<abstract xmlns="http://www.w3.org/1999/xhtml">We intend to unroll the surprizing properties of the Thue-Morse sequence with a harmonic analysis point of view, and mention in passing some related open questions.</abstract>

Questions Around the Thue-Morse Sequence

<abstract xmlns="http://www.w3.org/1999/xhtml">This paper concerns the problem of determining or estimating the maximal upper density of the sets of nonnegative integers <italic>S</italic> whose elements do not differ by an element of a given set <italic>M</italic> of positive integers. We find some exact values and some bounds for the maximal density when the elements of <italic>M</italic> are generalized Fibonacci numbers of odd order. The generalized Fibonacci sequence of order <italic>r</italic> is a generalization of the well known Fibonacci sequence, where instead of starting with two predetermined terms, we start with <italic>r</italic> predetermined terms and each term afterwards is the sum of <italic>r</italic> preceding terms. We also derive some new properties of the generalized Fibonacci sequence of order <italic>r</italic>. Furthermore, we discuss some related coloring parameters of distance graphs generated by the set <italic>M</italic>.</abstract>

Department of Mathematics, Indian Institute of Technology Patna

Department of Mathematics, Indian Institute of Technology Roorkee

Motzkin’s Maximal Density and Related Chromatic Numbers

<abstract xmlns="http://www.w3.org/1999/xhtml">A permuted van der Corput sequence <inline-formula><alternatives><inline-graphic xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="graphic/j_udt-2018-0003_eq_001.png"></inline-graphic><math xmlns:mml="http://www.w3.org/1998/Math/MathML"><msubsup><mi>S</mi><mi>b</mi><mi>σ</mi></msubsup></math><tex-math>$S_b^\sigma$</tex-math></alternatives></inline-formula> in base <italic>b</italic> is a one-dimensional, infinite sequence of real numbers in the interval [0, 1), generation of which involves a permutation σ of the set {0, 1,..., <italic>b −</italic> 1}. These sequences are known to have low discrepancy DN, i.e. <inline-formula><alternatives><inline-graphic xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="graphic/j_udt-2018-0003_eq_002.png"></inline-graphic><math xmlns:mml="http://www.w3.org/1998/Math/MathML"><mi>t</mi><mrow><mo>(</mo><mrow><msubsup><mi>S</mi><mi>b</mi><mi>σ</mi></msubsup></mrow><mo>)</mo></mrow><mo>:</mo><mo>=</mo><mtext>lim</mtext><mi> </mi><msub><mtext>sup</mtext><mrow><mi>N</mi><mo>→</mo><mo>∞</mo></mrow></msub><msub><mi>D</mi><mi>N</mi></msub><mrow><mo>(</mo><mrow><msubsup><mi>S</mi><mi>b</mi><mi>σ</mi></msubsup></mrow><mo>)</mo></mrow><mo>/</mo><mtext>log</mtext><mi> </mi><mi>N</mi></math><tex-math>$t\left({S_b^\sigma } \right): = {\rm{lim}}\,{\rm{sup}}_{N \to \infty } D_N \left({S_b^\sigma } \right)/{\rm{log}}\,N$</tex-math></alternatives></inline-formula> is finite. Restricting to prime bases <italic>p</italic> we present two families of generating permutations. We describe their elements as polynomials over finite fields 𝔽<italic>p</italic> in an explicit way. We use this characterization to obtain bounds for <inline-formula><alternatives><inline-graphic xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="graphic/j_udt-2018-0003_eq_003.png"></inline-graphic><math xmlns:mml="http://www.w3.org/1998/Math/MathML"><mi>t</mi><mrow><mo>(</mo><mrow><msubsup><mi>S</mi><mi>p</mi><mi>σ</mi></msubsup></mrow><mo>)</mo></mrow></math><tex-math>$t\left({S_p^\sigma } \right)$</tex-math></alternatives></inline-formula> for permutations <italic>σ</italic> in these families. We determine the best permutations in our first family and show that all permutations of the second family improve the distribution behavior of classical van der Corput sequences in the sense that <inline-formula><alternatives><inline-graphic xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="graphic/j_udt-2018-0003_eq_004.png"></inline-graphic><math xmlns:mml="http://www.w3.org/1998/Math/MathML"><mi>t</mi><mrow><mo>(</mo><mrow><msubsup><mi>S</mi><mi>p</mi><mi>σ</mi></msubsup></mrow><mo>)</mo></mrow><mo>&lt;</mo><mi>t</mi><mrow><mo>(</mo><mrow><msubsup><mi>S</mi><mi>p</mi><mrow><mi>i</mi><mi>d</mi></mrow></msubsup></mrow><mo>)</mo></mrow></math><tex-math>$t\left({S_p^\sigma } \right) &lt; t\left({S_p^{id} } \right)$</tex-math></alternatives></inline-formula>.</abstract>

