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The sixth International Conference on Uniform Distribution Theory (UDT 2018) CIRM, Luminy, Marseilles, France, October 1–5, 2018

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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

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

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

5 Artículos
Acceso abierto

On the Classification of Solutions of Quantum Functional Equations with Cyclic and Semi-Cyclic Supports

Publicado en línea: 30 Oct 2021
Páginas: 1 - 40

Resumen

Abstract

In this paper, we classify all solutions with cyclic and semi-cyclic semigroup supports of the functional equations arising from multiplication of quantum integers with fields of coefficients of characteristic zero. This also solves completely the classification problem proposed by Melvyn Nathanson and Yang Wang concerning the solutions, with semigroup supports which are not prime subsemigroups of ℕ, to these functional equations for the case of rational field of coefficients. As a consequence, we obtain some results for other problems raised by Nathanson concerning maximal solutions and extension of supports of solutions to these functional equations in the case where the semigroup supports are not prime subsemigroups of ℕ.

Palabras clave

  • quantum integer
  • quantum algebra
  • -series
  • semigroup
  • polynomial functional equation
  • cyclotomy

MSC 2010

  • 81R50
  • 11R18
  • 11T22
  • 11B13
  • 11C08
  • 39B05
Acceso abierto

Extreme Values of Euler-Kronecker Constants

Publicado en línea: 30 Oct 2021
Páginas: 41 - 52

Resumen

Abstract

In a family of Sn-fields (n ≤ 5), we show that except for a density zero set, the lower and upper bounds of the Euler-Kronecker constants are (n − 1) log log dK+ O(log log log dK) and loglog dK + O(log log log dK), resp., where dK is the absolute value of the discriminant of a number field K.

Palabras clave

  • Euler-Kronecker constants
  • Dedekind zeta functions
  • Logarithmic derivatives of -functions

MSC 2010

  • Primary: 11R42
  • Secondary: 11M41
Acceso abierto

Families of Well Approximable Measures

Publicado en línea: 30 Oct 2021
Páginas: 53 - 70

Resumen

Abstract

We provide an algorithm to approximate a finitely supported discrete measure μ by a measure νN corresponding to a set of N points so that the total variation between μ and νN has an upper bound. As a consequence if μ is a (finite or infinitely supported) discrete probability measure on [0, 1]d with a sufficient decay rate on the weights of each point, then μ can be approximated by νN with total variation, and hence star-discrepancy, bounded above by (log N)N1. Our result improves, in the discrete case, recent work by Aistleitner, Bilyk, and Nikolov who show that for any normalized Borel measure μ, there exist finite sets whose star-discrepancy with respect to μ is at most (logN)d12N1 {\left( {\log \,N} \right)^{d - {1 \over 2}}}{N^{ - 1}} . Moreover, we close a gap in the literature for discrepancy in the case d =1 showing both that Lebesgue is indeed the hardest measure to approximate by finite sets and also that all measures without discrete components have the same order of discrepancy as the Lebesgue measure.

Palabras clave

  • star-discrepancy
  • low-discrepancy sequences
  • Borel Measures
  • total variation

MSC 2010

  • 11K36
  • 11K38
  • 11J71
  • 11K06
Acceso abierto

The Inequality of Erdős-Turán-Koksma in the Terms of the Functions of the System Γ s

Publicado en línea: 30 Oct 2021
Páginas: 71 - 92

Resumen

Abstract

In the present paper the author uses the function system Γsconstructed in Cantor bases to show upper bounds of the extreme and star discrepancy of an arbitrary net in the terms of the trigonometric sum of this net with respect to the functions of this system. The obtained estimations are inequalities of the type of Erdős-Turán-Koksma. These inequalities are very suitable for studying of nets constructed in the same Cantor system.

Palabras clave

  • function system Γ constructed in Cantor bases
  • extreme and star discrepancy
  • inequalities of Erd˝ os-Tur´ an-Koksma

MSC 2010

  • 11K06
  • 11K31
  • 11K36
  • 65C05
Acceso abierto

Uniform Distribution of the Weighted Sum-of-Digits Functions

Publicado en línea: 30 Oct 2021
Páginas: 93 - 126

Resumen

Abstract

The higher-dimensional generalization of the weighted q-adic sum-of-digits functions sq,γ(n), n =0, 1, 2,..., covers several important cases of sequences investigated in the theory of uniformly distributed sequences, e.g., d-dimensional van der Corput-Halton or d-dimensional Kronecker sequences. We prove a necessary and sufficient condition for the higher-dimensional weighted q-adic sum-of-digits functions to be uniformly distributed modulo one in terms of a trigonometric product. As applications of our condition we prove some upper estimates of the extreme discrepancies of such sequences, and that the existence of distribution function g(x)= x implies the uniform distribution modulo one of the weighted q-adic sum-of-digits function sq,γ (n), n = 0, 1, 2,... We also prove the uniform distribution modulo one of related sequences h1sq, γ (n)+h2sq,γ (n +1), where h1 and h2 are integers such that h1 + h2 ≠ 0 and that the akin two-dimensional sequence sq,γ (n), sq,γ (n +1) cannot be uniformly distributed modulo one if q ≥ 3. The properties of the two-dimensional sequence sq,γ (n),sq,γ (n +1), n =0, 1, 2,..., will be instrumental in the proofs of the final section, where we show how the growth properties of the sequence of weights influence the distribution of values of the weighted sum-of-digits function which in turn imply a new property of the van der Corput sequence.

