<abstract xmlns="http://www.w3.org/1999/xhtml">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.</abstract>

maria-rita.iaco@liafa.univ-paris-diderot.fr

Institute of Information, Automation and Mathematics, Faculty of Food Technology, Slovak Technical University, Radlinského 12, SK-812 37 Bratislava,

IRIF (LIAFA), Bât. Sophie Germain, Univ. Paris Diderot, Paris 7, 8 Place Aurélie Nemours, F-75205 Paris Cedex 13,

Department of Mathematics, Slovak Academy of Sciences, Štefánikova 49, SK-814 73 Bratislava,

Institute for Analysis and Computational Number Theory, Graz University of Technology, Kopernikusgasse 24, A-8010 Graz,

Institute for Analysis and Computational Number Theory, Graz University of Technology, Steyrergasse 30, A-8010 Graz,

An Extremal Problem in Uniform Distribution Theory

<abstract xmlns="http://www.w3.org/1999/xhtml">We propose a notion of (<bold>t</bold>, <bold>e</bold>, <italic>s</italic>)-sequences in multiple bases, which unifies the Halton sequence and (<italic>t, s</italic>)-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.</abstract>

Tractability of Multivariate Integration Using Low-Discrepancy Sequences

<abstract xmlns="http://www.w3.org/1999/xhtml">Let Γ ⊂ ℝ<italic>s</italic> be a lattice obtained from a module in a totally real algebraic number field. Let ℛ(<bold><italic>θ</italic></bold>, <bold>N</bold>) be the error term in the lattice point problem for the parallelepiped [−<italic>θ</italic>1<italic>N</italic>1, <italic>θ</italic>1<italic>N</italic>1] × . . . × [−<italic>θs Ns</italic>, <italic>θs Ns</italic>]. In this paper, we prove that ℛ(<bold><italic>θ</italic></bold>, <bold>N</bold>)<italic>/σ</italic>(ℛ, <bold>N</bold>) has a Gaussian limiting distribution as <italic>N</italic>→∞, where <bold><italic>θ</italic></bold> = (<italic>θ</italic>1, . . . , <italic>θs</italic>) is a uniformly distributed random variable in [0, 1]<italic>s</italic>, <italic>N</italic> = <italic>N</italic>1 . . . . <italic>Ns</italic> and <italic>σ</italic>(ℛ, <bold>N</bold>) ≍ (log <italic>N</italic>)(<italic>s</italic>−1)/2. We obtain also a similar result for the low discrepancy sequence corresponding to Γ. The main tool is the <italic>S</italic>-unit theorem.</abstract>

On the Gaussian Limiting Distribution of Lattice Points in a Parallelepiped

<abstract xmlns="http://www.w3.org/1999/xhtml">The Bernoulli convolution associated to the real <italic>β</italic> > 1 and the probability vector (<italic>p</italic>0, . . . , <italic>pd</italic>−1) is a probability measure <italic>ηβ,p</italic> on ℝ, solution of the self-similarity relation <inline-formula><alternatives><inline-graphic xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="graphic/j_udt-2016-0015_eq_001.png"></inline-graphic><math xmlns:mml="http://www.w3.org/1998/Math/MathML" display="inline"><mi>η</mi><mo>=</mo><mstyle displaystyle="true"><msubsup><mo>∑</mo><mrow><mi>k</mi><mo>=</mo><mn>0</mn></mrow><mrow><mi>d</mi><mo>−</mo><mn>1</mn></mrow></msubsup><mrow><msub><mi>p</mi><mi>k</mi></msub><mo>⋅</mo><mi>η</mi><mi>○</mi><msubsup><mi>S</mi><mi>k</mi><mrow><mo>−</mo><mn>1</mn></mrow></msubsup></mrow></mstyle></math><tex-math>$\eta = \sum\nolimits_{k = 0}^{d - 1} {p_k \cdot \eta \circ S_k^{ - 1} } $</tex-math></alternatives></inline-formula>, where <inline-formula><alternatives><inline-graphic xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="graphic/j_udt-2016-0015_eq_002.png"></inline-graphic><math xmlns:mml="http://www.w3.org/1998/Math/MathML" display="inline"><msub><mi>S</mi><mi>k</mi></msub><mo stretchy="false">(</mo><mi>x</mi><mo stretchy="false">)</mo><mo>=</mo><mfrac><mrow><mi>x</mi><mo>+</mo><mi>k</mi></mrow><mi>β</mi></mfrac></math><tex-math>$S_k (x) = {{x + k} \over \beta }$</tex-math></alternatives></inline-formula>. If <italic>β</italic> is an integer or a Pisot algebraic number with finite Rényi expansion, <italic>ηβ,p</italic> is sofic and a Markov chain is naturally associated. If <italic>β</italic> = <italic>b</italic> ∈ ℕ and <inline-formula><alternatives><inline-graphic xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="graphic/j_udt-2016-0015_eq_003.png"></inline-graphic><math xmlns:mml="http://www.w3.org/1998/Math/MathML" display="inline"><msub><mi>p</mi><mn>0</mn></msub><mo>=</mo><mo>⋯</mo><mo>=</mo><msub><mi>p</mi><mrow><mi>d</mi><mo>−</mo><mn>1</mn></mrow></msub><mo>=</mo><mfrac><mn>1</mn><mi>d</mi></mfrac></math><tex-math>$p_0 = \cdots = p_{d - 1} = {1 \over d}$</tex-math></alternatives></inline-formula>, the study of <italic>ηb,p</italic> is close to the study of the order of growth of the number of representations in base <italic>b</italic> with digits in {0, 1, . . . , <italic>d</italic> − 1}. In the case <italic>b</italic> = 2 and <italic>d</italic> = 3 it has also something to do with the metric properties of the continued fractions.</abstract>

