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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 11 (2016): Issue 2 (December 2016)

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

Search

10 Articles
access type Open Access

An Extremal Problem in Uniform Distribution Theory

Published Online: 13 Jan 2017
Page range: 1 - 21

Abstract

Abstract

In this paper we consider an optimization problem for Cesàro means of bivariate functions. We apply methods from uniform distribution theory, calculus of variations and ideas from the theory of optimal transport.

Keywords

  • Uniform distribution
  • copula
  • Monge-Kantorovich problem
  • dual problem
  • -convex function
  • -subdifferential

MSC 2010

  • 11K06
  • 60E05
  • 60A10
  • 49K27
access type Open Access

Tractability of Multivariate Integration Using Low-Discrepancy Sequences

Published Online: 13 Jan 2017
Page range: 23 - 43

Abstract

Abstract

We propose a notion of (t, e, s)-sequences in multiple bases, which unifies the Halton sequence and (t, s)-sequences under one roof, and obtain an upper bound of their discrepancy consisting only of the leading term. By using this upper bound, we improve the tractability results currently known for the Halton sequence, the Niederreiter sequence, the Sobol’ sequence, and the generalized Faure sequence, and also give tractability results for the Xing-Niederreiter sequence and the Hofer-Niederreiter sequence, for which no results have been known so far.

Keywords

  • discrepancy
  • low-discrepancy sequences
  • multivariate integration
  • tractability
  • (, , )-sequences

MSC 2010

  • Primary 11K38
  • Secondary 11K06
access type Open Access

On the Gaussian Limiting Distribution of Lattice Points in a Parallelepiped

Published Online: 13 Jan 2017
Page range: 45 - 89

Abstract

Abstract

Let Γ ⊂ ℝs be a lattice obtained from a module in a totally real algebraic number field. Let ℛ(θ, N) be the error term in the lattice point problem for the parallelepiped [−θ1N1, θ1N1] × . . . × [−θs Ns, θs Ns]. In this paper, we prove that ℛ(θ, N)(ℛ, N) has a Gaussian limiting distribution as N→∞, where θ = (θ1, . . . , θs) is a uniformly distributed random variable in [0, 1]s, N = N1 . . . . Ns and σ(ℛ, N) ≍ (log N)(s−1)/2. We obtain also a similar result for the low discrepancy sequence corresponding to Γ. The main tool is the S-unit theorem.

Keywords

  • lattice points problem
  • low discrepancy sequences
  • totally real algebraic number field
  • central limit theorem

MSC 2010

  • Primary 11P21, 11K38, 11R80
  • Secondary 60F05
access type Open Access

Sofic Measures and Densities of Level Sets

Published Online: 13 Jan 2017
Page range: 91 - 124

Abstract

Abstract

The Bernoulli convolution associated to the real β > 1 and the probability vector (p0, . . . , pd−1) is a probability measure ηβ,p on ℝ, solution of the self-similarity relation η=k=0d1pkηSk1$\eta = \sum\nolimits_{k = 0}^{d - 1} {p_k \cdot \eta \circ S_k^{ - 1} } $, where Sk(x)=x+kβ$S_k (x) = {{x + k} \over \beta }$. If β is an integer or a Pisot algebraic number with finite Rényi expansion, ηβ,p is sofic and a Markov chain is naturally associated. If β = b ∈ ℕ and p0==pd1=1d$p_0 = \cdots = p_{d - 1} = {1 \over d}$, the study of ηb,p is close to the study of the order of growth of the number of representations in base b with digits in {0, 1, . . . , d − 1}. In the case b = 2 and d = 3 it has also something to do with the metric properties of the continued fractions.

Keywords

  • partition function
  • numeration system
  • radix expansion
  • Pisot scale
  • Bernoulli convolutions

MSC 2010

  • 11P99
  • 28XX
  • 15B48
access type Open Access

On the Constant in the Average Digit Sum for a Recurrence-Based Numeration

Published Online: 13 Jan 2017
Page range: 125 - 150

Abstract

Abstract

Given an integral, increasing, linear-recurrent sequence A with initial term 1, the greedy algorithm may be used on the terms of A to represent all positive integers. For large classes of recurrences, the average digit sum is known to equal cA log n+O(1), where cA is a positive constant that depends on A. This asymptotic result is re-proved with an elementary approach for a class of special recurrences larger than, or distinct from, that of former papers. The focus is on the constants cA for which, among other items, explicit formulas are provided and minimal values are found, or conjectured, for all special recurrences up to a certain order.