Palabras clave

  • uniform distribution
  • van der Corput sequence
  • higher-dimensional weighted sum-of-digits function
  • discrepancy
  • distribution function

MSC 2010

  • Primary 11K06
  • Secondary 11K31, 11J71
5 Artículos
Acceso abierto

On the Classification of Solutions of Quantum Functional Equations with Cyclic and Semi-Cyclic Supports

Publicado en línea: 30 Oct 2021
Páginas: 1 - 40

Resumen

Abstract

In this paper, we classify all solutions with cyclic and semi-cyclic semigroup supports of the functional equations arising from multiplication of quantum integers with fields of coefficients of characteristic zero. This also solves completely the classification problem proposed by Melvyn Nathanson and Yang Wang concerning the solutions, with semigroup supports which are not prime subsemigroups of ℕ, to these functional equations for the case of rational field of coefficients. As a consequence, we obtain some results for other problems raised by Nathanson concerning maximal solutions and extension of supports of solutions to these functional equations in the case where the semigroup supports are not prime subsemigroups of ℕ.

Palabras clave

  • quantum integer
  • quantum algebra
  • -series
  • semigroup
  • polynomial functional equation
  • cyclotomy

MSC 2010

  • 81R50
  • 11R18
  • 11T22
  • 11B13
  • 11C08
  • 39B05
Acceso abierto

Extreme Values of Euler-Kronecker Constants

Publicado en línea: 30 Oct 2021
Páginas: 41 - 52

Resumen

Abstract

In a family of Sn-fields (n ≤ 5), we show that except for a density zero set, the lower and upper bounds of the Euler-Kronecker constants are (n − 1) log log dK+ O(log log log dK) and loglog dK + O(log log log dK), resp., where dK is the absolute value of the discriminant of a number field K.

Palabras clave

  • Euler-Kronecker constants
  • Dedekind zeta functions
  • Logarithmic derivatives of -functions

MSC 2010

  • Primary: 11R42
  • Secondary: 11M41
Acceso abierto

Families of Well Approximable Measures

Publicado en línea: 30 Oct 2021
Páginas: 53 - 70

Resumen

Abstract

We provide an algorithm to approximate a finitely supported discrete measure μ by a measure νN corresponding to a set of N points so that the total variation between μ and νN has an upper bound. As a consequence if μ is a (finite or infinitely supported) discrete probability measure on [0, 1]d with a sufficient decay rate on the weights of each point, then μ can be approximated by νN with total variation, and hence star-discrepancy, bounded above by (log N)N1. Our result improves, in the discrete case, recent work by Aistleitner, Bilyk, and Nikolov who show that for any normalized Borel measure μ, there exist finite sets whose star-discrepancy with respect to μ is at most (logN)d12N1 {\left( {\log \,N} \right)^{d - {1 \over 2}}}{N^{ - 1}} . Moreover, we close a gap in the literature for discrepancy in the case d =1 showing both that Lebesgue is indeed the hardest measure to approximate by finite sets and also that all measures without discrete components have the same order of discrepancy as the Lebesgue measure.

Palabras clave

  • star-discrepancy
  • low-discrepancy sequences
  • Borel Measures
  • total variation

MSC 2010

  • 11K36
  • 11K38
  • 11J71
  • 11K06
Acceso abierto

The Inequality of Erdős-Turán-Koksma in the Terms of the Functions of the System Γ s

Publicado en línea: 30 Oct 2021
Páginas: 71 - 92

Resumen

Abstract

In the present paper the author uses the function system Γsconstructed in Cantor bases to show upper bounds of the extreme and star discrepancy of an arbitrary net in the terms of the trigonometric sum of this net with respect to the functions of this system. The obtained estimations are inequalities of the type of Erdős-Turán-Koksma. These inequalities are very suitable for studying of nets constructed in the same Cantor system.

Palabras clave

  • function system Γ constructed in Cantor bases
  • extreme and star discrepancy
  • inequalities of Erd˝ os-Tur´ an-Koksma

MSC 2010

  • 11K06
  • 11K31
  • 11K36
  • 65C05
Acceso abierto

Uniform Distribution of the Weighted Sum-of-Digits Functions

Publicado en línea: 30 Oct 2021
Páginas: 93 - 126

Resumen

Abstract

The higher-dimensional generalization of the weighted q-adic sum-of-digits functions sq,γ(n), n =0, 1, 2,..., covers several important cases of sequences investigated in the theory of uniformly distributed sequences, e.g., d-dimensional van der Corput-Halton or d-dimensional Kronecker sequences. We prove a necessary and sufficient condition for the higher-dimensional weighted q-adic sum-of-digits functions to be uniformly distributed modulo one in terms of a trigonometric product. As applications of our condition we prove some upper estimates of the extreme discrepancies of such sequences, and that the existence of distribution function g(x)= x implies the uniform distribution modulo one of the weighted q-adic sum-of-digits function sq,γ (n), n = 0, 1, 2,... We also prove the uniform distribution modulo one of related sequences h1sq, γ (n)+h2sq,γ (n +1), where h1 and h2 are integers such that h1 + h2 ≠ 0 and that the akin two-dimensional sequence sq,γ (n), sq,γ (n +1) cannot be uniformly distributed modulo one if q ≥ 3. The properties of the two-dimensional sequence sq,γ (n),sq,γ (n +1), n =0, 1, 2,..., will be instrumental in the proofs of the final section, where we show how the growth properties of the sequence of weights influence the distribution of values of the weighted sum-of-digits function which in turn imply a new property of the van der Corput sequence.

Palabras clave

  • uniform distribution
  • van der Corput sequence
  • higher-dimensional weighted sum-of-digits function
  • discrepancy
  • distribution function

MSC 2010

  • Primary 11K06
  • Secondary 11K31, 11J71

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