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

In this paper, we classify all solutions with cyclic and semi-cyclic semigroup supports of the functional equations arising from multiplication of quantum integers with fields of coefficients of characteristic zero. This also solves completely the classification problem proposed by Melvyn Nathanson and Yang Wang concerning the solutions, with semigroup supports which are not prime subsemigroups of ℕ, to these functional equations for the case of rational field of coefficients. As a consequence, we obtain some results for other problems raised by Nathanson concerning maximal solutions and extension of supports of solutions to these functional equations in the case where the semigroup supports are not prime subsemigroups of ℕ.
</abstract>

On the Classification of Solutions of Quantum Functional Equations with Cyclic and Semi-Cyclic Supports

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

In a family of <italic>Sn</italic>-fields (<italic>n ≤</italic> 5), we show that except for a density zero set, the lower and upper bounds of the Euler-Kronecker constants are <italic>−</italic>(<italic>n −</italic> 1) log log <italic>dK</italic>+ <italic>O</italic>(log log log <italic>dK</italic>) and loglog <italic>dK</italic> + <italic>O</italic>(log log log <italic>dK</italic>), resp., where <italic>dK</italic> is the absolute value of the discriminant of a number field <italic>K</italic>.
</abstract>

Extreme Values of Euler-Kronecker Constants

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

We provide an algorithm to approximate a finitely supported discrete measure <italic>μ</italic> by a measure <italic>νN</italic> corresponding to a set of <italic>N</italic> points so that the total variation between <italic>μ</italic> and <italic>νN</italic> has an upper bound. As a consequence if <italic>μ</italic> is a (finite or infinitely supported) discrete probability measure on [0, 1]<italic>d </italic>with a sufficient decay rate on the weights of each point, then <italic>μ</italic> can be approximated by <italic>νN</italic> with total variation, and hence star-discrepancy, bounded above by (log <italic>N</italic>)<italic>N−</italic>1. Our result improves, in the discrete case, recent work by Aistleitner, Bilyk, and Nikolov who show that for any normalized Borel measure <italic>μ</italic>, there exist finite sets whose star-discrepancy with respect to <italic>μ</italic> is at most <inline-formula>
<alternatives>
<inline-graphic xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="graphic/j_udt-2021-0003_eq_001.png"></inline-graphic>
<math xmlns:mml="http://www.w3.org/1998/Math/MathML" display="inline"><mrow><msup><mrow><mrow><mrow><mo>(</mo><mrow><mo>log</mo><mi> </mi><mi>N</mi></mrow><mo>)</mo></mrow></mrow></mrow><mrow><mi>d</mi><mo>−</mo><mfrac><mn>1</mn><mn>2</mn></mfrac></mrow></msup><msup><mrow><mi>N</mi></mrow><mrow><mo>−</mo><mn>1</mn></mrow></msup></mrow></math>
<tex-math>{\left( {\log \,N} \right)^{d - {1 \over 2}}}{N^{ - 1}}</tex-math>
</alternatives>
</inline-formula>. Moreover, we close a gap in the literature for discrepancy in the case <italic>d</italic> =1 showing both that Lebesgue is indeed the hardest measure to approximate by finite sets and also that all measures without discrete components have the same order of discrepancy as the Lebesgue measure.
</abstract>

Families of Well Approximable Measures

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

In the present paper the author uses the function system Γ<italic>ℬ</italic><italic>s</italic>constructed in Cantor bases to show upper bounds of the extreme and star discrepancy of an arbitrary net in the terms of the trigonometric sum of this net with respect to the functions of this system. The obtained estimations are inequalities of the type of Erdős-Turán-Koksma. These inequalities are very suitable for studying of nets constructed in the same Cantor system.
</abstract>

The Inequality of Erdős-Turán-Koksma in the Terms of the Functions of the System Γ ℬ<italic>s</italic>

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

The higher-dimensional generalization of the weighted <italic>q</italic>-adic sum-of-digits functions <italic>sq,γ</italic>(<italic>n</italic>), <italic>n</italic> =0, 1, 2,..., covers several important cases of sequences investigated in the theory of uniformly distributed sequences, e.g., <italic>d</italic>-dimensional van der Corput-Halton or <italic>d</italic>-dimensional Kronecker sequences. We prove a necessary and sufficient condition for the higher-dimensional weighted <italic>q</italic>-adic sum-of-digits functions to be uniformly distributed modulo one in terms of a trigonometric product. As applications of our condition we prove some upper estimates of the extreme discrepancies of such sequences, and that the existence of distribution function <italic>g</italic>(<italic>x</italic>)= <italic>x</italic> implies the uniform distribution modulo one of the weighted <italic>q</italic>-adic sum-of-digits function <italic>sq,γ</italic> (<italic>n</italic>), <italic>n</italic> = 0, 1, 2,... We also prove the uniform distribution modulo one of related sequences <italic>h</italic>1<italic>sq, γ</italic> (<italic>n</italic>)+<italic>h</italic>2<italic>sq,γ</italic> (<italic>n</italic> +1), where <italic>h</italic>1 and <italic>h</italic>2 are integers such that <italic>h</italic>1 + <italic>h</italic>2 ≠ 0 and that the akin two-dimensional sequence <italic>sq,γ</italic> (<italic>n</italic>), <italic>sq,γ</italic> (<italic>n</italic> +1) cannot be uniformly distributed modulo one if <italic>q ≥</italic> 3. The properties of the two-dimensional sequence <italic>sq,γ</italic> (<italic>n</italic>),s<italic>q,γ</italic> (<italic>n</italic> +1), <italic>n</italic> =0, 1, 2,..., will be instrumental in the proofs of the final section, where we show how the growth properties of the sequence of weights influence the distribution of values of the weighted sum-of-digits function which in turn imply a new property of the van der Corput sequence.
</abstract>

Department of Mathematics, Faculty of Sciences

Department of Mathematics and Informatics

Faculty of Economics of J. Selye University

Uniform Distribution of the Weighted Sum-of-Digits Functions

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

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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. 

Uniform distribution theory

Volume 16 (2021): Issue 1 (June 2021)

On the Classification of Solutions of Quantum Functional Equations with Cyclic and Semi-Cyclic Supports

Extreme Values of Euler-Kronecker Constants

Families of Well Approximable Measures

The Inequality of Erdős-Turán-Koksma in the Terms of the Functions of the System Γ _{ℬ_s}

Uniform Distribution of the Weighted Sum-of-Digits Functions

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All Volumes and Issues in this Journal

Uniform distribution theory

Volume 16 (2021): Issue 1 (June 2021)

On the Classification of Solutions of Quantum Functional Equations with Cyclic and Semi-Cyclic Supports

Extreme Values of Euler-Kronecker Constants

Families of Well Approximable Measures

The Inequality of Erdős-Turán-Koksma in the Terms of the Functions of the System Γ ℬs

Uniform Distribution of the Weighted Sum-of-Digits Functions

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All Volumes and Issues in this Journal

The Inequality of Erdős-Turán-Koksma in the Terms of the Functions of the System Γ _{ℬ_s}