published Online: 31 May 2022
Page range: 89 - 104

Abstract

Abstract

In this paper, we discuss some properties of completely irrational subspaces. We prove that there exist completely irrational subspaces that are badly approximable and, moreover, sets of such subspaces are winning in different senses. We get some bounds for Diophantine exponents of vectors that lie in badly approximable subspaces that are completely irrational; in particular, for any vector ξ from two-dimensional badly approximable completely irrational subspace of ℝd one has ω(ξ)5-12 \mathord{\buildrel{\lower3pt\hbox{$\scriptscriptstyle\frown$}}\over \omega } \left( \xi \right) \le {{\sqrt {5 - 1} } \over 2} . Besides that, some statements about the dimension of subspaces generated by best approximations to completely irrational subspace easily follow from properties that we discuss.

Keywords

  • badly approximable matrices
  • completely irrational subspaces
  • ()-games

MSC 2010

  • 11J13
Open Access

Density of Oscillating Sequences in the Real Line

Published Online: 31 May 2022
Page range: 105 - 130

Abstract

Abstract

In this paper we study the density in the real line of oscillating sequences of the form (g(k)F(kα))k, {\left( {g\left( k \right) \cdot F\left( {k\alpha } \right)} \right)_{k \in \mathbb{N}}}, where g is a positive increasing function and F a real continuous 1-periodic function. This extends work by Berend, Boshernitzan and Kolesnik [Distribution Modulo 1 of Some Oscillating Sequences I-III] who established differential properties on the function F ensuring that the oscillating sequence is dense modulo 1.

More precisely, when F has finitely many roots in [0, 1), we provide necessary and also sufficient conditions for the oscillating sequence under consideration to be dense in ℝ. All the results are stated in terms of the Diophantine properties of α, with the help of the theory of continued fractions.

Keywords

  • Diophantine approximation
  • oscillating sequences
  • irrationality measure
  • continued fractions
  • Ostrowski expansion

MSC 2010

  • 11J70
  • 11J82
  • 11B05
8 Articles
Open Access

The Seventh International Conference on Uniform Distribution Theory (UDT 2021)

Published Online: 31 May 2022
Page range: i - ii

Abstract

Open Access

Bounds on the size of Progression-Free Sets in ℤmn

Published Online: 31 May 2022
Page range: 1 - 10

Abstract

Abstract

In this note we give an overview of the currently known best lower and upper bounds on the size of a subset of ℤnm avoiding k-term arithmetic progression. We will focus on the case when the length of the forbidden progression is 3. We also formulate some open questions.

Keywords

  • progression-free sets
  • cap set problem
  • polynomial method

MSC 2010

  • 11B25
  • 05D99
Open Access

Products of Integers with Few Nonzero Digits

Published Online: 31 May 2022
Page range: 11 - 28

Abstract

Abstract

Let s(n) be the number of nonzero bits in the binary digital expansion of the integer n. We study, for fixed k, ℓ, m, the Diophantine system

s(ab)= k, s(a)= ℓ, and s(b)= m

in odd integer variables a, b.When k =2 or k = 3, we establish a bound on ab in terms of and m. While such a bound does not exist in the case of k =4, we give an upper bound for min{a, b} in terms of and m.

Keywords

  • sum of digits
  • digital expansion
  • factors

MSC 2010

  • Primary: 11A63
  • Secondary: 11B83
Open Access

On a Class of Lacunary Almost Newman Polynomials Modulo P and Density Theorems

Published Online: 31 May 2022
Page range: 29 - 54

Abstract

Abstract

The reduction modulo p of a family of lacunary integer polynomials, associated with the dynamical zeta function ζβ(z)of the β-shift, for β> 1 close to one, is investigated. We briefly recall how this family is correlated to the problem of Lehmer. A variety of questions is raised about their numbers of zeroes in 𝔽p and their factorizations, via Kronecker’s Average Value Theorem (viewed as an analog of classical Theorems of Uniform Distribution Theory). These questions are partially answered using results of Schinzel, revisited by Sawin, Shusterman and Stoll, and density theorems (Frobenius, Chebotarev, Serre, Rosen). These questions arise from the search for the existence of integer polynomials of Mahler measure > 1 less than the smallest Salem number 1.176280. Explicit connection with modular forms (or modular representations) of the numbers of zeroes of these polynomials in 𝔽p is obtained in a few cases. In general it is expected since it must exist according to the Langlands program.

