Pierre Liardet (1943–2014)

<abstract xmlns="http://www.w3.org/1999/xhtml">It is known that there is a constant <italic>c</italic> > 0 such that for every sequence <italic>x</italic>1, <italic>x</italic>2, . . . in [0, 1) we have for the star discrepancy <inline-formula><alternatives><inline-graphic xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="graphic/j_udt-2016-0001_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></math><tex-math>$D_N^* $</tex-math></alternatives></inline-formula> of the first <italic>N</italic> elements of the sequence that <inline-formula><alternatives><inline-graphic xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="graphic/j_udt-2016-0001_eq_002.png"></inline-graphic><math xmlns:mml="http://www.w3.org/1998/Math/MathML"><msubsup><mi>ND</mi><mi>N</mi><mo>*</mo></msubsup><mo>≥</mo><mi>c</mi><mo>⋅</mo><mi>log</mi><mo>⁡</mo><mi>N</mi></math><tex-math>$ND_N^* \ge c \cdot \log N$</tex-math></alternatives></inline-formula> holds for infinitely many <italic>N</italic>. Let <italic>c</italic>∗ be the supremum of all such <italic>c</italic> with this property. We show <italic>c</italic>∗ > 0.065664679 . . . , thereby slightly improving the estimates known until now.</abstract>

Institute of Financial Mathematics and Applied Number Theory, University Linz, Altenbergerstraße 69, 4040 Linz,

An Improved Bound for the Star Discrepancy of Sequences in the Unit Interval

<abstract xmlns="http://www.w3.org/1999/xhtml">The balancing numbers and the balancers were introduced by Behera et al. in the year 1999, which were obtained from a simple diophantine equation. The goal of this paper is to investigate the moduli for which all the residues appear with equal frequency with a single period in the sequence of balancing numbers. Also, it is claimed that, the balancing numbers are uniformly distributed modulo 2, and this holds for all other powers of 2 as well. Further, it is shown that the balancing numbers are not uniformly distributed over odd primes.</abstract>

Department of Mathematics, Veer Surendra Sai University of Technology, Burla, Sambalpur-768018,

Department of Mathematics, International Institute of Information Technology, Bhubaneswar-751003,

Uniform Distribution of the Sequence of Balancing Numbers Modulo <italic>m</italic>

<abstract xmlns="http://www.w3.org/1999/xhtml">Applying the theory of distribution functions of sequences we find the relative densities of the first digits also for sequences <italic>xn</italic> not satisfying Benford’s law. Especially for sequence <italic>xn</italic> = <italic>nr</italic>, <italic>n</italic> = 1, 2, . . . and <inline-formula><alternatives><inline-graphic xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="graphic/j_udt-2016-0003_eq_001.png"></inline-graphic><math xmlns:mml="http://www.w3.org/1998/Math/MathML"><msub><mi>x</mi><mi>n</mi></msub><mo>=</mo><msubsup><mi>p</mi><mi>n</mi><mi>r</mi></msubsup></math><tex-math>$x_n = p_n^r $</tex-math></alternatives></inline-formula>, <italic>n</italic> = 1, 2, . . ., where <italic>pn</italic> is the increasing sequence of all primes and <italic>r</italic> > 0 is an arbitrary real. We also add rate of convergence to such densities.</abstract>

Department of Business Administration, The International University of Kagoshima, 8-34-1 Sakanoue, Kagoshima-shi, 891-0197,

Mathematical Institute, Slovak Academy of Sciences, Štefánikova 49, SK-814 73 Bratislava,

Distribution of Leading Digits of Numbers

<abstract xmlns="http://www.w3.org/1999/xhtml">We study several pseudorandom properties of the Liouville function and the Möbius function of polynomials over a finite field. More precisely, we obtain bounds on their balancedness as well as their well-distribution measure, correlation measure, and linear complexity profile.</abstract>

On the Pseudorandomness of the Liouville Function of Polynomials over a Finite Field

<abstract xmlns="http://www.w3.org/1999/xhtml">We introduce the concept of strong normality by defining strong normal numbers and provide various properties of these numbers, including the fact that almost all real numbers are strongly normal.</abstract>

