rss_2.0Uniform distribution theory FeedSciendo RSS Feed for Uniform distribution theoryhttps://sciendo.com/journal/UDThttps://www.sciendo.comUniform distribution theory 's Coverhttps://sciendo-parsed-data-feed.s3.eu-central-1.amazonaws.com/61fc7f380bfe4f0ecbde536b/cover-image.jpg?X-Amz-Algorithm=AWS4-HMAC-SHA256&X-Amz-Date=20220927T195924Z&X-Amz-SignedHeaders=host&X-Amz-Expires=604800&X-Amz-Credential=AKIA6AP2G7AKP25APDM2%2F20220927%2Feu-central-1%2Fs3%2Faws4_request&X-Amz-Signature=4cd57dd0706bf1d8825306b45a520435c0c3194c14f15471470c99fdf868b12c200300The Seventh International Conference on Uniform Distribution Theory (UDT 2021) https://sciendo.com/article/10.2478/udt-2022-0004ARTICLE2022-05-31T00:00:00.000+00:00On a Class of Lacunary Almost Newman Polynomials Modulo and Density Theoremshttps://sciendo.com/article/10.2478/udt-2022-0007<abstract> <title style='display:none'>Abstract</title> <p>The reduction modulo <italic>p</italic> of a family of lacunary integer polynomials, associated with the dynamical zeta function <italic>ζ<sub>β</sub></italic>(<italic>z</italic>)of the <italic>β</italic>-shift, for <italic>β&gt;</italic> 1 close to one, is investigated. We briefly recall how this family is correlated to the problem of Lehmer. A variety of questions is raised about their numbers of zeroes in 𝔽<italic><sub>p</sub></italic> and their factorizations, via Kronecker’s Average Value Theorem (viewed as an analog of classical Theorems of Uniform Distribution Theory). These questions are partially answered using results of Schinzel, revisited by Sawin, Shusterman and Stoll, and density theorems (Frobenius, Chebotarev, Serre, Rosen). These questions arise from the search for the existence of integer polynomials of Mahler measure <italic>&gt;</italic> 1 less than the smallest Salem number 1.176280. Explicit connection with modular forms (or modular representations) of the numbers of zeroes of these polynomials in 𝔽<italic><sub>p</sub></italic> is obtained in a few cases. In general it is expected since it must exist according to the Langlands program.</p> </abstract>ARTICLE2022-05-31T00:00:00.000+00:00A Typical Number is Extremely Non-Normalhttps://sciendo.com/article/10.2478/udt-2022-0001<abstract> <title style='display:none'>Abstract</title> <p>Fix a positive integer <italic>N</italic> ≥ 2. For a real number <italic>x</italic> ∈ [0, 1] and a digit <italic>i</italic> ∈ {0, 1,..., <italic>N</italic> − 1}, let Π<italic><sub>i</sub></italic>(<italic>x, n</italic>) denote the frequency of the digit <italic>i</italic> among the first <italic>nN</italic>-adic digits of <italic>x</italic>. It is well-known that for a typical (in the sense of Baire) <italic>x</italic> ∈ [0, 1], the sequence of digit frequencies diverges as <italic>n</italic> →∞. In this paper we show that for any regular linear transformation <italic>T</italic> there exists a residual set of points <italic>x</italic> ∈ [0,1] such that the <italic>T</italic> -averaged version of the sequence (Π<italic><sub>i</sub></italic>(<italic>x, n</italic>))<italic><sub>n</sub></italic> also diverges significantly.