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

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

<abstract xmlns="http://www.w3.org/1999/xhtml">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 (<italic>e</italic>2πiαβn g(β))n∈𝕅, for a non-decreasing twice differentiable function <italic>g</italic> with a mild condition. This follows the result we prove in this paper that for a fixed non-zero real number <italic>α</italic> and almost all real numbers <italic>β></italic> 1 (alternatively, for a fixed real number <italic>β></italic> 1 and almost all real numbers <italic>α</italic>) and for all real polynomials <italic>Q</italic>(<italic>x</italic>), sequences (<italic>αβ</italic><italic>n</italic><italic>g</italic>(<italic>β</italic>)+ <italic>Q</italic>(<italic>n</italic>)) n∈𝕅 are uniformly distributed modulo 1.</abstract>

Higher Order Oscillation and Uniform Distribution

<abstract xmlns="http://www.w3.org/1999/xhtml">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 <italic>r</italic>-dimensional sequences of real numbers which are uniformly distributed modulo [0, 1)<italic>r</italic>.</abstract>

Département mathématiques et statistique

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

<abstract xmlns="http://www.w3.org/1999/xhtml">In this paper, we study the sequence (<italic>f</italic> (<italic>p</italic>n))n≥1,where <italic>p</italic>n is the <italic>n</italic>th prime number and <italic>f</italic> is a function of a class of slowly increasing functions including <italic>f</italic> (<italic>x</italic>)=log<italic>b x</italic><italic>r</italic> and <italic>f</italic> (<italic>x</italic>)=log<italic>b</italic>(<italic>x</italic> log <italic>x</italic>)<italic>r</italic>,where <italic>b</italic> ≥ 2 is an integer and <italic>r></italic> 0 is a real number. We give upper bounds of the discrepancy <inline-formula><alternatives><inline-graphic xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="graphic/j_udt-2019-0003_eq_001.png"></inline-graphic><math xmlns:mml="http://www.w3.org/1998/Math/MathML"><mrow><msubsup><mrow><mi>D</mi></mrow><mrow><msub><mrow><mi>N</mi></mrow><mi>i</mi></msub></mrow><mo>*</mo></msubsup><mrow><mo>(</mo><mrow><mi>f</mi><mrow><mo>(</mo><mrow><msub><mrow><mi>p</mi></mrow><mi>n</mi></msub></mrow><mo>)</mo></mrow><mo>,</mo><mi>g</mi></mrow><mo>)</mo></mrow></mrow></math><tex-math>D_{{N_i}}^*\left( {f\left( {{p_n}} \right),g} \right)</tex-math></alternatives></inline-formula> for a distribution function <italic>g</italic> and a sub-sequence (<italic>N</italic><italic>i</italic>)<italic>i</italic>≥1 of the natural numbers. Especially for <italic>f</italic> (<italic>x</italic>)= log<italic>b x</italic><italic>r</italic>, we obtain the effective results for an upper bound of <inline-formula><alternatives><inline-graphic xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="graphic/j_udt-2019-0003_eq_002.png"></inline-graphic><math xmlns:mml="http://www.w3.org/1998/Math/MathML"><mrow><msubsup><mrow><mi>D</mi></mrow><mrow><msub><mrow><mi>N</mi></mrow><mi>i</mi></msub></mrow><mo>*</mo></msubsup><mrow><mo>(</mo><mrow><mi>f</mi><mrow><mo>(</mo><mrow><msub><mrow><mi>p</mi></mrow><mi>n</mi></msub></mrow><mo>)</mo></mrow><mo></mo><mi>g</mi></mrow><mo>)</mo></mrow></mrow></math><tex-math>D_{{N_i}}^*\left( {f\left( {{p_n}} \right),g} \right)</tex-math></alternatives></inline-formula>.</abstract>

