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Volumen 17 (2022): Edición 1 (December 2022)

Volumen 16 (2021): Edición 2 (December 2021)

Volumen 16 (2021): Edición 1 (June 2021)

Volumen 15 (2020): Edición 2 (December 2020)

Volumen 15 (2020): Edición 1 (June 2020)

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

Volumen 13 (2018): Edición 2 (December 2018)

Volumen 13 (2018): Edición 1 (June 2018)

Volumen 12 (2017): Edición 2 (December 2017)

Volumen 12 (2017): Edición 1 (June 2017)

Volumen 11 (2016): Edición 2 (December 2016)

Volumen 11 (2016): Edición 1 (June 2016)

Detalles de la revista
Formato
Revista
eISSN
2309-5377
Publicado por primera vez
30 Dec 2013
Periodo de publicación
2 veces al año
Idiomas
Inglés

Buscar

Volumen 17 (2022): Edición 1 (December 2022)

Detalles de la revista
Formato
Revista
eISSN
2309-5377
Publicado por primera vez
30 Dec 2013
Periodo de publicación
2 veces al año
Idiomas
Inglés

Buscar

8 Artículos
Acceso abierto

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

Publicado en línea: 31 May 2022
Páginas: i - ii

Resumen

Acceso abierto

Bounds on the size of Progression-Free Sets in ℤmn

Publicado en línea: 31 May 2022
Páginas: 1 - 10

Resumen

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.

Palabras clave

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

MSC 2010

  • 11B25
  • 05D99
Acceso abierto

Products of Integers with Few Nonzero Digits

Publicado en línea: 31 May 2022
Páginas: 11 - 28

Resumen

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.

Palabras clave

  • sum of digits
  • digital expansion
  • factors

MSC 2010

  • Primary: 11A63
  • Secondary: 11B83
Acceso abierto

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

Publicado en línea: 31 May 2022
Páginas: 29 - 54

Resumen

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.

Palabras clave

  • 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
Acceso abierto

Insertion in Constructed Normal Numbers

Publicado en línea: 31 May 2022
Páginas: 55 - 76

Resumen

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).

Palabras clave

  • normal numbers
  • de Bruijn sequences
  • combinatorics on words

MSC 2010

  • 11K16
  • 05C45
  • 68R15
Acceso abierto

A Typical Number is Extremely Non-Normal

Publicado en línea: 31 May 2022
Páginas: 77 - 88

Resumen

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.

Palabras clave

  • normal numbers
  • digit frequencies
  • regular linear transformations

MSC 2010

  • 11K16
Acceso abierto

On Some Properties of Irrational Subspaces

Publicado en línea: 31 May 2022
Páginas: 89 - 104

Resumen

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.

Palabras clave

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

MSC 2010

  • 11J13
Acceso abierto

Density of Oscillating Sequences in the Real Line

Publicado en línea: 31 May 2022
Páginas: 105 - 130

Resumen

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.

Palabras clave

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

MSC 2010

  • 11J70
  • 11J82
  • 11B05
8 Artículos
Acceso abierto

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

Publicado en línea: 31 May 2022
Páginas: i - ii

Resumen

Acceso abierto

Bounds on the size of Progression-Free Sets in ℤmn

Publicado en línea: 31 May 2022
Páginas: 1 - 10

Resumen

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.

Palabras clave

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

MSC 2010

  • 11B25
  • 05D99
Acceso abierto

Products of Integers with Few Nonzero Digits

Publicado en línea: 31 May 2022
Páginas: 11 - 28

Resumen

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.

Palabras clave

  • sum of digits
  • digital expansion
  • factors

MSC 2010

  • Primary: 11A63
  • Secondary: 11B83
Acceso abierto

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

Publicado en línea: 31 May 2022
Páginas: 29 - 54

Resumen

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.

Palabras clave

  • 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
Acceso abierto

Insertion in Constructed Normal Numbers

Publicado en línea: 31 May 2022
Páginas: 55 - 76

Resumen

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).

Palabras clave

  • normal numbers
  • de Bruijn sequences
  • combinatorics on words

MSC 2010

  • 11K16
  • 05C45
  • 68R15
Acceso abierto

A Typical Number is Extremely Non-Normal

Publicado en línea: 31 May 2022
Páginas: 77 - 88

Resumen

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.

Palabras clave

  • normal numbers
  • digit frequencies
  • regular linear transformations

MSC 2010

  • 11K16
Acceso abierto

On Some Properties of Irrational Subspaces

Publicado en línea: 31 May 2022
Páginas: 89 - 104

Resumen

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.

Palabras clave

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

MSC 2010

  • 11J13
Acceso abierto

Density of Oscillating Sequences in the Real Line

Publicado en línea: 31 May 2022
Páginas: 105 - 130

Resumen

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.

Palabras clave

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

MSC 2010

  • 11J70
  • 11J82
  • 11B05

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