<abstract xmlns="http://www.w3.org/1999/xhtml">In Part I of this paper we studied the irregularities of distribution of binary sequences relative to short arithmetic progressions. First we introduced a quantitative measure for this property. Then we studied the typical and minimal values of this measure for binary sequences of a given length. In this paper our goal is to give constructive bounds for these minimal values.</abstract>

On Irregularities of Distribution of Binary Sequences Relative to Arithmetic Progressions, II (Constructive Bounds)

<abstract xmlns="http://www.w3.org/1999/xhtml">Quantization for a probability distribution refers to the idea of estimating a given probability by a discrete probability supported by a finite number of points. In this paper, firstly a general approach to this process is outlined using independent random variables and ergodic maps; these give asymptotically the optimal sets of <italic>n</italic>-means and the <italic>n</italic>th quantization errors for all positive integers <italic>n</italic>. Secondly two piecewise uniform distributions are considered on R: one with infinite number of pieces and one with finite number of pieces. For these two probability measures, we describe the optimal sets of <italic>n</italic>-means and the <italic>n</italic>th quantization errors for all <italic>n</italic> ∈ N. It is seen that for a uniform distribution with infinite number of pieces to determine the optimal sets of <italic>n</italic>-means for <italic>n ≥</italic> 2 one needs to know an optimal set of (<italic>n −</italic> 1)-means, but for a uniform distribution with finite number of pieces one can directly determine the optimal sets of <italic>n</italic>-means and the <italic>n</italic>th quantization errors for all <italic>n</italic> ∈ N.</abstract>

Indiana University-Purdue University Indianapolis

Optimal Quantization for Piecewise Uniform Distributions

<abstract xmlns="http://www.w3.org/1999/xhtml">Let <italic>dN</italic> = <italic>NDN</italic> (<italic>ω</italic>) be the discrepancy of the van der Corput sequence in base 2. We improve on the known bounds for the number of indices <italic>N</italic> such that <italic>dN ≤</italic> log <italic>N/</italic>100. Moreover, we show that the summatory function of <italic>dN</italic> satisfies an exact formula involving a 1-periodic, continuous function. Finally, we give a new proof of the fact that <italic>dN</italic> is invariant under digit reversal in base 2.</abstract>

Discrepancy Results for The Van Der Corput Sequence

<abstract xmlns="http://www.w3.org/1999/xhtml">We study sets of bounded remainder for the billiard on the unit square. In particular, we note that every convex set S whose boundary is twice continuously differentiable with positive curvature at every point, is a bounded remainder set for almost all starting angles <italic>a</italic> and every starting point x. We show that this assertion for a large class of sets does not hold for <italic>all</italic> irrational starting angles <italic>α.</italic></abstract>

Sets of Bounded Remainder for The Billiard on A Square

<abstract xmlns="http://www.w3.org/1999/xhtml">This paper addresses the question whether the <italic>LS</italic>-sequences constructed in [Car12] yield indeed a new family of low-discrepancy sequences. While it is well known that the case <italic>S</italic> = 0 corresponds to van der Corput sequences, we prove here that the case <italic>S</italic> = 1 can be traced back to symmetrized Kronecker sequences and moreover that for <italic>S ≥</italic> 2 none of these two types occurs anymore. In addition, our approach allows for an improved discrepancy bound for <italic>S</italic> = 1 and <italic>L</italic> arbitrary.</abstract>

On The Classification of <italic>Ls</italic>-Sequences

<abstract xmlns="http://www.w3.org/1999/xhtml">For a function <italic>f</italic> satisfying <italic>f</italic> (<italic>x</italic>) = <italic>o</italic>((log <italic>x</italic>) <italic>K</italic>), <italic>K ></italic> 0, and a sequence of numbers (<italic>qn</italic>) <italic>n</italic>, we prove by assuming several conditions on <italic>f</italic> that the sequence (<italic>αf</italic> (<italic>qn</italic>)) <italic>n≥n</italic>0 is uniformly distributed modulo one for any nonzero real number <italic>α</italic>. This generalises some former results due to Too, Goto and Kano where instead of (<italic>qn</italic>) <italic>n</italic> the sequence of primes was considered.</abstract>

Log-Like Functions and Uniform Distribution Modulo One

<abstract xmlns="http://www.w3.org/1999/xhtml">The goal of this overview article is to give a tangible presentation of the breakthrough works in discrepancy theory [3, 5] by M. B. Levin. These works provide proofs for the exact lower discrepancy bounds of Halton’s sequence and a certain class of (<italic>t, s</italic>)-sequences. Our survey aims at highlighting the major ideas of the proofs and we discuss further implications of the employed methods. Moreover, we derive extensions of Levin’s results.</abstract>

On M. B. Levin’s Proofs for The Exact Lower Discrepancy Bounds of Special Sequences and Point Sets (A Survey)

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. 