</p> </abstract>ARTICLE2022-05-31T00:00:00.000+00:00Density of Oscillating Sequences in the Real Linehttps://sciendo.com/article/10.2478/udt-2022-0003<abstract> <title style='display:none'>Abstract</title> <p>In this paper we study the density in the real line of oscillating sequences of the form <disp-formula> <alternatives> <graphic xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="graphic/j_udt-2022-0003_eq_001.png"/> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" display="block"><mml:mrow><mml:msub><mml:mrow><mml:mrow><mml:mrow><mml:mo>(</mml:mo><mml:mrow><mml:mi>g</mml:mi><mml:mrow><mml:mo>(</mml:mo><mml:mi>k</mml:mi><mml:mo>)</mml:mo></mml:mrow><mml:mo>⋅</mml:mo><mml:mi>F</mml:mi><mml:mrow><mml:mo>(</mml:mo><mml:mrow><mml:mi>k</mml:mi><mml:mi>α</mml:mi></mml:mrow><mml:mo>)</mml:mo></mml:mrow></mml:mrow><mml:mo>)</mml:mo></mml:mrow></mml:mrow></mml:mrow><mml:mrow><mml:mi>k</mml:mi><mml:mo>∈</mml:mo><mml:mi>ℕ</mml:mi></mml:mrow></mml:msub><mml:mo>,</mml:mo></mml:mrow></mml:math> <tex-math>{\left( {g\left( k \right) \cdot F\left( {k\alpha } \right)} \right)_{k \in \mathbb{N}}},</tex-math> </alternatives> </disp-formula> where <italic>g</italic> is a positive increasing function and <italic>F</italic> a real continuous 1-periodic function. This extends work by Berend, Boshernitzan and Kolesnik [Distribution Modulo 1 of Some Oscillating Sequences I-III] who established differential properties on the function <italic>F</italic> ensuring that the oscillating sequence is dense modulo 1.</p> <p>More precisely, when <italic>F</italic> has finitely many roots in [0, 1), we provide necessary and also sufficient conditions for the oscillating sequence under consideration to be dense in ℝ. All the results are stated in terms of the Diophantine properties of <italic>α</italic>, with the help of the theory of continued fractions.</p> </abstract>ARTICLE2022-05-31T00:00:00.000+00:00Insertion in Constructed Normal Numbershttps://sciendo.com/article/10.2478/udt-2022-0008<abstract> <title style='display:none'>Abstract</title> <p>Defined by Borel, a real number is normal to an integer base <italic>b</italic> ≥ 2 if in its base-<italic>b</italic> expansion every block of digits occurs with the same limiting frequency as every other block of the same length. We consider the problem of <italic>insertion</italic> in constructed base-<italic>b</italic> normal expansions to obtain normality to base (<italic>b</italic> + 1).</p> </abstract>ARTICLE2022-05-31T00:00:00.000+00:00Products of Integers with Few Nonzero Digitshttps://sciendo.com/article/10.2478/udt-2022-0006<abstract> <title style='display:none'>Abstract</title> <p>Let <italic>s</italic>(<italic>n</italic>) be the number of nonzero bits in the binary digital expansion of the integer <italic>n</italic>. We study, for fixed <italic>k, ℓ, m</italic>, the Diophantine system</p> <p><italic>s</italic>(<italic>ab</italic>)= <italic>k, s</italic>(<italic>a</italic>)= <italic>ℓ,</italic> and <italic>s</italic>(<italic>b</italic>)= <italic>m</italic></p> <p>in odd integer variables <italic>a, b</italic>.When <italic>k</italic> =2 or <italic>k</italic> = 3, we establish a bound on <italic>ab</italic> in terms of <italic>ℓ</italic> and <italic>m</italic>. While such a bound does not exist in the case of <italic>k</italic> =4, we give an upper bound for min{<italic>a, b</italic>} in terms of <italic>ℓ</italic> and <italic>m</italic>.</p> </abstract>ARTICLE2022-05-31T00:00:00.000+00:00On Some Properties of Irrational Subspaceshttps://sciendo.com/article/10.2478/udt-2022-0002<abstract> <title style='display:none'>Abstract</title> <p>In this paper, we discuss some properties of completely irrational subspaces. We prove that there exist completely irrational subspaces that are badly approximable and, moreover, sets of such subspaces are winning in different senses. We get some bounds for Diophantine exponents of vectors that lie in badly approximable subspaces that are completely irrational; in particular, for any vector <italic>ξ</italic> from two-dimensional badly approximable