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

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.
</abstract>

On the Distribution of <italic>αp</italic> Modulo One in Quadratic Number Fields

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

K. Mahler’s conjecture: There exists no <italic>ξ</italic> ∈ ℝ+ such that the fractional parts {<italic>ξ</italic>(3/2)<italic>n</italic>} satisfy 0 <italic>≤</italic> {<italic>ξ</italic>(3/2)<italic>n</italic>} &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>xn</italic> = {<italic>ξ</italic>(3/2)<italic>n</italic>}, <italic>n</italic> =1, 2,... has asymptotic distribution function <italic>c</italic>0(<italic>x</italic>), where <italic>c</italic>0(<italic>x</italic>)=1 for <italic>x</italic> ∈ (0, 1].
</abstract>

Mahler’s Conjecture on <italic>ξ</italic>(3/2)<italic>n</italic>mod 1

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

Let <italic>K </italic>be a number field, and let <italic>ℓ</italic> be a prime number. Fix some elements <italic>α</italic>1,...<italic>,αr</italic> of <italic>K× </italic>which generate a subgroup of <italic>K× </italic>of rank <italic>r</italic>. Let <italic>n</italic>1,...<italic>,nr</italic>, <italic>m</italic> be positive integers with <italic>m</italic> ⩾ <italic>ni</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>ℓm, 
<inline-formula>
<alternatives>
<inline-graphic xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="graphic/j_udt-2021-0008_ineq_001.png"></inline-graphic>
<math xmlns:mml="http://www.w3.org/1998/Math/MathML" display="inline"><mrow><mroot><mrow><msub><mi>α</mi><mn>1</mn></msub></mrow><mrow><msup><mi>ℓ</mi><mrow><msub><mi>n</mi><mn>1</mn></msub></mrow></msup></mrow></mroot><mo>,</mo><mo>…</mo><mo>,</mo><mroot><mrow><msub><mi>α</mi><mi>r</mi></msub></mrow><mrow><msup><mi>ℓ</mi><mrow><msub><mi>n</mi><mi>r</mi></msub></mrow></msup></mrow></mroot></mrow></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>ζℓm</italic>) for all <italic>n</italic>1,...<italic>, nr, 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.
</abstract>

