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

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

Search

9 Articles
access type Open Access

The Sixth International Conference on Uniform Distribution Theory (UDT 2018)

Published Online: 27 Mar 2020
Page range: i - x

Abstract

Abstract

This volume contains papers originally presented or inspired by the Sixth International Conference on Uniform Distribution Theory, which was held at CIRM in Luminy, Marseilles, France, October 1–5, 2016.

access type Open Access

Higher Order Oscillation and Uniform Distribution

Published Online: 27 Mar 2020
Page range: 1 - 10

Abstract

Abstract

It is known that the Möbius function in number theory is higher order oscillating. In this paper we show that there is another kind of higher order oscillating sequences in the form (e2πiαβn g(β))n∈𝕅, for a non-decreasing twice differentiable function g with a mild condition. This follows the result we prove in this paper that for a fixed non-zero real number α and almost all real numbers β> 1 (alternatively, for a fixed real number β> 1 and almost all real numbers α) and for all real polynomials Q(x), sequences (αβng(β)+ Q(n)) n∈𝕅 are uniformly distributed modulo 1.

Keywords

  • higher order oscillating sequence
  • uniformly distributed modulo 1 (u. d. mod 1)

MSC 2010

  • Primary 11K65, 37A35
  • Secondary 37A25, 11N05
access type Open Access

Partitioning the Set of Primes to Create r-Dimensional Sequences Which are Uniformly Distributed Modulo [0, 1)r

Published Online: 27 Mar 2020
Page range: 11 - 18

Abstract

Abstract

Expanding on our previous results, we show that by partitioning the set of primes into a finite number of subsets of roughly the same size, we can create r-dimensional sequences of real numbers which are uniformly distributed modulo [0, 1)r.

Keywords

  • Uniform distribution modulo one

MSC 2010

  • 11K16
  • 11J71
access type Open Access

Distribuion of Leading Digits of Numbers II

Published Online: 27 Mar 2020
Page range: 19 - 42

Abstract

Abstract

In this paper, we study the sequence (f (pn))n≥1,where pn is the nth prime number and f is a function of a class of slowly increasing functions including f (x)=logb xr and f (x)=logb(x log x)r,where b ≥ 2 is an integer and r> 0 is a real number. We give upper bounds of the discrepancy DNi*(f(pn),g)D_{{N_i}}^*\left( {f\left( {{p_n}} \right),g} \right) for a distribution function g and a sub-sequence (Ni)i≥1 of the natural numbers. Especially for f (x)= logb xr, we obtain the effective results for an upper bound of DNi*(f(pn)g)D_{{N_i}}^*\left( {f\left( {{p_n}} \right),g} \right).

Keywords

  • prime numbers
  • Benford’s law
  • distribution function
  • discrepancy
  • regularly varying function

MSC 2010

  • 11K06
  • 11K31
  • 11K38
access type Open Access

A Complete Classification of Digital (0,m, 3)-Nets and Digital (0, 2)-Sequences in Base 2

Published Online: 27 Mar 2020
Page range: 43 - 52

Abstract

Abstract

We give a complete classification of all matrices C1,C2,C3C3𝔽2m×m{C_3} \in \mathbb{F}_2^{m \times m} which generate a digital (0,m, 3)-net in base 2 and a complete classification of all matrices C1,C2C2𝔽2×{C_2} \in \mathbb{F}_2^{\mathbb{N} \times \mathbb{N}} which generate a digital (0, 2)-sequence in base 2.

Keywords

  • digital nets
  • digital sequences
  • classification of generating matrices

MSC 2010

  • 11K16
  • 11K38
access type Open Access

On the Intriguing Search for Good Permutations

Published Online: 27 Mar 2020
Page range: 53 - 86

Abstract

Abstract

The intriguing search for permutations that generate generalised van der Corput sequences with exceptionally small discrepancy forms an important part of the research work of Henri Faure. On the occasion of Henri’s 80th birthday we aim to survey (some of) his contributions over the last four decades which considerably improved our understanding of one-dimensional van der Corput sequences and inspired a lot of related work. We recall and compare the different approaches in the search for generalised van der Corput sequences with low discrepancy, i.e., using a single generating permutation versus using a sequence of permutations. Throughout, we collect, sharpen and extend open questions which all stem from the extensive work of Henri and his coworkers and which will hopefully inspire more work in the future.

Keywords

  • Van der Corput sequence
  • low discrepancy
  • permutation polynomial
  • Henri Faure

MSC 2010

  • 11K38
  • 11K06
  • 11T06
  • 05A05
access type Open Access

Notes on the Distribution of Roots Modulo a Prime of a Polynomial II

Published Online: 27 Mar 2020
Page range: 87 - 104

Abstract

Abstract

Let f (x) be a monic polynomial with integer coefficients and 0 ≤ r1 ≤ ··· ≤ rn<p its roots modulo a prime p. We generalize a conjecture on the distribution of roots ri with additional congruence relations riRi mod L from the case that f has no non-trivial linear relation among roots to the case that f has a non-trivial linear relation.