completely irrational subspace of ℝ<italic><sup>d</sup></italic> one has <inline-formula> <alternatives> <inline-graphic xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="graphic/j_udt-2022-0002_eq_001.png"/> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" display="inline"><mml:mrow><mml:mover accent="true"><mml:mi>ω</mml:mi><mml:mo>⌢</mml:mo></mml:mover><mml:mrow><mml:mo>(</mml:mo><mml:mi>ξ</mml:mi><mml:mo>)</mml:mo></mml:mrow><mml:mo>≤</mml:mo><mml:mfrac><mml:mrow><mml:msqrt><mml:mrow><mml:mn>5</mml:mn><mml:mo>-</mml:mo><mml:mn>1</mml:mn></mml:mrow></mml:msqrt></mml:mrow><mml:mn>2</mml:mn></mml:mfrac></mml:mrow></mml:math> <tex-math>\mathord{\buildrel{\lower3pt\hbox{$\scriptscriptstyle\frown$}}\over \omega } \left( \xi \right) \le {{\sqrt {5 - 1} } \over 2}</tex-math> </alternatives> </inline-formula>. Besides that, some statements about the dimension of subspaces generated by best approximations to completely irrational subspace easily follow from properties that we discuss.</p> </abstract>ARTICLE2022-05-31T00:00:00.000+00:00Bounds on the size of Progression-Free Sets in ℤhttps://sciendo.com/article/10.2478/udt-2022-0005<abstract> <title style='display:none'>Abstract</title> <p>In this note we give an overview of the currently known best lower and upper bounds on the size of a subset of ℤ<italic><sup>n</sup><sub>m</sub></italic> avoiding <italic>k</italic>-term arithmetic progression. We will focus on the case when the length of the forbidden progression is 3. We also formulate some open questions.</p> </abstract>ARTICLE2022-05-31T00:00:00.000+00:00Divisibility Parameters and the Degree of Kummer Extensions of Number Fieldshttps://sciendo.com/article/10.2478/udt-2021-0008<abstract> <title style='display:none'>Abstract</title> <p>Let <italic>K </italic>be a number field, and let <italic>ℓ</italic> be a prime number. Fix some elements <italic>α</italic><sub>1</sub>,...<italic>,α<sub>r</sub></italic> of <italic>K<sup>×</sup> </italic>which generate a subgroup of <italic>K<sup>×</sup> </italic>of rank <italic>r</italic>. Let <italic>n</italic><sub>1</sub>,...<italic>,n<sub>r</sub></italic>, <italic>m</italic> be positive integers with <italic>m</italic> ⩾ <italic>n<sub>i</sub></italic> for every <italic>i</italic>. We show that there exist computable parametric formulas (involving only a finite case distinction) to express the degree of the Kummer extension <italic>K</italic>(<italic>ζ</italic><sub>ℓ</sub><sup>m</sup>, <inline-formula> <alternatives> <inline-graphic xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="graphic/j_udt-2021-0008_ineq_001.png"/> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" display="inline"><mml:mrow><mml:mroot><mml:mrow><mml:msub><mml:mi>α</mml:mi><mml:mn>1</mml:mn></mml:msub></mml:mrow><mml:mrow><mml:msup><mml:mi>ℓ</mml:mi><mml:mrow><mml:msub><mml:mi>n</mml:mi><mml:mn>1</mml:mn></mml:msub></mml:mrow></mml:msup></mml:mrow></mml:mroot><mml:mo>,</mml:mo><mml:mo>…</mml:mo><mml:mo>,</mml:mo><mml:mroot><mml:mrow><mml:msub><mml:mi>α</mml:mi><mml:mi>r</mml:mi></mml:msub></mml:mrow><mml:mrow><mml:msup><mml:mi>ℓ</mml:mi><mml:mrow><mml:msub><mml:mi>n</mml:mi><mml:mi>r</mml:mi></mml:msub></mml:mrow></mml:msup></mml:mrow></mml:mroot></mml:mrow></mml:math> <tex-math>\root {{\ell ^{{n_1}}}} \of {{\alpha _1}} , \ldots ,\root {{\ell ^{{n_r}}}} \of {{\alpha _r}}</tex-math> </alternatives> </inline-formula> ) over <italic>K</italic>(<italic>ζ<sub>ℓ</sub><sup>m</sup></italic>) for all <italic>n</italic><sub>1</sub>,...