<abstract xmlns="http://www.w3.org/1999/xhtml">Let <italic>q</italic> be an integer greater than or equal to 2, and let <italic>S</italic><italic>q</italic>(<italic>n</italic>)denote the sum of digits of <italic>n</italic> in base <italic>q</italic>.For<disp-formula><alternatives><graphic xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="graphic/j_udt-2019-0010_eq_001.png"></graphic><math xmlns:mml="http://www.w3.org/1998/Math/MathML"><mrow><mi>α</mi><mo>=</mo><mrow><mo>[</mo><mrow><mn>0</mn><mo>;</mo><mover><mrow><mn>1</mn><mo>,</mo><mi>m</mi></mrow><mo>¯</mo></mover></mrow><mo>]</mo></mrow><mo>,</mo><mi> </mi><mi> </mi><mi> </mi><mi>m</mi><mo>≥</mo><mn>2</mn><mo>,</mo></mrow></math><tex-math>\alpha = \left[ {0;\overline {1,m} } \right],\,\,\,m \ge 2,</tex-math></alternatives></disp-formula>let <italic>S</italic>α(<italic>n</italic>) denote the sum of digits in the Ostrowski <italic>α</italic>-representation of <italic>n</italic>. Let <italic>m</italic>1,<italic>m</italic>2 ≥ 2 be integers with<disp-formula><alternatives><graphic xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="graphic/j_udt-2019-0010_eq_002.png"></graphic><math xmlns:mml="http://www.w3.org/1998/Math/MathML"><mrow><mo>gcd</mo><mrow><mo>(</mo><mrow><mi>q</mi><mo>-</mo><mn>1</mn><mo>,</mo><msub><mrow><mi>m</mi></mrow><mn>1</mn></msub></mrow><mo>)</mo></mrow><mo>=</mo><mo>gcd</mo><mrow><mo>(</mo><mrow><mi>m</mi><mo>,</mo><msub><mrow><mi>m</mi></mrow><mn>2</mn></msub></mrow><mo>)</mo></mrow><mo>=</mo><mn>1</mn></mrow></math><tex-math>\gcd \left( {q - 1,{m_1}} \right) = \gcd \left( {m,{m_2}} \right) = 1</tex-math></alternatives></disp-formula>We prove that there exists <italic>δ></italic> 0 such that for all integers <italic>r</italic>1,<italic>r</italic>2,<disp-formula><alternatives><graphic xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="graphic/j_udt-2019-0010_eq_003.png"></graphic><math xmlns:mml="http://www.w3.org/1998/Math/MathML"><mrow><mtable><mtr><mtd><mrow><mrow><mo>|</mo><mrow><mrow><mo>{</mo><mrow><mn>0</mn><mo>≤</mo><mi>n</mi><mo>&lt;</mo><mi>N</mi><mo>:</mo><msub><mrow><mi>S</mi></mrow><mi>q</mi></msub><mo>(</mo><mi>n</mi><mo>)</mo><mo>≡</mo><msub><mrow><mi>r</mi></mrow><mn>1</mn></msub><mrow><mo>(</mo><mrow><mo>mod</mo><mi> </mi><msub><mrow><mi>m</mi></mrow><mn>1</mn></msub></mrow><mo>)</mo></mrow><mo>,</mo><mi> </mi><mi> </mi><msub><mrow><mi>S</mi></mrow><mi>α</mi></msub><mo>(</mo><mi>n</mi><mo>)</mo><mo>≡</mo><msub><mrow><mi>r</mi></mrow><mn>2</mn></msub><mrow><mo>(</mo><mrow><mo>mod</mo><mi> </mi><msub><mrow><mi>m</mi></mrow><mn>2</mn></msub></mrow><mo>)</mo></mrow></mrow><mo>}</mo></mrow></mrow><mo>|</mo></mrow></mrow></mtd></mtr><mtr><mtd><mrow><mo>=</mo><mfrac><mi>N</mi><mrow><msub><mrow><mi>m</mi></mrow><mn>1</mn></msub><msub><mrow><mi>m</mi></mrow><mn>2</mn></msub></mrow></mfrac><mo>+</mo><mn>0</mn><mrow><mo>(</mo><mrow><msup><mrow><mi>N</mi></mrow><mrow><mn>1</mn><mo>-</mo><mi>δ</mi></mrow></msup></mrow><mo>)</mo></mrow><mo>.</mo></mrow></mtd></mtr></mtable></mrow></math><tex-math>\matrix{ {\left| {\left\{ {0 \le n &lt; N:{S_q}(n) \equiv {r_1}\left( {\bmod \,{m_1}} \right),\,\,{S_\alpha }(n) \equiv {r_2}\left( {\bmod \,{m_2}} \right)} \right\}} \right|} \cr { = {N \over {{m_1}{m_2}}} + 0\left( {{N^{1 - \delta }}} \right).} \cr } </tex-math></alternatives></disp-formula>The asymptotic relation implied by this equality was proved by Coquet, Rhin &amp; Toffin and the equality was proved for the case <inline-formula><alternatives><inline-graphic xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="graphic/j_udt-2019-0010_eq_004.png"></inline-graphic><math xmlns:mml="http://www.w3.org/1998/Math/MathML"><mrow><mi>α</mi><mo>=</mo><mrow><mo>[</mo><mover><mn>1</mn><mo>¯</mo></mover><mo>]</mo></mrow></mrow></math><tex-math>\alpha = \left[ {\bar 1} \right]</tex-math></alternatives></inline-formula> by Spiegelhofer.</abstract>