Keywords

  • distribution
  • polynomial
  • roots modulo a prime

MSC 2010

  • 11K
access type Open Access

The Distributional Asymptotics Mod 1 of (logb n)

Published Online: 27 Mar 2020
Page range: 105 - 122

Abstract

Abstract

This paper studies the distributional asymptotics of the slowly changing sequence of logarithms (logb n) with b ∈ 𝕅 \ {1}. It is known that (logbn) is not uniformly distributed modulo one, and its omega limit set is composed of a family of translated exponential distributions with constant log b. An improved upper estimate (logN/N\sqrt {\log N} /N) is obtained for the rate of convergence with respect to (w. r. t.)the Kantorovich metric on the circle, compared to the general results on rates of convergence for a class of slowly changing sequences in the author’s companion in-progress work. Moreover, a sharp rate of convergence (log N/N)w. r. t. the Kantorovich metric on the interval [0, 1], is derived. As a byproduct, the rate of convergence w.r.t. the discrepancy metric (or the Kolmogorov metric) turns out to be (log N/N) as well, which verifies that an upper bound for this rate derived in [Ohkubo, Y.—Strauch, O.: Distribution of leading digits of numbers, Unif. Distrib. Theory, 11 (2016), no.1, 23–45.] is sharp.

Keywords

  • Uniformly distributed modulo one sequence
  • slowly changing sequence
  • rate of convergence
  • Kantorovich metric
  • discrepancy
  • probability measures

MSC 2010

  • 11K06
  • 11K31
  • 60B10
  • 60E20
access type Open Access

Cyclotomic Expressions for Representation Functions

Published Online: 27 Mar 2020
Page range: 123 - 140

Abstract

Abstract

Given a subset A of the natural numbers 𝕅 = {0, 1, 2, ···} (resp. of the ring 𝕑/ N𝕑 of residue classes modulo a positive integer N), we introduce certain sums of roots of unity associated with A. We study some of their properties, and we use them to obtain new expressions for the classical functions that characterize A, i.e. of the representation function, the counting function and the characteristic function of A. We also give an example of computations of the representation function using such expressions.

Keywords

  • Representation function
  • exponential sums
  • cyclotomic fields

MSC 2010

  • 11B34
  • 11B75
  • 11L03
  • 11R18
9 Articles
access type Open Access

The Sixth International Conference on Uniform Distribution Theory (UDT 2018)

Published Online: 27 Mar 2020
Page range: i - x

Abstract

Abstract

This volume contains papers originally presented or inspired by the Sixth International Conference on Uniform Distribution Theory, which was held at CIRM in Luminy, Marseilles, France, October 1–5, 2016.

access type Open Access

Higher Order Oscillation and Uniform Distribution

Published Online: 27 Mar 2020
Page range: 1 - 10

Abstract

Abstract

It is known that the Möbius function in number theory is higher order oscillating. In this paper we show that there is another kind of higher order oscillating sequences in the form (e2πiαβn g(β))n∈𝕅, for a non-decreasing twice differentiable function g with a mild condition. This follows the result we prove in this paper that for a fixed non-zero real number α and almost all real numbers β> 1 (alternatively, for a fixed real number β> 1 and almost all real numbers α) and for all real polynomials Q(x), sequences (αβng(β)+ Q(n)) n∈𝕅 are uniformly distributed modulo 1.

Keywords

  • higher order oscillating sequence
  • uniformly distributed modulo 1 (u. d. mod 1)

MSC 2010

  • Primary 11K65, 37A35
  • Secondary 37A25, 11N05
access type Open Access

Partitioning the Set of Primes to Create r-Dimensional Sequences Which are Uniformly Distributed Modulo [0, 1)r

Published Online: 27 Mar 2020
Page range: 11 - 18

Abstract

Abstract

Expanding on our previous results, we show that by partitioning the set of primes into a finite number of subsets of roughly the same size, we can create r-dimensional sequences of real numbers which are uniformly distributed modulo [0, 1)r.

Keywords

  • Uniform distribution modulo one

MSC 2010

  • 11K16
  • 11J71
access type Open Access

Distribuion of Leading Digits of Numbers II

Published Online: 27 Mar 2020
Page range: 19 - 42

Abstract

Abstract

In this paper, we study the sequence (f (pn))n≥1,where pn is the nth prime number and f is a function of a class of slowly increasing functions including f (x)=logb xr and f (x)=logb(x log x)r,where b ≥ 2 is an integer and r> 0 is a real number. We give upper bounds of the discrepancy DNi*(f(pn),g)D_{{N_i}}^*\left( {f\left( {{p_n}} \right),g} \right) for a distribution function g and a sub-sequence (Ni)i≥1 of the natural numbers. Especially for f (x)= logb xr, we obtain the effective results for an upper bound of DNi*(f(pn)g)D_{{N_i}}^*\left( {f\left( {{p_n}} \right),g} \right).