<italic>, n<sub>r</sub>, m</italic>. This is achieved with a new method with respect to a previous work, namely we determine explicit formulas for the divisibility parameters which come into play.</p> </abstract>ARTICLE2022-02-02T00:00:00.000+00:00From Randomness in Two Symbols to Randomness in Three Symbolshttps://sciendo.com/article/10.2478/udt-2021-0010<abstract> <title style='display:none'>Abstract</title> <p>In 1909 Borel defined normality as a notion of randomness of the digits of the representation of a real number over certain base (fractional expansion). If we think of the representation of a number over a base as an infinite sequence of symbols from a finite alphabet <italic>A</italic>, we can define normality directly for words of symbols of <italic>A</italic>: A word <italic>x</italic> is normal to the alphabet <italic>A </italic>if every finite block of symbols from <italic>A </italic>appears with the same asymptotic frequency in <italic>x</italic> as every other block of the same length. Many examples of normal words have been found since its definition, being Champernowne in 1933 the first to show an explicit and simple instance. Moreover, it has been characterized how we can select subsequences of a normal word <italic>x</italic> preserving its normality, always leaving the alphabet <italic>A </italic>fixed. In this work we consider the dual problem which consists of inserting symbols in infinitely many positions of a given word, in such a way that normality is preserved. Specifically, given a symbol <italic>b</italic> that is not present in the original alphabet <italic>A </italic>and given a word <italic>x</italic> that is normal to the alphabet <italic>A </italic>we solve how to insert the symbol <italic>b</italic> in infinitely many positions of the word <italic>x</italic> such that the resulting word is normal to the expanded alphabet <italic>A ∪</italic>{<italic>b</italic>}.</p> </abstract>ARTICLE2022-02-02T00:00:00.000+00:00On the Distribution of Modulo One in Quadratic Number Fieldshttps://sciendo.com/article/10.2478/udt-2021-0006<abstract> <title style='display:none'>Abstract</title> <p>We investigate the distribution of <italic>αp</italic> modulo one in quadratic number fields 𝕂 with class number one, where <italic>p</italic> is restricted to prime elements in the ring of integers of 𝕂. Here we improve the relevant exponent 1/4 obtained by the first- and third-named authors for imaginary quadratic number fields [<italic>On the distribution of αp modulo one in imaginary quadratic number fields with class number one</italic>, J. Théor. Nombres Bordx. <bold>32 </bold>(2020), no. 3, 719–760]) and by the first- and second-named authors for real quadratic number fields [<italic>Diophantine approximation with prime restriction in real quadratic number fields</italic>, Math. Z. (2021)] to 7/22. This generalizes a result of Harman [<italic>Diophantine approximation with Gaussian primes</italic>, Q. J. Math. <bold>70 </bold>(2019), no. 4, 1505–1519] who obtained the same exponent 7/22 for ℚ (<italic>i</italic>) by extending his method which gave this exponent for ℚ [<italic>On the distribution of αp modulo one. II</italic>, Proc. London Math. Soc. <bold>72</bold>, (1996), no. 3, 241–260]. Our proof is based on an extension of Harman’s sieve method to arbitrary number fields. Moreover, we need an asymptotic evaluation of certain smooth sums over prime ideals appearing in the above-mentioned work by the first- and second-named authors, for which we use analytic properties of Hecke L-functions with Größencharacters.