Sofic Measures and Densities of Level Sets

<abstract xmlns="http://www.w3.org/1999/xhtml">Given an integral, increasing, linear-recurrent sequence <italic>A</italic> with initial term 1, the greedy algorithm may be used on the terms of <italic>A</italic> to represent all positive integers. For large classes of recurrences, the average digit sum is known to equal <italic>cA</italic> log <italic>n</italic>+<italic>O</italic>(1), where <italic>cA</italic> is a positive constant that depends on <italic>A</italic>. 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 <italic>cA</italic> 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.</abstract>

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

<abstract xmlns="http://www.w3.org/1999/xhtml">The spatial distribution of binomial coefficients in residue classes modulo prime powers is studied. It is proved <italic>inter alia</italic> that empirical distribution of the points (<italic>k,m</italic>)<italic>p</italic>−<italic>m</italic> with 0 ≤ <italic>k</italic> ≤ <italic>n &lt; pm</italic> and <inline-formula><alternatives><inline-graphic xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="graphic/j_udt-2016-0017_eq_001.png"></inline-graphic><math xmlns:mml="http://www.w3.org/1998/Math/MathML" display="inline"><mrow><mo>(</mo><mrow><mtable><mtr><mtd><mi>n</mi></mtd></mtr><mtr><mtd><mi>k</mi></mtd></mtr></mtable></mrow><mo>)</mo></mrow><mo>≡</mo><mi>a</mi><mo> </mo><msup><mrow><mo>(</mo><mrow><mi>mod</mi><mo>⁡</mo><mtext> </mtext><mi>p</mi></mrow><mo>)</mo></mrow><mi>s</mi></msup></math><tex-math>$\left( {\matrix{n \cr k \cr } } \right) \equiv a\left( {\bmod \;p} \right)^s $</tex-math></alternatives></inline-formula> (for (<italic>a, p</italic>) = 1) for <italic>m</italic>→∞ tends to the Hausdorff measure on the “<italic>p</italic>-adic Sierpiński gasket”, a fractals studied earlier by von Haeseler, Peitgen, and Skordev.</abstract>

Spatial Equidistribution of Binomial Coefficients Modulo Prime Powers

<abstract xmlns="http://www.w3.org/1999/xhtml">We continue the study, initiated with Imre Ruzsa, of the last non zero digit ℓ12(<italic>n</italic>!) of <italic>n</italic>! in base 12, showing that for any <italic>a</italic> ∈ {3, 4, 6, 8, 9}, the set of those integers <italic>n</italic> for which ℓ12(<italic>n</italic>!) = <italic>a</italic> is not 3-automatic.</abstract>

jean-marc.deshouillers@math.u-bordeaux.fr

Yet Another Footnote to <italic>the Least Non Zero Digit of n</italic>! <italic>in Base</italic> 12

<abstract xmlns="http://www.w3.org/1999/xhtml">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 <italic>Théorie des Nombres, Systèmes de Numération, Théorie Ergodique</italic> 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.</abstract>

Les Huit Premiers Travaux de Pierre Liardet

<abstract xmlns="http://www.w3.org/1999/xhtml">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 <italic>r</italic>-iterated arithmetic means of such denominators. These results are extended to <italic>r</italic>-iterated weighted means.</abstract>

Convergence Results for <italic>r</italic>-Iterated Means of the Denominators of the Lüroth Series

<abstract xmlns="http://www.w3.org/1999/xhtml">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 <italic>N</italic>” property, analogously to the summatory function of the Golay-Shapiro sequence. We prove here that the root <italic>N</italic> 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.</abstract>

On a Golay-Shapiro-Like Sequence

AHEAD OF PRINT

Volume 18 (2023): Issue 1 (July 2023)

Volume 17 (2022): Issue 2 (December 2022)

Volume 17 (2022): Issue 1 (May 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)

Uniform distribution theory

 The journal is published by:The BOKU-University of Natural Resources and Applied Life Sciences (Vienna, Austria) and Institute of Mathematics of the Slovak Academy of Sciences, (Bratislava, Slovakia)  The aim of this journal is to publish original research papers on various aspects of uniform distribution theory, especially theoretical and computational aspects of combinatorial, diophantine and probabilistic number theory. In particular we welcome submissions on the following topics:   Distribution of one dimensional and multidimensional sequences.  Distribution of binary sequences.  Distribution of integer sequences and sequences from groups and  Distribution problems in finite abstract sets.  Continuously uniform distributions.  Discrepancies.  Pseudo-random number generators.  Quasi-Monte Carlo integration.  Quasi-Monte Carlo methods in financial mathematics.  Theory of densities.  Theory of distribution functions of sequences.  Distribution of integer points in large domains.  Spectral properties of sequences.  Dynamics emerging from sequences.  Number theoretic ciphers and codes.  Combinatorial number theory.  Trigonometric sums.  Diophantine approximations and diophantine equations.  Continued fractions.  Quantum chaos.  In terms of the AMS subject classification, the main subject areas supported are:11K, 11J, 11B and 11L.  Company Registration Number: 00166791; ISSN 1336–913X; EV 4556/12; math.boku.ac.at/udt  Archiving  Sciendo archives the contents of this journal in Portico - digital long-term preservation service of scholarly books, journals and collections.  Plagiarism Policy  The editorial board is participating in a growing community of Similarity Check System's users in order to ensure that the content published is original and trustworthy. Similarity Check is a medium that allows for comprehensive manuscripts screening, aimed to eliminate plagiarism and provide a high standard and quality peer-review process. 