<abstract xmlns="http://www.w3.org/1999/xhtml"> This volume contains papers originally presented or inspired by the Fifth International Conference on Uniform Distribution Theory which was held in Sopron, Hungary, July 5-8, 2016.</abstract>

The Fifth International Conference on Uniform Distribution Theory (UDT 2016) Sopron, Hungary, July 5–8, 2016

<abstract xmlns="http://www.w3.org/1999/xhtml"> The notion of symmetrization, also known as Davenport’s reflection principle, is well known in the area of the discrepancy theory and quasi- Monte Carlo (QMC) integration. In this paper we consider applying a symmetrization technique to a certain class of QMC point sets called digital nets over ℤb. Although symmetrization has been recognized as a geometric technique in the multi-dimensional unit cube, we give another look at symmetrization as a geometric technique in a compact totally disconnected abelian group with dyadic arithmetic operations. Based on this observation we generalize the notion of symmetrization from base 2 to an arbitrary base b ∈ ℕ, b ≥ 2. Subsequently, we study the QMC integration error of symmetrized digital nets over ℤb in a reproducing kernel Hilbert space. The result can be applied to component-by-component construction or Korobov construction for finding good symmetrized (higher order) polynomial lattice rules which achieve high order convergence of the integration error for smooth integrands at the expense of an exponential growth of the number of points with the dimension. Moreover, we consider two-dimensional symmetrized Hammersley point sets in prime base b, and prove that the minimum Dick weight is large enough to achieve the best possible order of Lp discrepancy for all 1 ≤ p &lt; ∞.</abstract>

The Bb-adic Symmetrization of Digital Nets for Quasi-Monte Carlo Integration

<abstract xmlns="http://www.w3.org/1999/xhtml"> Zeckendorf's theorem states that every positive integer can be decomposed uniquely as a sum of nonconsecutive Fibonacci numbers. The distribution of the number of summands converges to a Gaussian, and the individual measures on gajw between summands for m € [Fn,Fn+1) converge to geometric decay for almost all m as n→ ∞. While similar results are known for many other recurrences, previous work focused on proving Gaussianity for the number of summands or the average gap measure. We derive general conditions, which are easily checked, that yield geometric decay in the individual gap measures of generalized Zerkendorf decompositions attached to many linear recurrence relations. </abstract>

Individual Gap Measures from Generalized Zeckendorf Degompositions

<abstract xmlns="http://www.w3.org/1999/xhtml"> In this paper we compute the p-adic valuation of exponential sums associated to binomials F(X) = aXd₁ + bXd₂ over Fp. In particular, its p-adic valuation is constant for a, b ∈ F∗p . As a byproduct of our results, we obtain a lower bound for the sizes of value sets of binomials over Fq.</abstract>

p-adic Valuation of Exponential Sums in One Variable Associated to Binomials

<abstract xmlns="http://www.w3.org/1999/xhtml"> In 1964 K. F. Roth initiated the study of irregularities of distribution of binary sequences relative to arithmetic progressions and since that numerous papers have been written on this subject. In the applications one needs binary sequences which are well distributed relative to arithmetic progressions, in particular, in cryptography one needs binary sequences whose short subsequences are also well-distributed relative to arithmetic progressions. Thus we introduce weighted measures of pseudorandomness of binary sequences to study this property. We study the typical and minimal values of this measure for binary sequences of a given length.</abstract>

On Irregularities of Distribution of Binary Sequences Relative to Arithmetic Progressions, I. (General Results)

<abstract xmlns="http://www.w3.org/1999/xhtml"> This paper deals with the distribution of αζn mod 1, where α ≠ 0, ζ > 1 are fixed real numbers and n runs through the positive integers. Denote by ‖·‖ the distance to the nearest integer. We investigate the case of αζn all lying in prescribed small intervals modulo 1 for all large n, with focus on the case ‖αζn‖ ≤ ɛ for small ɛ > 0. We are particularly interested in what we call cardinality gap phenomena. For example for fixed ζ > 1 and small ɛ > 0 there are at most countably many values of α such that ‖αζn‖ ≤ ɛ for all large n, whereas larger ɛ induces an uncountable set. We investigate the value of ‖ at which the gap occurs. We will pay particular attention to the case of algebraic and, more specific, rational ζ > 1. Results concerning Pisot and Salem numbers such as some contribution to Mahler’s 3/2-problem are implicitly deduced. We study similar questions for fixed α ≠ 0 as well. </abstract>

Integral Powers of Numbers in Small Intervals Modulo 1: The Cardinality Gap Phenomenon

<abstract xmlns="http://www.w3.org/1999/xhtml"> We consider strictly increasing sequences (an)n≥1 of integers and sequences of fractional parts ({anα})n≥1 where α ∈ R. We show that a small additive energy of (an)n≥1 implies that for almost all α the sequence ({anα})n≥1 has large discrepancy. We prove a general result, provide various examples, and show that the converse assertion is not necessarily true. </abstract>

Additive Energy and Irregularities of Distribution

<abstract xmlns="http://www.w3.org/1999/xhtml"> Continuing the previous paper, we give several data on the distribution of roots modulo primes of an irreducible polynomial, and based on them, we propose problems on the distribution.</abstract>

Statistical Distribution of Roots of a Polynomial Modulo Primes II

<abstract xmlns="http://www.w3.org/1999/xhtml"> The paper deals with a generalisation of uniform distribution. The analogues of Weyl’s criterion are derived.</abstract>

Uniform Distribution with Respect to Density

<abstract xmlns="http://www.w3.org/1999/xhtml"> In this article , we show that the set of Mills constants (real numbers M such that [M3ⁿ] is prime for all n ≥ 0) is the increasing limit of sets homeomorphic to the triadic Cantor’s set. More generally, for a given function ϕ and a set A of integers, we studying the Mills set Mϕ(A) = {α ∈ ℝ/ ∀n ∈ ℕ, [ϕn(α)] ∈ A} (where ϕn = ϕ∘...∘ϕ n times). We show that, under certain assumptions over ϕ and A, for all real w > infMϕ(A) the set Mϕ(A) ∩ [2, w] is homeomorphic to the triadic Cantor’s set. </abstract>

Une Propriété Topologique de Certains Ensembles de Mills

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. 