Keywords

  • numeration
  • digit sum
  • average
  • recurrence

MSC 2010

  • 11A63
  • 11B37
access type Open Access

Spatial Equidistribution of Binomial Coefficients Modulo Prime Powers

Published Online: 13 Jan 2017
Page range: 151 - 161

Abstract

Abstract

The spatial distribution of binomial coefficients in residue classes modulo prime powers is studied. It is proved inter alia that empirical distribution of the points (k,m)pm with 0 ≤ kn < pm and (nk)a(modp)s$\left( {\matrix{n \cr k \cr } } \right) \equiv a\left( {\bmod \;p} \right)^s $ (for (a, p) = 1) for m→∞ tends to the Hausdorff measure on the “p-adic Sierpiński gasket”, a fractals studied earlier by von Haeseler, Peitgen, and Skordev.

Keywords

  • Binomial coefficients
  • equidistribution

MSC 2010

  • Primary 11B65
  • Secondary 11A63
access type Open Access

Yet Another Footnote to the Least Non Zero Digit of n! in Base 12

Published Online: 13 Jan 2017
Page range: 163 - 167

Abstract

Abstract

We continue the study, initiated with Imre Ruzsa, of the last non zero digit ℓ12(n!) of n! in base 12, showing that for any a ∈ {3, 4, 6, 8, 9}, the set of those integers n for which ℓ12(n!) = a is not 3-automatic.

Keywords

  • radix representation
  • automatic sequences
  • significant digit
  • factorial
  • base 12

MSC 2010

  • 11A63
  • 11B85
access type Open Access

Les Huit Premiers Travaux de Pierre Liardet

Published Online: 13 Jan 2017
Page range: 169 - 177

Abstract

Abstract

Ce texte est une présentation résumée des huit premiers travaux de Pierre Liardet. Il reprend l’exposé donné à l’Université de Savoie Mont Blanc (Le Bourget-du-Lac) lors du colloque Théorie des Nombres, Systèmes de Numération, Théorie Ergodique les 28 et 29 septembre 2015, un colloque inspiré par les mathématiques de Pierre Liardet.

Le premier texte publié par Pierre Liardet l’a été en 1969 dans les Comptes Rendus de l’Académie des Sciences de Paris, il est intitulé “Transformations rationnelles laissant stables certains ensembles de nombres algébriques”, avec Madeleine Ventadoux comme coauteur. Ils étendent des résultats de Gérard Rauzy.

Dans la lignée de ces premiers travaux, il s’est attaqué à une conjecture de Władysław Narkiewicz sur les transformations polynomiales et rationnelles. En 1976, avec Ken K. Kubota, il a finalement réfuté cette conjecture.

Il a ensuite obtenu des résultats précurseurs sur une conjecture de Serge Lang, qui sont très souvent cités. Nous donnerons un bref survol des résultats qui ont suivi cette percée significative.

Keywords

  • Transformations polynomiales
  • transformations rationnelles
  • stabilité d’ensembles algébriques
  • conjecture de Narkiewicz
  • nombres de Pisot - Vijayaraghavan
  • propriété de Northcott
  • topologies hilbertiennes
  • équations diophantiennes exponentielles
  • points rationnels sur un groupe algébrique
  • géométrie diophantienne
  • conjecture de Lang
  • conjecture de Mordell-Lang
  • intersections exceptionnelles

MSC 2010

  • Primary 11–03, 12–03, 14–03
  • Secundary 11D41, 11R04, 11R06, 11R09, 11R18, 11R99, 12D10, 12D99, 12E05, 12E99, 12F05, 12F99, 14E05, 14G05
access type Open Access

Convergence Results for r-Iterated Means of the Denominators of the Lüroth Series

Published Online: 13 Jan 2017
Page range: 179 - 203

Abstract

Abstract

In the present paper we extend two classic asymptotic results concerning convergence in probability and convergence in distribution for the denominators of the Lüroth series and obtain new theorems concerning the same two kinds of convergence for the r-iterated arithmetic means of such denominators. These results are extended to r-iterated weighted means.

Keywords

  • Lüroth series
  • convergence in probability
  • convergence in distribution
  • iterated arithmetic means
  • iterated weighted means

MSC 2010

  • Primary 60F05
  • Secondary 11K55
access type Open Access

On a Golay-Shapiro-Like Sequence

Published Online: 13 Jan 2017
Page range: 205 - 210

Abstract

Abstract

A recent paper by P. Lafrance, N. Rampersad, and R. Yee studies the sequence of occurrences of 10 as a scattered subsequence in the binary expansion of integers. They prove in particular that the summatory function of this sequence has the “root N” property, analogously to the summatory function of the Golay-Shapiro sequence. We prove here that the root N property does not hold if we twist the sequence by powers of a complex number of modulus one, hence showing a fundamental difference with the Golay-Shapiro sequence.

