<abstract xmlns="http://www.w3.org/1999/xhtml">Let <italic>α</italic> be an irrational real number; the behaviour of the sum <italic>SN</italic> (<italic>α</italic>):= (<italic>−</italic>1)[<italic>α</italic>] +(<italic>−</italic>1)[2<italic>α</italic>] + <italic>···</italic> +(<italic>−</italic>1)[<italic>Nα</italic>] depends on the continued fraction expansion of <italic>α/</italic>2. Since the continued fraction expansion of <inline-formula><alternatives><inline-graphic xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="graphic/j_udt-2020-0007_eq_001.png"></inline-graphic><math xmlns:mml="http://www.w3.org/1998/Math/MathML"><mrow><msqrt><mn>2</mn></msqrt><mo>/</mo><mn>2</mn></mrow></math><tex-math>\sqrt 2 /2</tex-math></alternatives></inline-formula> has bounded partial quotients, <inline-formula><alternatives><inline-graphic xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="graphic/j_udt-2020-0007_eq_002.png"></inline-graphic><math xmlns:mml="http://www.w3.org/1998/Math/MathML"><mrow><msub><mrow><mi>S</mi></mrow><mi>N</mi></msub><mrow><mo>(</mo><mrow><msqrt><mn>2</mn></msqrt></mrow><mo>)</mo></mrow><mo>=</mo><mi>O</mi><mrow><mo>(</mo><mrow><mo>log</mo><mrow><mo>(</mo><mi>N</mi><mo>)</mo></mrow></mrow><mo>)</mo></mrow></mrow></math><tex-math>{S_N}\left( {\sqrt 2 } \right) = O\left( {\log \left( N \right)} \right)</tex-math></alternatives></inline-formula> and this bound is best possible. The partial quotients of the continued fraction expansion of <italic>e</italic> grow slowly and thus <inline-formula><alternatives><inline-graphic xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="graphic/j_udt-2020-0007_eq_003.png"></inline-graphic><math xmlns:mml="http://www.w3.org/1998/Math/MathML"><mrow><msub><mrow><mi>S</mi></mrow><mi>N</mi></msub><mrow><mo>(</mo><mrow><mn>2</mn><mi>e</mi></mrow><mo>)</mo></mrow><mo>=</mo><mi>O</mi><mrow><mo>(</mo><mrow><mfrac><mrow><mo>log</mo><msup><mrow><mrow><mrow><mo>(</mo><mi>N</mi><mo>)</mo></mrow></mrow></mrow><mn>2</mn></msup></mrow><mrow><mo>log</mo><mi> </mi><mo>log</mo><msup><mrow><mrow><mrow><mo>(</mo><mi>N</mi><mo>)</mo></mrow></mrow></mrow><mn>2</mn></msup></mrow></mfrac></mrow><mo>)</mo></mrow></mrow></math><tex-math>{S_N}\left( {2e} \right) = O\left( {{{\log {{\left( N \right)}^2}} \over {\log \,\log {{\left( N \right)}^2}}}} \right)</tex-math></alternatives></inline-formula>, again best possible. The partial quotients of the continued fraction expansion of <italic>e/</italic>2 behave similarly as those of <italic>e</italic>. Surprisingly enough <inline-formula><alternatives><inline-graphic xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="graphic/j_udt-2020-0007_eq_004.png"></inline-graphic><math xmlns:mml="http://www.w3.org/1998/Math/MathML"><mrow><msub><mrow><mi>S</mi></mrow><mi>N</mi></msub><mrow><mo>(</mo><mi>e</mi><mo>)</mo></mrow><mo>=</mo><mi>O</mi><mrow><mo>(</mo><mrow><mfrac><mrow><mo>log</mo><mrow><mo>(</mo><mi>N</mi><mo>)</mo></mrow></mrow><mrow><mo>log</mo><mi> </mi><mo>log</mo><mrow><mo>(</mo><mi>N</mi><mo>)</mo></mrow></mrow></mfrac></mrow><mo>)</mo></mrow></mrow></math><tex-math>1188</tex-math></alternatives></inline-formula>.</abstract>

A curiosity About (<italic>−</italic>1)[<italic>e</italic>] +(<italic>−</italic>1)[2<italic>e</italic>] + <italic>···</italic> +(<italic>−</italic>1)[<italic>Ne</italic>]