Joint Distribution in Residue Classes of the Base-<italic>q</italic> and Ostrowski Digital Sums

<abstract xmlns="http://www.w3.org/1999/xhtml">It was proved by Jang et al. that various chains of one-parameter distributions converge to Benford’s law. We study chains of truncated distributions and propose another approach, using a recent convergence result of the Lerch transcendent function, to proving that they converge to Benford’s law for initial Beta distributions with parameters <italic>α</italic> and 1.</abstract>

Chains of Truncated Beta Distributions and Benford’s Law

<abstract xmlns="http://www.w3.org/1999/xhtml">Expansion complexity and maximum order complexity are both finer measures of pseudorandomness than the linear complexity which is the most prominent quality measure for cryptographic sequences. The expected value of the <italic>N</italic>th maximum order complexity is of order of magnitude log <italic>N</italic> whereas it is easy to find families of sequences with <italic>N</italic>th expansion complexity exponential in log <italic>N</italic>. This might lead to the conjecture that the maximum order complexity is a finer measure than the expansion complexity. However, in this paper we provide two examples, the Thue-Morse sequence and the Rudin-Shapiro sequence with very small expansion complexity but very large maximum order complexity. More precisely, we prove explicit formulas for their <italic>N</italic> th maximum order complexity which are both of the largest possible order of magnitude <italic>N</italic>. We present the result on the Rudin-Shapiro sequence in a more general form as a formula for the maximum order complexity of certain pattern sequences.</abstract>

On the Maximum Order Complexity of the Thue-Morse and Rudin-Shapiro Sequence

<abstract xmlns="http://www.w3.org/1999/xhtml">Let <italic>X</italic> and <italic>Y</italic> be nonempty finite subsets of 𝕑 and <italic>X</italic> +<italic>Y</italic> its sumset. The structures of <italic>X</italic> and <italic>Y</italic> when <italic>r</italic>(<italic>X, Y</italic> ):= |<italic>X</italic> +<italic>Y</italic> |−|<italic>X</italic>|−|<italic>Y</italic> | is small have been widely studied; in particular the Generalized Freiman’s 3<italic>k</italic> − 4 Theorem describes <italic>X</italic> and <italic>Y</italic> when <italic>r</italic>(<italic>X, Y</italic> ) ≤ min{|<italic>X</italic>|, |<italic>Y</italic> |} − 4. However, not too much is known about <italic>X</italic> and <italic>Y</italic> when <italic>r</italic>(<italic>X, Y</italic> ) <italic>></italic> min{|<italic>X</italic>|, |<italic>Y</italic> |} − 4. In this paper we study the structure of <italic>X</italic> and <italic>Y</italic> for arbitrary <italic>r</italic>(<italic>X, Y</italic> ).</abstract>

On Freiman’s 3<italic>k</italic> − 4 Theorem

<abstract xmlns="http://www.w3.org/1999/xhtml">We show that for an arbitrary sequence of intervals <italic>I</italic>n with constant length c, there exist real numbers <italic>β</italic> such that for all n <italic>β</italic>n belongs to <italic>I</italic>n modulo one.</abstract>

Sur Les Parties Fractionnaires Des Suites (<italic>β</italic><italic>n</italic>)<italic>n</italic>≥1