</p> </abstract>ARTICLE2022-02-02T00:00:00.000+00:00Balance and Pattern Distribution of Sequences Derived from Pseudorandom Subsets of ℤhttps://sciendo.com/article/10.2478/udt-2021-0009<abstract> <title style='display:none'>Abstract</title> <p>Let <italic>q</italic> be a positive integer and <inline-formula> <alternatives> <inline-graphic xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="graphic/j_udt-2021-0009_ineq_001.png"/> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" display="inline"><mml:mrow><mml:mi>𝒮</mml:mi><mml:mo>=</mml:mo><mml:mo>{</mml:mo><mml:msub><mml:mi>x</mml:mi><mml:mn>0</mml:mn></mml:msub><mml:mo>,</mml:mo><mml:msub><mml:mi>x</mml:mi><mml:mn>1</mml:mn></mml:msub><mml:mo>,</mml:mo><mml:mo>⋯</mml:mo><mml:mo>,</mml:mo><mml:msub><mml:mi>x</mml:mi><mml:mrow><mml:mi>T</mml:mi><mml:mo>−</mml:mo><mml:mn>1</mml:mn></mml:mrow></mml:msub><mml:mo>}</mml:mo><mml:mo>⊆</mml:mo><mml:msub><mml:mi>ℤ</mml:mi><mml:mi>q</mml:mi></mml:msub><mml:mo>=</mml:mo><mml:mo>{</mml:mo><mml:mn>0</mml:mn><mml:mo>,</mml:mo><mml:mn>1</mml:mn><mml:mo>,</mml:mo><mml:mo>…</mml:mo><mml:mo>,</mml:mo><mml:mi>q</mml:mi><mml:mo>−</mml:mo><mml:mn>1</mml:mn><mml:mo>}</mml:mo></mml:mrow></mml:math> <tex-math>{\scr S} = \{{x_0},{x_1}, \cdots ,{x_{T - 1}}\}\subseteq {{\rm{\mathbb Z}}_q} = \{0,1, \ldots ,q - 1\}</tex-math> </alternatives> </inline-formula> with <inline-formula> <alternatives> <inline-graphic xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="graphic/j_udt-2021-0009_ineq_002.png"/> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" display="inline"><mml:mrow><mml:mn>0</mml:mn><mml:mo>≤</mml:mo><mml:msub><mml:mi>x</mml:mi><mml:mn>0</mml:mn></mml:msub><mml:mo>&lt;</mml:mo><mml:msub><mml:mi>x</mml:mi><mml:mn>1</mml:mn></mml:msub><mml:mo>&lt;</mml:mo><mml:mo>⋯</mml:mo><mml:mo>&lt;</mml:mo><mml:msub><mml:mi>x</mml:mi><mml:mrow><mml:mi>T</mml:mi><mml:mo>−</mml:mo><mml:mn>1</mml:mn></mml:mrow></mml:msub><mml:mo>≤</mml:mo><mml:mi>q</mml:mi><mml:mo>−</mml:mo><mml:mn>1.</mml:mn></mml:mrow></mml:math> <tex-math>0 \le {x_0} &lt; {x_1} &lt;\cdots&lt; {x_{T - 1}} \le q - 1.</tex-math> </alternatives> </inline-formula> . We derive from <italic>S </italic>three (finite) sequences: (1) For an integer <italic>M ≥ </italic>2let (<italic>s<sub>n</sub></italic>)be the <italic>M</italic>-ary sequence defined by <italic>s<sub>n</sub> ≡ x<sub>n</sub></italic><sub>+1</sub> <italic>− x<sub>n</sub></italic> mod <italic>M, n</italic> =0<italic/>, 1,...<italic>,T −</italic> 2<italic>.</italic></p> <p>(2) For an integer <italic>m ≥</italic> 2let (<italic>t<sub>n</sub></italic>) be the binary sequence defined by <inline-formula> <alternatives> <inline-graphic xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="graphic/j_udt-2021-0009_ineq_003.png"/> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" display="inline"><mml:mrow><mml:mtable><mml:mtr><mml:mtd><mml:mrow><mml:msub><mml:mi>s</mml:mi><mml:mi>n</mml:mi></mml:msub><mml:mo>≡</mml:mo><mml:msub><mml:mi>x</mml:mi><mml:mrow><mml:mi>n</mml:mi><mml:mo>+</mml:mo><mml:mn>1</mml:mn></mml:mrow></mml:msub><mml:mo>−</mml:mo><mml:msub><mml:mi>x</mml:mi><mml:mi>n</mml:mi></mml:msub><mml:mtext> </mml:mtext><mml:mi>mod</mml:mi><mml:mtext> </mml:mtext><mml:mi>M</mml:mi><mml:mo>,</mml:mo></mml:mrow></mml:mtd><mml:mtd><mml:mrow><mml:mi>n</mml:mi><mml:mo>=</mml:mo><mml:mn>0</mml:mn><mml:mo>,</mml:mo><mml:mn>1</mml:mn><mml:mo>,</mml:mo><mml:mo>⋯</mml:mo><mml:mo>,</mml:mo><mml:mi>T</mml:mi><mml:mo>−</mml:mo><mml:mn>2.</mml:mn></mml:mrow></mml:mtd></mml:mtr></mml:mtable></mml:mrow></mml:math> <tex-math>\matrix{{{s_n} \equiv {x_{n + 1}} - {x_n}\,\bmod \,M,} &amp;#38; {n = 0,1, \cdots ,T - 2.