On the Discrepancy of Two Families of Permuted Van der Corput Sequences

<abstract xmlns="http://www.w3.org/1999/xhtml">The star discrepancy <inline-formula><alternatives><inline-graphic xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="graphic/j_udt-2018-0004_eq_001.png"></inline-graphic><math xmlns:mml="http://www.w3.org/1998/Math/MathML"><msubsup><mi>D</mi><mi>N</mi><mo>*</mo></msubsup><mrow><mo>(</mo><mi>𝒫</mi><mo>)</mo></mrow></math><tex-math>$D_N^* \left( {\cal P} \right)$</tex-math></alternatives></inline-formula> is a quantitative measure for the irregularity of distribution of a finite point set <italic>𝒫</italic> in the multi-dimensional unit cube which is intimately related to the integration error of quasi-Monte Carlo algorithms. It is known that for every integer <italic>N ≥</italic> 2 there are point sets <italic>𝒫</italic> in [0, 1)<italic>d</italic> with <italic>|𝒫|</italic> = <italic>N</italic> and <inline-formula><alternatives><inline-graphic xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="graphic/j_udt-2018-0004_eq_002.png"></inline-graphic><math xmlns:mml="http://www.w3.org/1998/Math/MathML"><msubsup><mi>D</mi><mi>N</mi><mo>*</mo></msubsup><mrow><mo>(</mo><mi>𝒫</mi><mo>)</mo></mrow><mo>=</mo><mi>O</mi><mrow><mo>(</mo><mrow><msup><mrow><mrow><mo>(</mo><mrow><mo>log</mo><mi> </mi><mi>N</mi></mrow><mo>)</mo></mrow></mrow><mrow><mi>d</mi><mo>-</mo><mn>1</mn></mrow></msup><mo>/</mo><mi>N</mi></mrow><mo>)</mo></mrow></math><tex-math>$D_N^* \left( {\cal P} \right) = O\left( {\left( {\log \,N} \right)^{d - 1} /N} \right)$</tex-math></alternatives></inline-formula>. However, for small <italic>N</italic> compared to the dimension <italic>d</italic> this asymptotically excellent bound is useless (e.g., for <italic>N ≤</italic> e<italic>d</italic>−1).In 2001 it has been shown by Heinrich, Novak, Wasilkowski and Woźniakowski that for every integer <italic>N ≥</italic> 2there exist point sets <italic>𝒫</italic> in [0, 1)<italic>d</italic> with <italic>|𝒫|</italic> = <italic>N</italic> and <inline-formula><alternatives><inline-graphic xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="graphic/j_udt-2018-0004_eq_003.png"></inline-graphic><math xmlns:mml="http://www.w3.org/1998/Math/MathML"><msubsup><mi>D</mi><mi>N</mi><mo>*</mo></msubsup><mrow><mo>(</mo><mi>𝒫</mi><mo>)</mo></mrow><mo>≤</mo><mi>C</mi><msqrt><mrow><mi>d</mi><mo>/</mo><mi>N</mi></mrow></msqrt></math><tex-math>$D_N^* \left( {\cal P} \right) \le C\sqrt {d/N}$</tex-math></alternatives></inline-formula>. Although not optimal in an asymptotic sense in <italic>N</italic>, this upper bound has a much better (and even optimal) dependence on the dimension <italic>d</italic>.Unfortunately the result by Heinrich et al. and also later variants thereof by other authors are pure existence results and until now no explicit construction of point sets with the above properties is known. Quite recently Löbbe studied lacunary subsequences of Kronecker’s (<italic>n</italic><bold>α</bold>)-sequence and showed a metrical discrepancy bound of the form <inline-formula><alternatives><inline-graphic xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="graphic/j_udt-2018-0004_eq_004.png"></inline-graphic><math xmlns:mml="http://www.w3.org/1998/Math/MathML"><mi>C</mi><msqrt><mrow><mi>d</mi><mrow><mo>(</mo><mrow><mo>log</mo><mi> </mi><mi>d</mi></mrow><mo>)</mo></mrow><mo>/</mo><mi>N</mi></mrow></msqrt></math><tex-math>$C\sqrt {d\left({\log \,d} \right)/N}$</tex-math></alternatives></inline-formula> with implied absolute constant <italic>C></italic> 0 independent of <italic>N</italic> and <italic>d</italic>.In this paper we show a corresponding result for digital Kronecker sequences, which are a non-archimedean analog of classical Kronecker sequences.</abstract>

Metrical Star Discrepancy Bounds for Lacunary Subsequences of Digital Kronecker-Sequences and Polynomial Tractability

<abstract xmlns="http://www.w3.org/1999/xhtml">We introduce a hybridization of digital sequences with uniformly distributed sequences in the domain of <italic>b</italic>-adic integers, ℤ<italic>b,b</italic> ∈ℕ \ {1}, by using such sequences as input for generating matrices. The generating matrices are then naturally required to have finite row-lengths. We exhibit some relations of the ‘classical’ digital method to our extended version, and also give several examples of new constructions with their respective quality assessments in terms of <italic>t</italic>, <bold>T</bold> and discrepancy.</abstract>

An Extension of the Digital Method Based on <italic>b</italic>-Adic Integers

<abstract xmlns="http://www.w3.org/1999/xhtml">The aim of the present paper is to provide the background to construct linear recurring sequences with uniform distribution modulo 2<italic>s</italic>. The theory is developed and an algorithm based on the achieved results is given. The constructed sequences may have arbitrary large period length depending only on the computational power of the used machines.</abstract>

Construction of Uniformly Distributed Linear Recurring Sequences Modulo Powers of 2

<abstract xmlns="http://www.w3.org/1999/xhtml">In this paper, we prove a discrete analogue of Voronin’s early finite-dimensional approximation result with respect to terms from a given Beatty sequence and make use of Taylor approximation in order to derive a weak universality statement.</abstract>

athanasios.sourmelidis@mathematik.uni-wuerzburg.de

Weak Universality Theorem on the Approximation of Analytic Functions by Shifts of the Riemann Zeta-Function from a Beatty Sequence

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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. 