Keywords

  • binary expansion
  • digital sequence
  • Rudin-Shapiro sequence
  • summatory function

MSC 2010

  • Primary: 11K16
  • Secondary: 11B85
10 Articles
access type Open Access

An Extremal Problem in Uniform Distribution Theory

Published Online: 13 Jan 2017
Page range: 1 - 21

Abstract

Abstract

In this paper we consider an optimization problem for Cesàro means of bivariate functions. We apply methods from uniform distribution theory, calculus of variations and ideas from the theory of optimal transport.

Keywords

  • Uniform distribution
  • copula
  • Monge-Kantorovich problem
  • dual problem
  • -convex function
  • -subdifferential

MSC 2010

  • 11K06
  • 60E05
  • 60A10
  • 49K27
access type Open Access

Tractability of Multivariate Integration Using Low-Discrepancy Sequences

Published Online: 13 Jan 2017
Page range: 23 - 43

Abstract

Abstract

We propose a notion of (t, e, s)-sequences in multiple bases, which unifies the Halton sequence and (t, s)-sequences under one roof, and obtain an upper bound of their discrepancy consisting only of the leading term. By using this upper bound, we improve the tractability results currently known for the Halton sequence, the Niederreiter sequence, the Sobol’ sequence, and the generalized Faure sequence, and also give tractability results for the Xing-Niederreiter sequence and the Hofer-Niederreiter sequence, for which no results have been known so far.

Keywords

  • discrepancy
  • low-discrepancy sequences
  • multivariate integration
  • tractability
  • (, , )-sequences

MSC 2010

  • Primary 11K38
  • Secondary 11K06
access type Open Access

On the Gaussian Limiting Distribution of Lattice Points in a Parallelepiped

Published Online: 13 Jan 2017
Page range: 45 - 89

Abstract

Abstract

Let Γ ⊂ ℝs be a lattice obtained from a module in a totally real algebraic number field. Let ℛ(θ, N) be the error term in the lattice point problem for the parallelepiped [−θ1N1, θ1N1] × . . . × [−θs Ns, θs Ns]. In this paper, we prove that ℛ(θ, N)(ℛ, N) has a Gaussian limiting distribution as N→∞, where θ = (θ1, . . . , θs) is a uniformly distributed random variable in [0, 1]s, N = N1 . . . . Ns and σ(ℛ, N) ≍ (log N)(s−1)/2. We obtain also a similar result for the low discrepancy sequence corresponding to Γ. The main tool is the S-unit theorem.

Keywords

  • lattice points problem
  • low discrepancy sequences
  • totally real algebraic number field
  • central limit theorem

MSC 2010

  • Primary 11P21, 11K38, 11R80
  • Secondary 60F05
access type Open Access

Sofic Measures and Densities of Level Sets

Published Online: 13 Jan 2017
Page range: 91 - 124

Abstract

Abstract

The Bernoulli convolution associated to the real β > 1 and the probability vector (p0, . . . , pd−1) is a probability measure ηβ,p on ℝ, solution of the self-similarity relation η=k=0d1pkηSk1$\eta = \sum\nolimits_{k = 0}^{d - 1} {p_k \cdot \eta \circ S_k^{ - 1} } $, where Sk(x)=x+kβ$S_k (x) = {{x + k} \over \beta }$. If β is an integer or a Pisot algebraic number with finite Rényi expansion, ηβ,p is sofic and a Markov chain is naturally associated. If β = b ∈ ℕ and p0==pd1=1d$p_0 = \cdots = p_{d - 1} = {1 \over d}$, the study of ηb,p is close to the study of the order of growth of the number of representations in base b with digits in {0, 1, . . . , d − 1}. In the case b = 2 and d = 3 it has also something to do with the metric properties of the continued fractions.

Keywords

  • partition function
  • numeration system
  • radix expansion
  • Pisot scale
  • Bernoulli convolutions

MSC 2010

  • 11P99
  • 28XX
  • 15B48
access type Open Access

On the Constant in the Average Digit Sum for a Recurrence-Based Numeration

Published Online: 13 Jan 2017
Page range: 125 - 150

Abstract

Abstract

Given an integral, increasing, linear-recurrent sequence A with initial term 1, the greedy algorithm may be used on the terms of A to represent all positive integers. For large classes of recurrences, the average digit sum is known to equal cA log n+O(1), where cA is a positive constant that depends on A. This asymptotic result is re-proved with an elementary approach for a class of special recurrences larger than, or distinct from, that of former papers. The focus is on the constants cA for which, among other items, explicit formulas are provided and minimal values are found, or conjectured, for all special recurrences up to a certain order.