}\cr}</tex-math> </alternatives> </inline-formula> <italic>n</italic> =0<italic/>, 1,...<italic>,T −</italic> 2<italic>.</italic></p> <p>(3) Let (<italic>u<sub>n</sub></italic>) be the characteristic sequence of <italic>S</italic>, <inline-formula> <alternatives> <inline-graphic xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="graphic/j_udt-2021-0009_ineq_004.png"/> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" display="inline"><mml:mrow><mml:mtable><mml:mtr><mml:mtd><mml:mrow><mml:msub><mml:mi>t</mml:mi><mml:mi>n</mml:mi></mml:msub><mml:mo>=</mml:mo><mml:mrow><mml:mo>{</mml:mo><mml:mrow><mml:mtable columnalign="left"><mml:mtr columnalign="left"><mml:mtd columnalign="left"><mml:mn>1</mml:mn></mml:mtd><mml:mtd columnalign="left"><mml:mrow><mml:mtext>if</mml:mtext><mml:mtext> </mml:mtext><mml:mn>1</mml:mn><mml:mo>≤</mml:mo><mml:msub><mml:mi>x</mml:mi><mml:mrow><mml:mi>n</mml:mi><mml:mo>+</mml:mo><mml:mn>1</mml:mn></mml:mrow></mml:msub><mml:mo>−</mml:mo><mml:msub><mml:mi>x</mml:mi><mml:mi>n</mml:mi></mml:msub><mml:mo>≤</mml:mo><mml:mi>m</mml:mi><mml:mo>−</mml:mo><mml:mn>1</mml:mn><mml:mo>,</mml:mo></mml:mrow></mml:mtd></mml:mtr><mml:mtr columnalign="left"><mml:mtd columnalign="left"><mml:mrow><mml:mn>0</mml:mn><mml:mo>,</mml:mo></mml:mrow></mml:mtd><mml:mtd columnalign="left"><mml:mrow><mml:mtext>otherwise</mml:mtext><mml:mo>,</mml:mo></mml:mrow></mml:mtd></mml:mtr></mml:mtable></mml:mrow></mml:mrow></mml:mrow></mml:mtd><mml:mtd><mml:mrow><mml:mi>n</mml:mi><mml:mo>=</mml:mo><mml:mn>0</mml:mn><mml:mo>,</mml:mo><mml:mn>1</mml:mn><mml:mo>,</mml:mo><mml:mo>…</mml:mo><mml:mo>,</mml:mo><mml:mi>T</mml:mi><mml:mo>−</mml:mo><mml:mn>2.</mml:mn></mml:mrow></mml:mtd></mml:mtr></mml:mtable></mml:mrow></mml:math> <tex-math>\matrix{{{t_n} = \left\{{\matrix{1 \hfill &amp;#38; {{\rm{if}}\,1 \le {x_{n + 1}} - {x_n} \le m - 1,} \hfill\cr{0,} \hfill &amp;#38; {{\rm{otherwise}},} \hfill\cr}} \right.} &amp;#38; {n = 0,1, \ldots ,T - 2.}\cr}</tex-math> </alternatives> </inline-formula> <italic>n</italic> =0<italic/>, 1,...<italic>,q −</italic> 1.</p> <p>We study the balance and pattern distribution of the sequences (<italic>s<sub>n</sub></italic>), (<italic>t<sub>n</sub></italic>)and (<italic>u<sub>n</sub></italic>). For sets <italic>S </italic>with desirable pseudorandom properties, more precisely, sets with low correlation measures, we show the following:</p> <p>(1) The sequence (<italic>s<sub>n</sub></italic>) is (asymptotically) balanced and has uniform pattern distribution if <italic>T </italic>is of smaller order of magnitude than <italic>q</italic>.</p> <p>(2) The sequence (<italic>t<sub>n</sub></italic>) is balanced and has uniform pattern distribution if <italic>T </italic>is approximately <inline-formula> <alternatives> <inline-graphic xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="graphic/j_udt-2021-0009_ineq_005.png"/> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" display="inline"><mml:mrow><mml:mtable><mml:mtr><mml:mtd><mml:mrow><mml:msub><mml:mi>u</mml:mi><mml:mi>n</mml:mi></mml:msub><mml:mo>=</mml:mo><mml:mrow><mml:mo>{</mml:mo><mml:mrow><mml:mtable columnalign="left"><mml:mtr columnalign="left"><mml:mtd columnalign="left"><mml:mn>1</mml:mn></mml:mtd><mml:mtd columnalign="left"><mml:mrow><mml:mtext>if</mml:mtext><mml:mtext> </mml:mtext><mml:mi>n</mml:mi><mml:mo>∈</mml:mo><mml:mi>𝒮</mml:mi><mml:mo>,</mml:mo></mml:mrow></mml:mtd></mml:mtr><mml:mtr columnalign="left"><mml:mtd columnalign="left"><mml:mrow><mml:mn>0</mml:mn><mml:mo>,</mml:mo></mml:mrow></mml:mtd><mml:mtd columnalign="left"><mml:mrow><mml:mtext>otherwise</mml:mtext><mml:mo>,</mml:mo></mml:mrow></mml:mtd></mml:mtr></mml:mtable></mml:mrow></mml:mrow></mml:mrow></mml:mtd><mml:mtd><mml:mrow><mml:mi>n</mml:mi><mml:mo>=</mml:mo><mml:mn>0</mml:mn><mml:mo>,</mml:mo><mml:mn>1</mml:mn><mml:mo>,</mml:mo><mml:mo>…</mml:mo><mml:mo>,</mml:mo><mml:mi>q</mml:mi><mml:mo>−</mml:mo><mml:mn>1.