<abstract xmlns="http://www.w3.org/1999/xhtml">Let <italic>X</italic>1,<italic>X</italic>2,... be i.i.d. absolutely continuous random variables, let <inline-formula><alternatives><inline-graphic xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="graphic/j_udt-2019-0015_eq_001.png"></inline-graphic><math xmlns:mml="http://www.w3.org/1998/Math/MathML"><mrow><msub><mrow><mi>S</mi></mrow><mi>k</mi></msub><mo>=</mo><msubsup><mo>∑</mo><mrow><mi>j</mi><mo>=</mo><mn>1</mn></mrow><mi>k</mi></msubsup><mrow><msub><mrow><mi>X</mi></mrow><mi>j</mi></msub></mrow></mrow></math><tex-math>{S_k} = \sum\nolimits_{j = 1}^k {{X_j}}</tex-math></alternatives></inline-formula> (mod 1) and let <italic>D*</italic><italic>N</italic> denote the star discrepancy of the sequence (<italic>S</italic><italic>k</italic>)1≤<italic>k</italic>≤<italic>N</italic>. We determine the limit distribution of <inline-formula><alternatives><inline-graphic xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="graphic/j_udt-2019-0015_eq_002.png"></inline-graphic><math xmlns:mml="http://www.w3.org/1998/Math/MathML"><mrow><msqrt><mi>N</mi></msqrt><msubsup><mrow><mi>D</mi></mrow><mi>N</mi><mo>*</mo></msubsup></mrow></math><tex-math>\sqrt N D_N^*</tex-math></alternatives></inline-formula> and the weak limit of the sequence <inline-formula><alternatives><inline-graphic xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="graphic/j_udt-2019-0015_eq_003.png"></inline-graphic><math xmlns:mml="http://www.w3.org/1998/Math/MathML"><mrow><msqrt><mi>N</mi></msqrt><mrow><mo>(</mo><mrow><msub><mrow><mi>F</mi></mrow><mi>N</mi></msub><mo>(</mo><mi>t</mi><mo>)</mo><mo>-</mo><mi>t</mi></mrow><mo>)</mo></mrow></mrow></math><tex-math>\sqrt N \left( {{F_N}(t) - t} \right)</tex-math></alternatives></inline-formula> in the Skorohod space <italic>D</italic>[0, 1], where <italic>F</italic><italic>N</italic> (<italic>t</italic>) denotes the empirical distribution function of the sequence (<italic>S</italic><italic>k</italic>)1≤<italic>k</italic>≤<italic>N</italic>.</abstract>

On the Discrepancy of Random Walks on the Circle

<abstract xmlns="http://www.w3.org/1999/xhtml">Flat tori are analyzed in the context of an intrinsic Fourier-analytic approach to electrostatics on Riemannian manifolds, introduced by one of the authors in 1984 and previously developed for compact hyperbolic manifolds. The approach covers a large class of repelling laws, but does not naturally include laws with singularities at the origin, for which possible accommodations are discussed in the final section of the paper.</abstract>

Stable Configurations of Repelling Points on Flat Tori

<abstract xmlns="http://www.w3.org/1999/xhtml">In the last decades many results have been proved on pseudo-randomness of binary sequences. In this series our goal is to show that using many of these results one can also construct large families of quasi-random, pseudo-random and strongly pseudo-random graphs. Indeed, it will be proved that if the first row of the adjacency matrix of a circulant graph forms a binary sequence which possesses certain pseudorandom properties (and there are many large families of binary sequences known with these properties), then the graph is quasi-random, pseudo-random or strongly pseudo-random, respectively. In particular, here in Part I we will construct large families of quasi-random graphs along these lines. (In Parts II and III we will present and study constructions for pseudo-random and <italic>strongly</italic> pseudo-random graphs, respectively.)</abstract>

Quasi-Random Graphs, Pseudo-Random Graphs and Pseudorandom Binary Sequences, I. (Quasi-Random Graphs)

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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 14 (2019): Issue 2 (December 2019)

Joint Distribution in Residue Classes of the Base-q and Ostrowski Digital Sums

Chains of Truncated Beta Distributions and Benford’s Law

On the Maximum Order Complexity of the Thue-Morse and Rudin-Shapiro Sequence

On Freiman’s 3k − 4 Theorem

Sur Les Parties Fractionnaires Des Suites (βⁿ)_n_≥1

On the Discrepancy of Random Walks on the Circle

Stable Configurations of Repelling Points on Flat Tori

Quasi-Random Graphs, Pseudo-Random Graphs and Pseudorandom Binary Sequences, I. (Quasi-Random Graphs)

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