The Seventh International Conference on Uniform Distribution Theory (UDT 2021) 

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

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>nm</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.
</abstract>

Department of Computer Science and Information Theory

Budapest University of Technology and Economics

Bounds on the size of Progression-Free Sets in ℤ<italic>mn</italic>

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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
<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>
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>.
</abstract>

Products of Integers with Few Nonzero Digits

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The reduction modulo <italic>p</italic> of a family of lacunary integer polynomials, associated with the dynamical zeta function <italic>ζβ</italic>(<italic>z</italic>)of the <italic>β</italic>-shift, for <italic>β></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>p</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>></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>p</italic> is obtained in a few cases. In general it is expected since it must exist according to the Langlands program.
</abstract>

On a Class of Lacunary Almost Newman Polynomials Modulo <italic>P</italic> and Density Theorems

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

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).
</abstract>

Insertion in Constructed Normal Numbers

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

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>i</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>i</italic>(<italic>x, n</italic>))<italic>n</italic> also diverges significantly.
</abstract>

A Typical Number is Extremely Non-Normal

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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>d</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"></inline-graphic>
<math xmlns:mml="http://www.w3.org/1998/Math/MathML" display="inline"><mrow><mover accent="true"><mi>ω</mi><mo>⌢</mo></mover><mrow><mo>(</mo><mi>ξ</mi><mo>)</mo></mrow><mo>≤</mo><mfrac><mrow><msqrt><mrow><mn>5</mn><mo>-</mo><mn>1</mn></mrow></msqrt></mrow><mn>2</mn></mfrac></mrow></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.
</abstract>

On Some Properties of Irrational Subspaces

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

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"></graphic>
<math xmlns:mml="http://www.w3.org/1998/Math/MathML" display="block"><mrow><msub><mrow><mrow><mrow><mo>(</mo><mrow><mi>g</mi><mrow><mo>(</mo><mi>k</mi><mo>)</mo></mrow><mo>⋅</mo><mi>F</mi><mrow><mo>(</mo><mrow><mi>k</mi><mi>α</mi></mrow><mo>)</mo></mrow></mrow><mo>)</mo></mrow></mrow></mrow><mrow><mi>k</mi><mo>∈</mo><mi>ℕ</mi></mrow></msub><mo>,</mo></mrow></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.
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.
</abstract>

ioannis.tsokanos@postgrad.manchester.ac.uk

Density of Oscillating Sequences in the Real Line

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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 17 (2022): Issue 1 (May 2022)

The Seventh International Conference on Uniform Distribution Theory (UDT 2021)

Bounds on the size of Progression-Free Sets in ℤ_mⁿ

Products of Integers with Few Nonzero Digits

On a Class of Lacunary Almost Newman Polynomials Modulo P and Density Theorems

Insertion in Constructed Normal Numbers

A Typical Number is Extremely Non-Normal

On Some Properties of Irrational Subspaces

Density of Oscillating Sequences in the Real Line

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