</mml:mn></mml:mrow></mml:mtd></mml:mtr></mml:mtable></mml:mrow></mml:math> <tex-math>\matrix{{{u_n} = \left\{{\matrix{1 \hfill &amp;#38; {{\rm{if}}\,n \in {\scr S},} \hfill\cr{0,} \hfill &amp;#38; {{\rm{otherwise}},} \hfill\cr}} \right.} &amp;#38; {n = 0,1, \ldots ,q - 1.}\cr}</tex-math> </alternatives> </inline-formula> .</p> <p>(3) The sequence (<italic>u<sub>n</sub></italic>) is balanced and has uniform pattern distribution if <italic>T </italic>is approximately <sup><italic>q</italic></sup><sub>2</sub>.</p> <p>These results are motivated by earlier results for the sets of quadratic residues and primitive roots modulo a prime. We unify these results and derive many further (asymptotically) balanced sequences with uniform pattern distribution from pseudorandom subsets.</p> </abstract>ARTICLE2022-02-02T00:00:00.000+00:00AO. Univ.-Prof. Dr. Reinhard Winkler (1964–2021) An Obituaryhttps://sciendo.com/article/10.2478/udt-2021-0011ARTICLE2022-02-02T00:00:00.000+00:00Mahler’s Conjecture on (3/2)mod 1https://sciendo.com/article/10.2478/udt-2021-0007<abstract> <title style='display:none'>Abstract</title> <p>K. Mahler’s conjecture: There exists no <italic>ξ</italic> ∈ ℝ<sup>+</sup> such that the fractional parts {<italic>ξ</italic>(3/2)<sup><italic>n</italic></sup>} satisfy 0 <italic>≤</italic> {<italic>ξ</italic>(3/2)<sup><italic>n</italic></sup>} &lt; 1/2 for all <italic>n</italic> = 0, 1, 2,... Such a <italic>ξ</italic>, if exists, is called a Mahler’s <italic>Z</italic>-number. In this paper we prove that if <italic>ξ</italic> is a <italic>Z</italic>-number, then the sequence <italic>x<sub>n</sub></italic> = {<italic>ξ</italic>(3/2)<sup><italic>n</italic></sup>}, <italic>n</italic> =1, 2,... has asymptotic distribution function <italic>c</italic><sub>0</sub>(<italic>x</italic>), where <italic>c</italic><sub>0</sub>(<italic>x</italic>)=1 for <italic>x</italic> ∈ (0, 1].</p> </abstract>ARTICLE2022-02-02T00:00:00.000+00:00Extreme Values of Euler-Kronecker Constantshttps://sciendo.com/article/10.2478/udt-2021-0002<abstract> <title style='display:none'>Abstract</title> <p>In a family of <italic>S<sub>n</sub></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>d<sub>K</sub></italic>+ <italic>O</italic>(log log log <italic>d<sub>K</sub></italic>) and loglog <italic>d<sub>K</sub></italic> + <italic>O</italic>(log log log <italic>d<sub>K</sub></italic>), resp., where <italic>d<sub>K</sub></italic> is the absolute value of the discriminant of a number field <italic>K</italic>.</p> </abstract>ARTICLE2021-10-30T00:00:00.000+00:00On the Classification of Solutions of Quantum Functional Equations with Cyclic and Semi-Cyclic Supportshttps://sciendo.com/article/10.2478/udt-2021-0001<abstract> <title style='display:none'>Abstract</title> <p>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.