Keywords

  • numeration
  • digit sum
  • average
  • recurrence

MSC 2010

  • 11A63
  • 11B37
access type Open Access

Spatial Equidistribution of Binomial Coefficients Modulo Prime Powers

Published Online: 13 Jan 2017
Page range: 151 - 161

Abstract

Abstract

The spatial distribution of binomial coefficients in residue classes modulo prime powers is studied. It is proved inter alia that empirical distribution of the points (k,m)pm with 0 ≤ kn < pm and (nk)a(modp)s$\left( {\matrix{n \cr k \cr } } \right) \equiv a\left( {\bmod \;p} \right)^s $ (for (a, p) = 1) for m→∞ tends to the Hausdorff measure on the “p-adic Sierpiński gasket”, a fractals studied earlier by von Haeseler, Peitgen, and Skordev.

Keywords

  • Binomial coefficients
  • equidistribution

MSC 2010

  • Primary 11B65
  • Secondary 11A63
access type Open Access

Yet Another Footnote to the Least Non Zero Digit of n! in Base 12

Published Online: 13 Jan 2017
Page range: 163 - 167

Abstract

Abstract

We continue the study, initiated with Imre Ruzsa, of the last non zero digit ℓ12(n!) of n! in base 12, showing that for any a ∈ {3, 4, 6, 8, 9}, the set of those integers n for which ℓ12(n!) = a is not 3-automatic.

Keywords

  • radix representation
  • automatic sequences
  • significant digit
  • factorial
  • base 12

MSC 2010

  • 11A63
  • 11B85
access type Open Access

Les Huit Premiers Travaux de Pierre Liardet

Published Online: 13 Jan 2017
Page range: 169 - 177

Abstract

Abstract

Ce texte est une présentation résumée des huit premiers travaux de Pierre Liardet. Il reprend l’exposé donné à l’Université de Savoie Mont Blanc (Le Bourget-du-Lac) lors du colloque Théorie des Nombres, Systèmes de Numération, Théorie Ergodique les 28 et 29 septembre 2015, un colloque inspiré par les mathématiques de Pierre Liardet.

Le premier texte publié par Pierre Liardet l’a été en 1969 dans les Comptes Rendus de l’Académie des Sciences de Paris, il est intitulé “Transformations rationnelles laissant stables certains ensembles de nombres algébriques”, avec Madeleine Ventadoux comme coauteur. Ils étendent des résultats de Gérard Rauzy.

Dans la lignée de ces premiers travaux, il s’est attaqué à une conjecture de Władysław Narkiewicz sur les transformations polynomiales et rationnelles. En 1976, avec Ken K. Kubota, il a finalement réfuté cette conjecture.

Il a ensuite obtenu des résultats précurseurs sur une conjecture de Serge Lang, qui sont très souvent cités. Nous donnerons un bref survol des résultats qui ont suivi cette percée significative.

Keywords

  • Transformations polynomiales
  • transformations rationnelles
  • stabilité d’ensembles algébriques
  • conjecture de Narkiewicz
  • nombres de Pisot - Vijayaraghavan
  • propriété de Northcott
  • topologies hilbertiennes
  • équations diophantiennes exponentielles
  • points rationnels sur un groupe algébrique
  • géométrie diophantienne
  • conjecture de Lang
  • conjecture de Mordell-Lang
  • intersections exceptionnelles

MSC 2010

  • Primary 11–03, 12–03, 14–03
  • Secundary 11D41, 11R04, 11R06, 11R09, 11R18, 11R99, 12D10, 12D99, 12E05, 12E99, 12F05, 12F99, 14E05, 14G05
access type Open Access

Convergence Results for r-Iterated Means of the Denominators of the Lüroth Series

Published Online: 13 Jan 2017
Page range: 179 - 203

Abstract

Abstract

In the present paper we extend two classic asymptotic results concerning convergence in probability and convergence in distribution for the denominators of the Lüroth series and obtain new theorems concerning the same two kinds of convergence for the r-iterated arithmetic means of such denominators. These results are extended to r-iterated weighted means.

Keywords

  • Lüroth series
  • convergence in probability
  • convergence in distribution
  • iterated arithmetic means
  • iterated weighted means

MSC 2010

  • Primary 60F05
  • Secondary 11K55
access type Open Access

On a Golay-Shapiro-Like Sequence

Published Online: 13 Jan 2017
Page range: 205 - 210

Abstract

Abstract

A recent paper by P. Lafrance, N. Rampersad, and R. Yee studies the sequence of occurrences of 10 as a scattered subsequence in the binary expansion of integers. They prove in particular that the summatory function of this sequence has the “root N” property, analogously to the summatory function of the Golay-Shapiro sequence. We prove here that the root N property does not hold if we twist the sequence by powers of a complex number of modulus one, hence showing a fundamental difference with the Golay-Shapiro sequence.

Keywords

  • binary expansion
  • digital sequence
  • Rudin-Shapiro sequence
  • summatory function

MSC 2010

  • Primary: 11K16
  • Secondary: 11B85

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