Dép. math. et stat., Université Laval, Québec G1V 0A6,

Computer Algebra Department, Eötvös Loránd University, Pázmány Péter Sétány I/C,

On Strong Normality

<abstract xmlns="http://www.w3.org/1999/xhtml">Let <italic>n ≥</italic> 2 be an integer and denote by <italic>θn</italic> the real root in (0, 1) of the trinomial <italic>Gn</italic>(<italic>X</italic>) = −1 + <italic>X</italic> + <italic>Xn</italic>. The sequence of Perron numbers <inline-formula><alternatives><inline-graphic xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="graphic/j_udt-2016-0006_eq_001.png"></inline-graphic><math xmlns:mml="http://www.w3.org/1998/Math/MathML"><msub><mrow><mo>(</mo><msubsup><mi>θ</mi><mi>n</mi><mrow><mo>−</mo><mn>1</mn></mrow></msubsup><mo>)</mo></mrow><mrow><mi>n</mi><mo>≥</mo><mn>2</mn></mrow></msub></math><tex-math>$(\theta _n^{ - 1} )_{n \ge 2} $</tex-math></alternatives></inline-formula> tends to 1. We prove that the Conjecture of Lehmer is true for <inline-formula><alternatives><inline-graphic xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="graphic/j_udt-2016-0006_eq_002.png"></inline-graphic><math xmlns:mml="http://www.w3.org/1998/Math/MathML"><mo>{</mo><msubsup><mi>θ</mi><mi>n</mi><mrow><mo>−</mo><mn>1</mn></mrow></msubsup><mo>|</mo><mi>n</mi><mo>≥</mo><mn>2</mn><mo>}</mo></math><tex-math>$\{ \theta _n^{ - 1} |n \ge 2\} $</tex-math></alternatives></inline-formula> by the direct method of Poincaré asymptotic expansions (divergent formal series of functions) of the roots <italic>θn</italic>, <italic>zj,n</italic>, of <italic>Gn</italic>(<italic>X</italic>) lying in <italic>|z|</italic> &lt; 1, as a function of <italic>n, j</italic> only. This method, not yet applied to Lehmer’s problem up to the knowledge of the author, is successfully introduced here. It first gives the asymptotic expansion of the Mahler measures <inline-formula><alternatives><inline-graphic xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="graphic/j_udt-2016-0006_eq_003.png"></inline-graphic><math xmlns:mml="http://www.w3.org/1998/Math/MathML"><mtext>M</mtext><mo>(</mo><msub><mi>G</mi><mi>n</mi></msub><mo>)</mo><mo>=</mo><mtext>M</mtext><mo>(</mo><msub><mi>θ</mi><mi>n</mi></msub><mo>)</mo><mo>=</mo><mtext>M</mtext><mo>(</mo><msubsup><mi>θ</mi><mi>n</mi><mrow><mo>−</mo><mn>1</mn></mrow></msubsup><mo>)</mo></math><tex-math>${\rm{M}}(G_n ) = {\rm{M}}(\theta _n ) = {\rm{M}}(\theta _n^{ - 1} )$</tex-math></alternatives></inline-formula> of the trinomials <italic>Gn</italic> as a function of <italic>n</italic> only, without invoking Smyth’s Theorem, and their unique limit point above the smallest Pisot number. Comparison is made with Smyth’s, Boyd’s and Flammang’s previous results. By this method we obtain a direct proof that the conjecture of Schinzel-Zassenhaus is true for <inline-formula><alternatives><inline-graphic xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="graphic/j_udt-2016-0006_eq_004.png"></inline-graphic><math xmlns:mml="http://www.w3.org/1998/Math/MathML"><mo>{</mo><msubsup><mi>θ</mi><mi>n</mi><mrow><mo>−</mo><mn>1</mn></mrow></msubsup><mo>|</mo><mi>n</mi><mo>≥</mo><mn>2</mn><mo>}</mo></math><tex-math>$\{ \theta _n^{ - 1} |n \ge 2\} $</tex-math></alternatives></inline-formula>, with a minoration of the house <inline-graphic xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="graphic/j_udt-2016-0006_eq_005.png"></inline-graphic>, and a minoration of the Mahler measure M(<italic>Gn</italic>) better than Dobrowolski’s one. The angular regularity of the roots of <italic>Gn</italic>, near the unit circle, and limit equidistribution of the conjugates, for <italic>n</italic> tending to infinity (in the sense of Bilu, Petsche, Pritsker), towards the Haar measure on the unit circle, are described in the context of the Erdős-Turán-Amoroso-Mignotte theory, with uniformly bounded discrepancy functions.</abstract>

On the Conjecture of Lehmer, Limit Mahler Measure of Trinomials and Asymptotic Expansions