Keywords

  • lacunary integer polynomial
  • zeroes
  • factorization
  • Lehmer’s problem
  • Chebotarev density theorem
  • Frobenius density theorem
  • number of zeroes modulo

MSC 2010

  • 11C08
  • 11R09
  • 11R45
  • 12E05
  • 13P05
Open Access

Insertion in Constructed Normal Numbers

Published Online: 31 May 2022
Page range: 55 - 76

Abstract

Abstract

Defined by Borel, a real number is normal to an integer base b ≥ 2 if in its base-b expansion every block of digits occurs with the same limiting frequency as every other block of the same length. We consider the problem of insertion in constructed base-b normal expansions to obtain normality to base (b + 1).

Keywords

  • normal numbers
  • de Bruijn sequences
  • combinatorics on words

MSC 2010

  • 11K16
  • 05C45
  • 68R15
Open Access

A Typical Number is Extremely Non-Normal

Published Online: 31 May 2022
Page range: 77 - 88

Abstract

Abstract

Fix a positive integer N ≥ 2. For a real number x ∈ [0, 1] and a digit i ∈ {0, 1,..., N − 1}, let Πi(x, n) denote the frequency of the digit i among the first nN-adic digits of x. It is well-known that for a typical (in the sense of Baire) x ∈ [0, 1], the sequence of digit frequencies diverges as n →∞. In this paper we show that for any regular linear transformation T there exists a residual set of points x ∈ [0,1] such that the T -averaged version of the sequence (Πi(x, n))n also diverges significantly.

Keywords

  • normal numbers
  • digit frequencies
  • regular linear transformations

MSC 2010

  • 11K16
Open Access

On Some Properties of Irrational Subspaces

Published Online: 31 May 2022
Page range: 89 - 104

Abstract

Abstract

In this paper, we discuss some properties of completely irrational subspaces. We prove that there exist completely irrational subspaces that are badly approximable and, moreover, sets of such subspaces are winning in different senses. We get some bounds for Diophantine exponents of vectors that lie in badly approximable subspaces that are completely irrational; in particular, for any vector ξ from two-dimensional badly approximable completely irrational subspace of ℝd one has ω(ξ)5-12 \mathord{\buildrel{\lower3pt\hbox{$\scriptscriptstyle\frown$}}\over \omega } \left( \xi \right) \le {{\sqrt {5 - 1} } \over 2} . Besides that, some statements about the dimension of subspaces generated by best approximations to completely irrational subspace easily follow from properties that we discuss.

Keywords

  • badly approximable matrices
  • completely irrational subspaces
  • ()-games

MSC 2010

  • 11J13
Open Access

Density of Oscillating Sequences in the Real Line

Published Online: 31 May 2022
Page range: 105 - 130

Abstract

Abstract

In this paper we study the density in the real line of oscillating sequences of the form (g(k)F(kα))k, {\left( {g\left( k \right) \cdot F\left( {k\alpha } \right)} \right)_{k \in \mathbb{N}}}, where g is a positive increasing function and F a real continuous 1-periodic function. This extends work by Berend, Boshernitzan and Kolesnik [Distribution Modulo 1 of Some Oscillating Sequences I-III] who established differential properties on the function F ensuring that the oscillating sequence is dense modulo 1.

More precisely, when F has finitely many roots in [0, 1), we provide necessary and also sufficient conditions for the oscillating sequence under consideration to be dense in ℝ. All the results are stated in terms of the Diophantine properties of α, with the help of the theory of continued fractions.

Keywords

  • Diophantine approximation
  • oscillating sequences
  • irrationality measure
  • continued fractions
  • Ostrowski expansion

MSC 2010

  • 11J70
  • 11J82
  • 11B05

Plan your remote conference with Sciendo