Divisibility Parameters and the Degree of Kummer Extensions of Number Fields

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

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"></inline-graphic>
<math xmlns:mml="http://www.w3.org/1998/Math/MathML" display="inline"><mrow><mi>𝒮</mi><mo>=</mo><mo>{</mo><msub><mi>x</mi><mn>0</mn></msub><mo>,</mo><msub><mi>x</mi><mn>1</mn></msub><mo>,</mo><mo>⋯</mo><mo>,</mo><msub><mi>x</mi><mrow><mi>T</mi><mo>−</mo><mn>1</mn></mrow></msub><mo>}</mo><mo>⊆</mo><msub><mi>ℤ</mi><mi>q</mi></msub><mo>=</mo><mo>{</mo><mn>0</mn><mo>,</mo><mn>1</mn><mo>,</mo><mo>…</mo><mo>,</mo><mi>q</mi><mo>−</mo><mn>1</mn><mo>}</mo></mrow></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"></inline-graphic>
<math xmlns:mml="http://www.w3.org/1998/Math/MathML" display="inline"><mrow><mn>0</mn><mo>≤</mo><msub><mi>x</mi><mn>0</mn></msub><mo>&lt;</mo><msub><mi>x</mi><mn>1</mn></msub><mo>&lt;</mo><mo>⋯</mo><mo>&lt;</mo><msub><mi>x</mi><mrow><mi>T</mi><mo>−</mo><mn>1</mn></mrow></msub><mo>≤</mo><mi>q</mi><mo>−</mo><mn>1.</mn></mrow></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>sn</italic>)be the <italic>M</italic>-ary sequence defined by <italic>sn ≡ xn</italic>+1 <italic>− xn</italic> mod <italic>M, n</italic> =0<italic></italic>, 1,...<italic>,T −</italic> 2<italic>.</italic>
(2) For an integer <italic>m ≥</italic> 2let (<italic>tn</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"></inline-graphic>
<math xmlns:mml="http://www.w3.org/1998/Math/MathML" display="inline"><mrow><mtable><mtr><mtd><mrow><msub><mi>s</mi><mi>n</mi></msub><mo>≡</mo><msub><mi>x</mi><mrow><mi>n</mi><mo>+</mo><mn>1</mn></mrow></msub><mo>−</mo><msub><mi>x</mi><mi>n</mi></msub><mtext> </mtext><mi>mod</mi><mtext> </mtext><mi>M</mi><mo>,</mo></mrow></mtd><mtd><mrow><mi>n</mi><mo>=</mo><mn>0</mn><mo>,</mo><mn>1</mn><mo>,</mo><mo>⋯</mo><mo>,</mo><mi>T</mi><mo>−</mo><mn>2.</mn></mrow></mtd></mtr></mtable></mrow></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></italic>, 1,...<italic>,T −</italic> 2<italic>.</italic>
(3) Let (<italic>un</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"></inline-graphic>
<math xmlns:mml="http://www.w3.org/1998/Math/MathML" display="inline"><mrow><mtable><mtr><mtd><mrow><msub><mi>t</mi><mi>n</mi></msub><mo>=</mo><mrow><mo>{</mo><mrow><mtable columnalign="left"><mtr columnalign="left"><mtd columnalign="left"><mn>1</mn></mtd><mtd columnalign="left"><mrow><mtext>if</mtext><mtext> </mtext><mn>1</mn><mo>≤</mo><msub><mi>x</mi><mrow><mi>n</mi><mo>+</mo><mn>1</mn></mrow></msub><mo>−</mo><msub><mi>x</mi><mi>n</mi></msub><mo>≤</mo><mi>m</mi><mo>−</mo><mn>1</mn><mo>,</mo></mrow></mtd></mtr><mtr columnalign="left"><mtd columnalign="left"><mrow><mn>0</mn><mo>,</mo></mrow></mtd><mtd columnalign="left"><mrow><mtext>otherwise</mtext><mo>,</mo></mrow></mtd></mtr></mtable></mrow></mrow></mrow></mtd><mtd><mrow><mi>n</mi><mo>=</mo><mn>0</mn><mo>,</mo><mn>1</mn><mo>,</mo><mo>…</mo><mo>,</mo><mi>T</mi><mo>−</mo><mn>2.</mn></mrow></mtd></mtr></mtable></mrow></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></italic>, 1,...<italic>,q −</italic> 1.
We study the balance and pattern distribution of the sequences (<italic>sn</italic>), (<italic>tn</italic>)and (<italic>un</italic>). For sets <italic>S </italic>with desirable pseudorandom properties, more precisely, sets with low correlation measures, we show the following:
(1) The sequence (<italic>sn</italic>) is (asymptotically) balanced and has uniform pattern distribution if <italic>T </italic>is of smaller order of magnitude than <italic>q</italic>.
(2) The sequence (<italic>tn</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"></inline-graphic>
<math xmlns:mml="http://www.w3.org/1998/Math/MathML" display="inline"><mrow><mtable><mtr><mtd><mrow><msub><mi>u</mi><mi>n</mi></msub><mo>=</mo><mrow><mo>{</mo><mrow><mtable columnalign="left"><mtr columnalign="left"><mtd columnalign="left"><mn>1</mn></mtd><mtd columnalign="left"><mrow><mtext>if</mtext><mtext> </mtext><mi>n</mi><mo>∈</mo><mi>𝒮</mi><mo>,</mo></mrow></mtd></mtr><mtr columnalign="left"><mtd columnalign="left"><mrow><mn>0</mn><mo>,</mo></mrow></mtd><mtd columnalign="left"><mrow><mtext>otherwise</mtext><mo>,</mo></mrow></mtd></mtr></mtable></mrow></mrow></mrow></mtd><mtd><mrow><mi>n</mi><mo>=</mo><mn>0</mn><mo>,</mo><mn>1</mn><mo>,</mo><mo>…</mo><mo>,</mo><mi>q</mi><mo>−</mo><mn>1.</mn></mrow></mtd></mtr></mtable></mrow></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>
.
(3) The sequence (<italic>un</italic>) is balanced and has uniform pattern distribution if <italic>T </italic>is approximately <italic>q</italic>2.
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.
</abstract>

Research Center for Number Theory and Its Applications

Johann Radon Institute for Computational and Applied Mathematics

Balance and Pattern Distribution of Sequences Derived from Pseudorandom Subsets of ℤ<italic>q</italic>

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

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>}.
</abstract>

From Randomness in Two Symbols to Randomness in Three Symbols

AO. Univ.-Prof. Dr. Reinhard Winkler (1964–2021) An Obituary

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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 16 (2021): Issue 2 (December 2021)

On the Distribution of αp Modulo One in Quadratic Number Fields

Mahler’s Conjecture on ξ(3/2)ⁿmod 1

Divisibility Parameters and the Degree of Kummer Extensions of Number Fields

Balance and Pattern Distribution of Sequences Derived from Pseudorandom Subsets of ℤ_q

From Randomness in Two Symbols to Randomness in Three Symbols

AO. Univ.-Prof. Dr. Reinhard Winkler (1964–2021) An Obituary

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All Volumes and Issues in this Journal

Uniform distribution theory

Volume 16 (2021): Issue 2 (December 2021)

On the Distribution of αp Modulo One in Quadratic Number Fields

Mahler’s Conjecture on ξ(3/2)nmod 1

Divisibility Parameters and the Degree of Kummer Extensions of Number Fields

Balance and Pattern Distribution of Sequences Derived from Pseudorandom Subsets of ℤq

From Randomness in Two Symbols to Randomness in Three Symbols

AO. Univ.-Prof. Dr. Reinhard Winkler (1964–2021) An Obituary

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All Volumes and Issues in this Journal

Mahler’s Conjecture on ξ(3/2)ⁿmod 1

Balance and Pattern Distribution of Sequences Derived from Pseudorandom Subsets of ℤ_q