<abstract xmlns="http://www.w3.org/1999/xhtml">For a set <italic>A</italic> of positive integers <italic>a</italic>1<italic>&lt; a</italic>2<italic>&lt; · · ·</italic>, let <italic><underline>d</underline></italic>(<italic>A</italic>), <inline-formula><alternatives><inline-graphic xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="graphic/j_udt-2016-0007_eq_001.png"></inline-graphic><math xmlns:mml="http://www.w3.org/1998/Math/MathML"><mover><mi>d</mi><mo>¯</mo></mover><mo>(</mo><mi>A</mi><mo>)</mo></math><tex-math>$\overline d (A)$</tex-math></alternatives></inline-formula> denote its lower and upper asymptotic densities. The gap density is defined as <inline-formula><alternatives><inline-graphic xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="graphic/j_udt-2016-0007_eq_002.png"></inline-graphic><math xmlns:mml="http://www.w3.org/1998/Math/MathML"><mi>λ</mi><mo>(</mo><mi>A</mi><mo>)</mo><mo>=</mo><mtext>lim</mtext><mo>⁡</mo><mtext> </mtext><msub><mtext>sup</mtext><mrow><mi>n</mi><mo>→</mo><mi>∞</mi></mrow></msub><mfrac><mrow><msub><mi>a</mi><mrow><mi>n</mi><mo>+</mo><mn>1</mn></mrow></msub></mrow><mrow><msub><mi>a</mi><mi>n</mi></msub></mrow></mfrac></math><tex-math>$\lambda (A) = \lim \;{\rm sup} _{n \to \infty } {{a_{n + 1} } \over {a_n }}$</tex-math></alternatives></inline-formula>. The paper investigates the class 𝒢(<italic>α, β, γ</italic>) of all sets <italic>A</italic> with <italic><underline>d</underline></italic>(<italic>A</italic>) = <italic>α</italic>, <inline-formula><alternatives><inline-graphic xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="graphic/j_udt-2016-0007_eq_003.png"></inline-graphic><math xmlns:mml="http://www.w3.org/1998/Math/MathML"><mover><mi>d</mi><mo>¯</mo></mover><mo>(</mo><mi>A</mi><mo>)</mo><mo>=</mo><mi>β</mi></math><tex-math>$\overline d (A) = \beta $</tex-math></alternatives></inline-formula> and <italic>λ</italic>(<italic>A</italic>) = <italic>γ</italic> for given <italic>α, β, γ</italic> with 0 <italic>≤ α ≤ β ≤</italic> 1 <italic>≤ γ</italic> and <italic>αγ ≤ β</italic>. Using the classical dyadic mapping <inline-formula><alternatives><inline-graphic xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="graphic/j_udt-2016-0007_eq_004.png"></inline-graphic><math xmlns:mml="http://www.w3.org/1998/Math/MathML"><mi>ϱ</mi><mo>(</mo><mi>A</mi><mo>)</mo><mo>=</mo><mstyle><msubsup><mo>∑</mo><mrow><mi>n</mi><mo>=</mo><mn>1</mn></mrow><mi>∞</mi></msubsup><mrow><mfrac><mrow><msub><mi>χ</mi><mi>A</mi></msub><mo>(</mo><mi>n</mi><mo>)</mo></mrow><mrow><msup><mn>2</mn><mi>n</mi></msup></mrow></mfrac></mrow></mstyle></math><tex-math>$\varrho (A) = \sum\nolimits_{n = 1}^\infty {{{\chi _A (n)} \over {2^n }}} $</tex-math></alternatives></inline-formula>, where <italic>χA</italic> is the characteristic function of <italic>A</italic>, the main result of the paper states that the ϱ-image set ϱ<italic>𝒢</italic>(<italic>α, β, γ</italic>) has the Hausdorff dimension
<disp-formula><alternatives><graphic xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="graphic/j_udt-2016-0007_eq_005.png"></graphic><math xmlns:mml="http://www.w3.org/1998/Math/MathML"><mi>dim</mi><mo>ϱ</mo><mi>𝒢</mi><mo>(</mo><mi>α</mi><mo>,</mo><mi>β</mi><mo>,</mo><mi>γ</mi><mo>)</mo><mo>=</mo><mi>min</mi><mo>⁡</mo><mrow><mo>{</mo><mrow><mi>δ</mi><mo>(</mo><mi>α</mi><mo>)</mo><mo>,</mo><mi>δ</mi><mo>(</mo><mi>β</mi><mo>)</mo><mo>,</mo><mfrac><mn>1</mn><mi>γ</mi></mfrac><munder><mrow><mi>max</mi><mo>⁡</mo></mrow><mrow><mi>σ</mi><mo>∈</mo><mo>[</mo><mi>α</mi><mi>γ</mi><mo>,</mo><mi>β</mi><mo>]</mo></mrow></munder><mi>δ</mi><mo>(</mo><mi>σ</mi><mo>)</mo></mrow><mo>}</mo></mrow><mo>,</mo></math><tex-math>$$\dim \varrho \cal {G}(\alpha ,\beta ,\gamma ) = \min \left\{ {\delta (\alpha ),\delta (\beta ), { 1 \over \gamma }\mathop {\max }\limits_{\sigma \in [\alpha \gamma ,\beta ]} \delta (\sigma )} \right\},$$</tex-math></alternatives></disp-formula>
where <italic>δ</italic> is the entropy function
<disp-formula><alternatives><graphic xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="graphic/j_udt-2016-0007_eq_006.png"></graphic><math xmlns:mml="http://www.w3.org/1998/Math/MathML"><mi>δ</mi><mo>(</mo><mi>x</mi><mo>)</mo><mo>=</mo><mo>−</mo><mi>x</mi><mo> </mo><msub><mi>log</mi><mn>2</mn></msub><mo> </mo><mi>x</mi><mo>−</mo><mo>(</mo><mn>1</mn><mo>−</mo><mi>x</mi><mo>)</mo><mo> </mo><msub><mi>log</mi><mn>2</mn></msub><mo> </mo><mo>(</mo><mn>1</mn><mo>−</mo><mi>x</mi><mo>)</mo><mo>.</mo></math><tex-math>$$\delta (x) = - x\log _2 x - (1 - x)\;\log _2 (1 - x).$$</tex-math></alternatives></disp-formula></abstract>