<abstract xmlns="http://www.w3.org/1999/xhtml">Both the Thue–Morse and Rudin–Shapiro sequences are not suitable sequences for cryptography since their expansion complexity is small and their correlation measure of order 2 is large. These facts imply that these sequences are highly predictable despite the fact that they have a large maximum order complexity. Sun and Winterhof (2019) showed that the Thue–Morse sequence along squares keeps a large maximum order complexity. Since, by Christol’s theorem, the expansion complexity of this rarefied sequence is no longer bounded, this provides a potentially better candidate for cryptographic applications. Similar results are known for the Rudin–Shapiro sequence and more general pattern sequences. In this paper we generalize these results to any polynomial subsequence (instead of squares) and thereby answer an open problem of Sun and Winterhof. We conclude this paper by some open problems.</abstract>

On the Maximum Order Complexity of Thue–Morse and Rudin–Shapiro Sequences along Polynomial Values

<abstract xmlns="http://www.w3.org/1999/xhtml">Given a countably infinite group <italic>G</italic> acting on some space <italic>X</italic>, an increasing family of finite subsets <italic>Gn</italic>, <italic>x</italic>∈ <italic>X</italic> and a function <italic>f</italic> over <italic>X</italic> we consider the sums <italic>Sn</italic>(<italic>f, x</italic>) = ∑<italic>g∈Gnf</italic>(<italic>gx</italic>). The asymptotic behaviour of <italic>Sn</italic>(<italic>f, x</italic>) is a delicate problem that was studied under various settings. In the following paper we study this problem when <italic>G</italic> is a specific lattice in SL (2, ℤ ) acting on the projective line and <italic>Gn</italic> are chosen using the word metric. The asymptotic distribution is calculated and shown to be tightly connected to Minkowski’s question mark function. We proceed to show that the limit distribution is stationary with respect to a random walk on <italic>G</italic> defined by a specific measure <italic>µ</italic>. We further prove a stronger result stating that the asymptotic distribution is the limit point for any probability measure over <italic>X</italic> pushed forward by the convolution power <italic>µ∗n</italic>.</abstract>

Word Metric, Stationary Measure and Minkowski’s Question Mark Function

<abstract xmlns="http://www.w3.org/1999/xhtml">In 1986, Proinov published an explicit lower bound for the diaphony of finite and infinite sequences of points contained in the <italic>d−</italic>dimensional unit cube [Proinov, P. D.:<italic>On irregularities of distribution</italic>, C. R. Acad. Bulgare Sci. <bold>39</bold> (1986), no. 9, 31–34]. However, his widely cited paper does not contain the proof of this result but simply states that this will appear elsewhere. To the best of our knowledge, this proof was so far only available in a monograph of Proinov written in Bulgarian [Proinov, P. D.: <italic>Quantitative Theory of Uniform Distribution and Integral Approximation</italic>, University of Plovdiv, Bulgaria (2000)]. The first contribution of our paper is to give a self contained version of Proinov’s proof in English. Along the way, we improve the explicit asymptotic constants implementing recent, and corrected results of [Hinrichs, A.—Markhasin, L.: <italic>On lower bounds for the</italic> ℒ2<italic>-discrepancy</italic>, J. Complexity <bold>27</bold> (2011), 127–132.] and [Hinrichs, A.—Larcher, G.: <italic>An improved lower bound for the</italic> ℒ2<italic>-discrepancy</italic>, J. Complexity <bold>34</bold> (2016), 68–77]. (The corrections are due to a note in [Hinrichs, A.—Larcher, G. <italic>An improved lower bound for the</italic> ℒ2<italic>-discrepancy</italic>, J. Complexity <bold>34</bold> (2016), 68–77].) Finally, as a main result, we use the method of Proinov to derive an explicit lower bound for the dyadic diaphony of finite and infinite sequences in a similar fashion.</abstract>