Keywords

  • prime numbers
  • Benford’s law
  • distribution function
  • discrepancy
  • regularly varying function

MSC 2010

  • 11K06
  • 11K31
  • 11K38
access type Open Access

A Complete Classification of Digital (0,m, 3)-Nets and Digital (0, 2)-Sequences in Base 2

Published Online: 27 Mar 2020
Page range: 43 - 52

Abstract

Abstract

We give a complete classification of all matrices C1,C2,C3C3𝔽2m×m{C_3} \in \mathbb{F}_2^{m \times m} which generate a digital (0,m, 3)-net in base 2 and a complete classification of all matrices C1,C2C2𝔽2×{C_2} \in \mathbb{F}_2^{\mathbb{N} \times \mathbb{N}} which generate a digital (0, 2)-sequence in base 2.

Keywords

  • digital nets
  • digital sequences
  • classification of generating matrices

MSC 2010

  • 11K16
  • 11K38
access type Open Access

On the Intriguing Search for Good Permutations

Published Online: 27 Mar 2020
Page range: 53 - 86

Abstract

Abstract

The intriguing search for permutations that generate generalised van der Corput sequences with exceptionally small discrepancy forms an important part of the research work of Henri Faure. On the occasion of Henri’s 80th birthday we aim to survey (some of) his contributions over the last four decades which considerably improved our understanding of one-dimensional van der Corput sequences and inspired a lot of related work. We recall and compare the different approaches in the search for generalised van der Corput sequences with low discrepancy, i.e., using a single generating permutation versus using a sequence of permutations. Throughout, we collect, sharpen and extend open questions which all stem from the extensive work of Henri and his coworkers and which will hopefully inspire more work in the future.

Keywords

  • Van der Corput sequence
  • low discrepancy
  • permutation polynomial
  • Henri Faure

MSC 2010

  • 11K38
  • 11K06
  • 11T06
  • 05A05
access type Open Access

Notes on the Distribution of Roots Modulo a Prime of a Polynomial II

Published Online: 27 Mar 2020
Page range: 87 - 104

Abstract

Abstract

Let f (x) be a monic polynomial with integer coefficients and 0 ≤ r1 ≤ ··· ≤ rn<p its roots modulo a prime p. We generalize a conjecture on the distribution of roots ri with additional congruence relations riRi mod L from the case that f has no non-trivial linear relation among roots to the case that f has a non-trivial linear relation.

Keywords

  • distribution
  • polynomial
  • roots modulo a prime

MSC 2010

  • 11K
access type Open Access

The Distributional Asymptotics Mod 1 of (logb n)

Published Online: 27 Mar 2020
Page range: 105 - 122

Abstract

Abstract

This paper studies the distributional asymptotics of the slowly changing sequence of logarithms (logb n) with b ∈ 𝕅 \ {1}. It is known that (logbn) is not uniformly distributed modulo one, and its omega limit set is composed of a family of translated exponential distributions with constant log b. An improved upper estimate (logN/N\sqrt {\log N} /N) is obtained for the rate of convergence with respect to (w. r. t.)the Kantorovich metric on the circle, compared to the general results on rates of convergence for a class of slowly changing sequences in the author’s companion in-progress work. Moreover, a sharp rate of convergence (log N/N)w. r. t. the Kantorovich metric on the interval [0, 1], is derived. As a byproduct, the rate of convergence w.r.t. the discrepancy metric (or the Kolmogorov metric) turns out to be (log N/N) as well, which verifies that an upper bound for this rate derived in [Ohkubo, Y.—Strauch, O.: Distribution of leading digits of numbers, Unif. Distrib. Theory, 11 (2016), no.1, 23–45.] is sharp.

Keywords

  • Uniformly distributed modulo one sequence
  • slowly changing sequence
  • rate of convergence
  • Kantorovich metric
  • discrepancy
  • probability measures

MSC 2010

  • 11K06
  • 11K31
  • 60B10
  • 60E20
access type Open Access

Cyclotomic Expressions for Representation Functions

Published Online: 27 Mar 2020
Page range: 123 - 140

Abstract

Abstract

Given a subset A of the natural numbers 𝕅 = {0, 1, 2, ···} (resp. of the ring 𝕑/ N𝕑 of residue classes modulo a positive integer N), we introduce certain sums of roots of unity associated with A. We study some of their properties, and we use them to obtain new expressions for the classical functions that characterize A, i.e. of the representation function, the counting function and the characteristic function of A. We also give an example of computations of the representation function using such expressions.

Keywords

  • Representation function
  • exponential sums
  • cyclotomic fields

MSC 2010

  • 11B34
  • 11B75
  • 11L03
  • 11R18

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