The International University of Kagoshima

Distribuion of Leading Digits of Numbers II

<abstract xmlns="http://www.w3.org/1999/xhtml">We give a complete classification of all matrices <italic>C</italic>1,<italic>C</italic>2,<italic>C</italic>3 ∈ <inline-formula><alternatives><inline-graphic xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="graphic/j_udt-2019-0004_eq_001.png"></inline-graphic><math xmlns:mml="http://www.w3.org/1998/Math/MathML"><mrow><msub><mrow><mi>C</mi></mrow><mn>3</mn></msub><mo>∈</mo><msubsup><mrow><mi>𝔽</mi></mrow><mn>2</mn><mrow><mi>m</mi><mo>×</mo><mi>m</mi></mrow></msubsup></mrow></math><tex-math>{C_3} \in \mathbb{F}_2^{m \times m}</tex-math></alternatives></inline-formula> which generate a digital (0,<italic>m,</italic> 3)-net in base 2 and a complete classification of all matrices <italic>C</italic>1,<italic>C</italic>2 ∈ <inline-formula><alternatives><inline-graphic xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="graphic/j_udt-2019-0004_eq_002.png"></inline-graphic><math xmlns:mml="http://www.w3.org/1998/Math/MathML"><mrow><msub><mrow><mi>C</mi></mrow><mn>2</mn></msub><mo>∈</mo><msubsup><mrow><mi>𝔽</mi></mrow><mn>2</mn><mrow><mi>ℕ</mi><mo>×</mo><mi>ℕ</mi></mrow></msubsup></mrow></math><tex-math>{C_2} \in \mathbb{F}_2^{\mathbb{N} \times \mathbb{N}}</tex-math></alternatives></inline-formula> which generate a digital (0, 2)-sequence in base 2.</abstract>

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

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

On the Intriguing Search for Good Permutations

<abstract xmlns="http://www.w3.org/1999/xhtml">Let <italic>f</italic> (<italic>x</italic>) be a monic polynomial with integer coefficients and 0 ≤ <italic>r</italic>1 ≤ ··· ≤ <italic>r</italic>n<italic>&lt;p</italic> its roots modulo a prime <italic>p</italic>. We generalize a conjecture on the distribution of roots <italic>r</italic><italic>i</italic> with additional congruence relations <italic>r</italic><italic>i</italic> ≡ <italic>R</italic><italic>i</italic> mod <italic>L</italic> from the case that <italic>f</italic> has no non-trivial linear relation among roots to the case that <italic>f</italic> has a non-trivial linear relation.</abstract>

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

<abstract xmlns="http://www.w3.org/1999/xhtml">This paper studies the distributional asymptotics of the slowly changing sequence of logarithms (log<italic>b n</italic>) with <italic>b</italic> ∈ 𝕅 \ {1}. It is known that (logb<italic>n</italic>) is not uniformly distributed modulo one, and its omega limit set is composed of a family of translated exponential distributions with constant log <italic>b.</italic> An improved upper estimate (<inline-formula><alternatives><inline-graphic xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="graphic/j_udt-2019-0007_eq_001.png"></inline-graphic><math xmlns:mml="http://www.w3.org/1998/Math/MathML"><mrow><msqrt><mrow><mo>log</mo><mi>N</mi></mrow></msqrt><mo>/</mo><mi>N</mi></mrow></math><tex-math>\sqrt {\log N} /N</tex-math></alternatives></inline-formula>) is obtained for the rate of convergence with respect to (w. r. t.)the <italic>Kantorovich metric</italic> 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 <italic>N/N</italic>)w. r. t. the <italic>Kantorovich metric</italic> on the interval [0, 1], is derived. As a byproduct, the rate of convergence w.r.t. the <italic>discrepancy metric</italic> (or the <italic>Kolmogorov metric</italic>) turns out to be (log <italic>N/N</italic>) as well, which verifies that an upper bound for this rate derived in [O<sc>hkubo</sc>, Y.—S<sc>trauch</sc>, O.: <italic>Distribution of leading digits of numbers</italic>, Unif. Distrib. Theory, <bold>11</bold> (2016), no.1, 23–45.] is sharp.</abstract>

The Distributional Asymptotics Mod 1 of (log<italic>b n</italic>)

<abstract xmlns="http://www.w3.org/1999/xhtml">Given a subset <italic>A</italic> of the natural numbers 𝕅 = {0, 1, 2, ···} (resp. of the ring 𝕑<italic>/ N</italic>𝕑 of residue classes modulo a positive integer <italic>N</italic>), we introduce certain sums of roots of unity associated with <italic>A</italic>. We study some of their properties, and we use them to obtain new expressions for the classical functions that characterize <italic>A</italic>, i.e. of the representation function, the counting function and the characteristic function of <italic>A</italic>. We also give an example of computations of the representation function using such expressions.</abstract>

Cyclotomic Expressions for Representation Functions

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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

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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. 

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

Uniform distribution theory

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

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

Higher Order Oscillation and Uniform Distribution

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

Distribuion of Leading Digits of Numbers II

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

On the Intriguing Search for Good Permutations

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

The Distributional Asymptotics Mod 1 of (log_b n)

Cyclotomic Expressions for Representation Functions

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