</p> </abstract>ARTICLE2021-10-30T00:00:00.000+00:00Uniform Distribution of the Weighted Sum-of-Digits Functionshttps://sciendo.com/article/10.2478/udt-2021-0005<abstract> <title style='display:none'>Abstract</title> <p>The higher-dimensional generalization of the weighted <italic>q</italic>-adic sum-of-digits functions <italic>s<sub>q,γ</sub></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>s<sub>q,γ</sub></italic> (<italic>n</italic>), <italic>n</italic> = 0, 1, 2,... We also prove the uniform distribution modulo one of related sequences <italic>h</italic><sub>1</sub><italic>s<sub>q, γ</sub></italic> (<italic>n</italic>)+<italic>h</italic><sub>2</sub><italic>s<sub>q,γ</sub></italic> (<italic>n</italic> +1), where <italic>h</italic><sub>1</sub> and <italic>h</italic><sub>2</sub> are integers such that <italic>h</italic><sub>1</sub> + <italic>h</italic><sub>2</sub> ≠ 0 and that the akin two-dimensional sequence <italic>s<sub>q,γ</sub></italic> (<italic>n</italic>), <italic>s<sub>q,γ</sub></italic> (<italic>n</italic> +1) cannot be uniformly distributed modulo one if <italic>q ≥</italic> 3. The properties of the two-dimensional sequence <italic>s<sub>q,γ</sub></italic> (<italic>n</italic>),s<italic><sub>q,γ</sub></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.</p> </abstract>ARTICLE2021-10-30T00:00:00.000+00:00The Inequality of Erdős-Turán-Koksma in the Terms of the Functions of the System Γ https://sciendo.com/article/10.2478/udt-2021-0004<abstract> <title style='display:none'>Abstract</title> <p>In the present paper the author uses the function system Γ<sub><italic>ℬ</italic><sub><italic>s</italic></sub></sub>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.</p> </abstract>ARTICLE2021-10-30T00:00:00.000+00:00Families of Well Approximable Measureshttps://sciendo.com/article/10.2478/udt-2021-0003<abstract> <title style='display:none'>Abstract</title> <p>We provide an algorithm to approximate a finitely supported discrete measure <italic>μ</italic> by a measure <italic>ν<sub>N</sub></italic> corresponding to a set of <italic>N</italic> points so that the total variation between <italic>μ</italic> and <italic>ν<sub>N</sub></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><sup>d </sup></italic>with a sufficient decay rate on the weights of each point, then <italic>μ</italic> can be approximated by <italic>ν<sub>N</sub></italic> with total variation, and hence star-discrepancy, bounded above by (log <italic>N</italic>)<italic>N<sup>−</sup></italic><sup>1</sup>. 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"/> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" display="inline"><mml:mrow><mml:msup><mml:mrow><mml:mrow><mml:mrow><mml:mo>(</mml:mo><mml:mrow><mml:mo>log</mml:mo><mml:mi> </mml:mi><mml:mi>N</mml:mi></mml:mrow><mml:mo>)</mml:mo></mml:mrow></mml:mrow></mml:mrow><mml:mrow><mml:mi>d</mml:mi><mml:mo>−</mml:mo><mml:mfrac><mml:mn>1</mml:mn><mml:mn>2</mml:mn></mml:mfrac></mml:mrow></mml:msup><mml:msup><mml:mrow><mml:mi>N</mml:mi></mml:mrow><mml:mrow><mml:mo>−</mml:mo><mml:mn>1</mml:mn></mml:mrow></mml:msup></mml:mrow></mml: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.</p> </abstract>ARTICLE2021-10-30T00:00:00.000+00:00Word Metric, Stationary Measure and Minkowski’s Question Mark Functionhttps://sciendo.com/article/10.2478/udt-2020-0009<abstract><title style='display:none'>Abstract</title><p>Given a countably infinite group <italic>G</italic> acting on some space <italic>X</italic>, an increasing family of finite subsets <italic>G<sub>n</sub></italic>, <italic>x</italic>∈ <italic>X</italic> and a function <italic>f</italic> over <italic>X</italic> we consider the sums <italic>S<sub>n</sub></italic>(<italic>f, x</italic>) = ∑<italic><sub>g∈Gn</sub>f</italic>(<italic>gx</italic>). The asymptotic behaviour of <italic>S<sub>n</sub></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>G<sub>n</sub></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<sup>∗n</sup></italic>.</p></abstract>ARTICLE2020-12-25T00:00:00.000+00:00en-us-1