On Proinov’s Lower Bound for the Diaphony

<abstract xmlns="http://www.w3.org/1999/xhtml">We give a heuristic argument predicting that the number <italic>N∗</italic>(<italic>T</italic>) of rationals <italic>p/q</italic> on Cantor’s middle thirds set <italic>C</italic> such that gcd(<italic>p, q</italic>)=1 and <italic>q ≤ T</italic>, has asymptotic growth <italic>O</italic>(<italic>Td</italic>+<italic>ε</italic>), for <italic>d</italic> = dim <italic>C</italic>. Our heuristic is related to similar heuristics and conjectures proposed by Fishman and Simmons. We also describe extensive numerical computations supporting this heuristic. Our heuristic predicts a similar asymptotic if <italic>C</italic> is replaced with any similar fractal with a description in terms of missing digits in a base expansion. Interest in the growth of <italic>N∗</italic> (<italic>T</italic>)is motivated by a problem of Mahler on intrinsic Diophantine approximation on <italic>C</italic>.</abstract>

The Distribution of Rational Numbers on Cantor’s Middle Thirds Set

<abstract xmlns="http://www.w3.org/1999/xhtml">Let <italic>p ≥</italic> 3 be a prime, <inline-formula><alternatives><inline-graphic xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="graphic/j_udt-2020-0012_eq_002.png"></inline-graphic><math xmlns:mml="http://www.w3.org/1998/Math/MathML"><mrow><mi>S</mi><mo>⊆</mo><msubsup><mrow><mi>𝔽</mi></mrow><mi>p</mi><mn>2</mn></msubsup></mrow></math><tex-math>S \subseteq \mathbb{F}_p^2</tex-math></alternatives></inline-formula> a nonempty set, and <inline-formula><alternatives><inline-graphic xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="graphic/j_udt-2020-0012_eq_003.png"></inline-graphic><math xmlns:mml="http://www.w3.org/1998/Math/MathML"><mrow><mi>w</mi><mo>:</mo><msubsup><mrow><mi>𝔽</mi></mrow><mi>p</mi><mn>2</mn></msubsup><mo>→</mo><mi>R</mi></mrow></math><tex-math>w:\mathbb{F}_p^2 \to R</tex-math></alternatives></inline-formula> a function with supp <italic>w</italic> = <italic>S</italic>. Applying an uncertainty inequality due to András Bíró and the present author, we show that there are at most <inline-formula><alternatives><inline-graphic xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="graphic/j_udt-2020-0012_eq_004.png"></inline-graphic><math xmlns:mml="http://www.w3.org/1998/Math/MathML"><mrow><mfrac><mn>1</mn><mn>2</mn></mfrac><mrow><mo>|</mo><mi>S</mi><mo>|</mo></mrow></mrow></math><tex-math>{1 \over 2}\left| S \right|</tex-math></alternatives></inline-formula> directions in <inline-formula><alternatives><inline-graphic xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="graphic/j_udt-2020-0012_eq_005.png"></inline-graphic><math xmlns:mml="http://www.w3.org/1998/Math/MathML"><mrow><msubsup><mrow><mi>𝔽</mi></mrow><mi>p</mi><mn>2</mn></msubsup></mrow></math><tex-math>\mathbb{F}_p^2</tex-math></alternatives></inline-formula> such that for every line <italic>l</italic> in any of these directions, one has
<disp-formula><alternatives><graphic xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="graphic/j_udt-2020-0012_eq_006.png"></graphic><math xmlns:mml="http://www.w3.org/1998/Math/MathML"><mrow><munder><mo>∑</mo><mrow><mi>z</mi><mo>∈</mo><mi>l</mi></mrow></munder><mrow><mi>w</mi><mrow><mo>(</mo><mi>z</mi><mo>)</mo></mrow><mo>=</mo><mfrac><mn>1</mn><mi>p</mi></mfrac><munder><mo>∑</mo><mrow><mi>z</mi><mo>∈</mo><msubsup><mrow><mi>𝔽</mi></mrow><mi>p</mi><mn>2</mn></msubsup></mrow></munder><mrow><mi>w</mi><mrow><mo>(</mo><mi>z</mi><mo>)</mo></mrow><mo>,</mo></mrow></mrow></mrow></math><tex-math>\sum\limits_{z \in l} {w\left( z \right) = {1 \over p}\sum\limits_{z \in \mathbb{F}_p^2} {w\left( z \right),} }</tex-math></alternatives></disp-formula>
except if <italic>S</italic> itself is a line and <italic>w</italic> is constant on <italic>S</italic> (in which case all, but one direction have the property in question). The bound <inline-formula><alternatives><inline-graphic xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="graphic/j_udt-2020-0012_eq_007.png"></inline-graphic><math xmlns:mml="http://www.w3.org/1998/Math/MathML"><mrow><mfrac><mn>1</mn><mn>2</mn></mfrac><mrow><mo>|</mo><mi>S</mi><mo>|</mo></mrow></mrow></math><tex-math>{1 \over 2}\left| S \right|</tex-math></alternatives></inline-formula> is sharp.As an application, we give a new proof of a result of Rédei-Megyesi about the number of directions determined by a set in a finite affine plane.</abstract>