Department of Mathematics, University of Ostrava & Centre of Excellence IT4Innovations-Division UO-IRAFM, 30. dubna 22, 701 03 Ostrava 1,

Department of Mathematics, University of Ostrava, 30. dubna 22, Ostrava, 701 03,

On Hausdorff Dimensions Related to Sets with Given Asymptotic and Gap Densities

<abstract xmlns="http://www.w3.org/1999/xhtml">The paper gives conditions for a sequence of fractional parts of real numbers <inline-formula><alternatives><inline-graphic xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="graphic/j_udt-2016-0008_eq_001.png"></inline-graphic><math xmlns:mml="http://www.w3.org/1998/Math/MathML"><msubsup><mrow><mo>(</mo><mrow><mo>{</mo><msub><mi>a</mi><mi>n</mi></msub><mi>x</mi><mo>}</mo></mrow><mo>)</mo></mrow><mrow><mi>n</mi><mo>=</mo><mn>1</mn></mrow><mi>∞</mi></msubsup></math><tex-math>$\left( {\{ a_n x\} } \right)_{n = 1}^\infty $</tex-math></alternatives></inline-formula> to satisfy a pair correlation estimate. Here <italic>x</italic> is a fixed nonzero real number and <inline-formula><alternatives><inline-graphic xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="graphic/j_udt-2016-0008_eq_002.png"></inline-graphic><math xmlns:mml="http://www.w3.org/1998/Math/MathML"><msubsup><mrow><mo>(</mo><mrow><msub><mi>a</mi><mi>n</mi></msub></mrow><mo>)</mo></mrow><mrow><mi>n</mi><mo>=</mo><mn>1</mn></mrow><mi>∞</mi></msubsup></math><tex-math>$\left( {a_n } \right)_{n = 1}^\infty $</tex-math></alternatives></inline-formula> is a random walk on the integers.</abstract>

Pair Correlations and Random Walks on the Integers

<abstract xmlns="http://www.w3.org/1999/xhtml">There are various methods how to describe and characterize distribution of elements of sets of positive integers. One of the most interesting is that using the set of all distribution functions of the corresponding ratio block sequence introduced in [Strauch, O.—Tóth, J.T.: Publ. Math. Debrecen <bold>58</bold> (2001), no. 4, 751–778]. In the present paper we give some sufficient conditions under which this set is small in a metric sense. As a corollary we obtain a new characterization of the case of asymptotic distribution.</abstract>

On Small Sets of Distribution Functions of Ratio Block Sequences

<abstract xmlns="http://www.w3.org/1999/xhtml">The aim of this paper is to study the connection between different properties related to <italic>β</italic>-expansions. In particular, the relation between two conditions, both ensuring purely discrete spectrum of the odometer, is analyzed. The first one is the so-called Hypothesis B for the <italic>G</italic>-odometers and the second one is denoted by (QM) and it has been introduced in the framework of tilings associated to Pisot <italic>β</italic>-numerations.</abstract>

maria-rita.iaco@liafa.univ-paris-diderot.fr

IRIF, CNRS UMR 8243, Université Paris Diderot – Paris 7, Case 7014, 75205 Paris Cedex 13,

Graz University of Technology, Institute for Analysis and Number Theory, Steyrergasse 30, 8010 Graz,

Linear Recursive Odometers and Beta-Expansions

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. 