Point Distribution and Perfect Directions in <inline-formula><alternatives><inline-graphic xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="graphic/j_udt-2020-0012_eq_001.png"/><math xmlns:mml="http://www.w3.org/1998/Math/MathML"><mrow><msubsup><mrow><mi>𝔽</mi></mrow><mi>p</mi><mn>2</mn></msubsup></mrow></math><tex-math>\mathbb{F}_p^2</tex-math></alternatives></inline-formula>

<abstract xmlns="http://www.w3.org/1999/xhtml">Motivated by the maximal average distance of uniformly distributed sequences we consider some extremal problems for functionals of type
<disp-formula><alternatives><graphic xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="graphic/j_udt-2020-0013_eq_001.png"></graphic><math xmlns:mml="http://www.w3.org/1998/Math/MathML"><mrow><msub><mrow><mi>μ</mi></mrow><mi>C</mi></msub><mo>↦</mo><mrow><msubsup><mo>∫</mo><mn>0</mn><mn>1</mn></msubsup><mrow><msub><mrow><mrow><mrow><msubsup><mo>∫</mo><mn>0</mn><mn>1</mn></msubsup><mrow><mi>F</mi><mi>d</mi></mrow></mrow></mrow></mrow><mi>μ</mi></msub><msub><mrow></mrow><mi>C</mi></msub><mo>,</mo></mrow></mrow></mrow></math><tex-math>{\mu _C} \mapsto \int_0^1 {{{\int_0^1 {Fd} }_\mu }_C,}</tex-math></alternatives></disp-formula>
where <italic>µC</italic> is a copula measure and <italic>F</italic> is a Riemann integrable function on [0, 1]2 of a specific type. Such problems have been considered in [4] and are of interest in the study of limit points of two uniformly distributed sequences.</abstract>

Dipartimento di Matematica e Fisica “Ennio De Giorgi”

On Extremal Problems for Pairs of Uniformly Distributed Sequences and Integrals with Respect to Copula Measures

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Uniform distribution theory

 The journal is published by: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 15 (2020): Issue 2 (December 2020)

A curiosity About (−1)^[^e^] +(−1)^[2^e^] + ··· +(−1)^[^Ne^]

On the Maximum Order Complexity of Thue–Morse and Rudin–Shapiro Sequences along Polynomial Values

Word Metric, Stationary Measure and Minkowski’s Question Mark Function

On Proinov’s Lower Bound for the Diaphony

The Distribution of Rational Numbers on Cantor’s Middle Thirds Set

Point Distribution and Perfect Directions in $𝔽_{p}^{2}$ \mathbb{F}_p^2

On Extremal Problems for Pairs of Uniformly Distributed Sequences and Integrals with Respect to Copula Measures

Chercher

Tous les volumes et éditions dans cette revue

Uniform distribution theory

Volume 15 (2020): Issue 2 (December 2020)

A curiosity About (−1)[e] +(−1)[2e] + ··· +(−1)[Ne]

On the Maximum Order Complexity of Thue–Morse and Rudin–Shapiro Sequences along Polynomial Values

Word Metric, Stationary Measure and Minkowski’s Question Mark Function

On Proinov’s Lower Bound for the Diaphony

The Distribution of Rational Numbers on Cantor’s Middle Thirds Set

Point Distribution and Perfect Directions in 𝔽p2\mathbb{F}_p^2

On Extremal Problems for Pairs of Uniformly Distributed Sequences and Integrals with Respect to Copula Measures

Chercher

Tous les volumes et éditions dans cette revue

A curiosity About (−1)^[^e^] +(−1)^[2^e^] + ··· +(−1)^[^Ne^]

Point Distribution and Perfect Directions in $𝔽_{p}^{2}$ \mathbb{F}_p^2