The Journal of Slovak Academy of Sciences

Uniform distribution theory

<abstract>
<title style='display:none'>Abstract</title>
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>

Abstract

Keywords

MSC 2010

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

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Uniform distribution theory ist in den folgenden Services indiziert: <UL> <LI>Baidu Scholar </LI> <LI>CNKI Scholar (China National Knowledge Infrastructure) </LI> <LI>CNPIEC - cnpLINKer </LI> <LI>Dimensions </LI> <LI>EBSCO </LI> <LI>ExLibris </LI> <LI>Google Scholar </LI> <LI>J-Gate </LI> <LI>JournalGuide </LI> <LI>JournalTOCs </LI> <LI>KESLI-NDSL (Korean National Discovery for Science Leaders) </LI> <LI>Mathematical Reviews (MathSciNet) </LI> <LI>MyScienceWork </LI> <LI>Naver Academic </LI> <LI>Naviga (Softweco) </LI> <LI>QOAM (Quality Open Access Market) </LI> <LI>ReadCube </LI> <LI>SCILIT </LI> <LI>Semantic Scholar </LI> <LI>TDNet </LI> <LI>Ulrich's Periodicals Directory/ulrichsweb </LI> <LI>WanFang Data </LI> <LI>WorldCat (OCLC) </LI> <LI>X-MOL </LI> <LI>zbMATH Open </LI></UL>Additionally, the journal is registered and indexed in the Crossref database.

MANAGING EDITORS Karpenkov, Oleg (Liverpool) <A href="mailto:karpenk@liv.ac.uk">karpenk@liv.ac.uk</A> Lattice geometry, multidimensional continued fractions, geometry of numbers. Nair, Radhakrishnan (Liverpool) <A href="mailto:nair@liverpool.ac.uk">nair@liverpool.ac.uk</A> Ergodic theoretic aspects of uniform distribution, issues related to pointwise convergence, metrical theory of uniform distribution, exceptional sets, exponential sums, distribution of primes, densities and combinatorial number theory. Pillichshammer, Friedrich (Linz) <A href="mailto:friedrich.pillichshammer@jku.at">friedrich.pillichshammer@jku.at</A> Discrepancy, digital nets, quasi-Monte Carlo integration, lattice rules, tractability of high-dimensional problems. EDITORIAL BOARD Akiyama, Shigeki (Tsukuba) <A href="mailto:akiyama@math.tsukuba.ac.jp">akiyama@math.tsukuba.ac.jp</A> Dynamics emerging from sequences, continued fractions, interplay between symbolic dynamics and number theory, spectral properties of sequences. Allouche, Jean-Paul (Paris) <A href="mailto:jean-paul.allouche@imj-prg.fr">jean-paul.allouche@imj-prg.fr</A> Combinatorial number theory, continued fractions, automatic sequences. Baláž, Vladimír (Bratislava) <A href="mailto:vladimir.balaz@stuba.sk">vladimir.balaz@stuba.sk</A> Theory of densities, summation methods. Berkes, István (Graz) <A href="mailto:berkes@tugraz.at">berkes@tugraz.at</A> Discrepancies, distribution of one dimensional and multidimensional sequences. Bugeaud, Yann (Strasbourg) <A href="mailto:bugeaud@math.unistra.fr">bugeaud@math.unistra.fr</A> Diophantine approximation, diophantine equations, continued fraction, distribution modulo one. Dick, Josef (Sydney) <A href="mailto:josef.dick@unsw.edu.au">josef.dick@unsw.edu.au</A> Quasi-Monte Carlo methods, digital nets and sequences, lattice rules, geometric discrepancy, uniform distribution on the sphere. Drmota, Michael (Vienna) <A href="mailto:michael.drmota@tuwien.ac.at">michael.drmota@tuwien.ac.at</A> Continuous uniform distribution, discrepancies, distribution of one dimensional and multidimensional sequences. Dubickas, Artūras (Vilnius) <A href="mailto:arturas.dubickas@mif.vu.lt">arturas.dubickas@mif.vu.lt</A> Distribution modulo one, diophantine approximation. Faure, Henri (Marseille) <A href="mailto:henri.faure@univ-amu.fr">henri.faure@univ-amu.fr</A> Distribution of one dimensional and multidimensional sequences, discrepancies, quasi-Monte Carlo integration. Giuliano Antonini, Rita (Pisa) <A href="mailto:giuliano@dm.unipi.it">giuliano@dm.unipi.it</A> Theory of densities, distribution functions of sequences, summation methods, continued fractions. Grabner, Peter (Graz) <A href="mailto:peter.grabner@tugraz.at">peter.grabner@tugraz.at</A> Point distributions on manifolds, extremal energy point sets, dynamical and fractal aspects of uniformly distributed sequences. Grekos, Georges (Saint-Etienne) <A href="mailto:grekos@univ-st-etienne.fr">grekos@univ-st-etienne.fr</A> Theory of densities, combinatorial number theory. Grozdanov, Vasil (Blagoevgrad) <A href="mailto:vassgrozdanov@yahoo.com">vassgrozdanov@yahoo.com</A> Well distributed sequences and nets, discrepancy and diaphony, applications of uniformly distributed sequences. Gyarmati, Katalin (Budapest) <A href="mailto:gykati@cs.elte.hu">gykati@cs.elte.hu</A> Pseudorandomness, combinatorial number theory. Konyagin, Sergei (Moscow) <A href="mailto:konyagin23@gmail.com">konyagin23@gmail.com</A> Pseudorandom numbers generators, combinatorial number theory. Kraaikamp, Cor (Delft) <A href="mailto:c.kraaikamp@ewi.tudelft.nl">c.kraaikamp@ewi.tudelft.nl</A> Metric properties of number theoretic expansions: beta-expansions, one and multi-dimensional continued fraction algorithms, Lüroth- and Engel series. Kritzer, Peter (Linz) <A href="mailto:peter.kritzer@oeaw.ac.at">peter.kritzer@oeaw.ac.at</A> Discrepancy, digital nets, lattice rules, quasi-Monte Carlo methods, tractability of multivariate problems. Kühleitner, Manfred (Vienna)) (†) <A href="mailto:kleitner@boku.ac.at">kleitner@boku.ac.at</A> Lattice points in bodies, the asymptotic theory of arithmetic functions, divisor problems. Larcher, Gerhard (Linz) <A href="mailto:Gerhard.Larcher@jku.at">Gerhard.Larcher@jku.at</A> Irregularities of distribution, discrepancy, diophantine approximation, low-discrepancy sequences, quasi-Monte Carlo methods, applications in finance. Lev, Vsevolod (Haifa) <A href="mailto:seva@math.haifa.ac.il">seva@math.haifa.ac.il</A> Combinatorial number theory, additive combinatorics. Liardet, Pierre (Marseille) (†) Luca, Florian (Johannesburg) <A href="mailto:Florian.Luca@wits.ac.za">Florian.Luca@wits.ac.za</A> Distributions of binary sequences, combinatorial number theory, diophantine approximations and diophantine equations, continued fractions. Mauduit, Christian (Marseille) (†) Mišík, Ladislav (Ostrava) <A href="mailto:ladislav.misik@osu.cz">ladislav.misik@osu.cz</A> The theory of generalized densities, measures on sets of integers. Niederreiter, Harald (Linz, Salzburg) <A href="mailto:ghnied@gmail.com">ghnied@gmail.com</A> <A href="mailto:harald.niederreiter@oeaw.ac.at">harald.niederreiter@oeaw.ac.at</A> All parts of uniform distribution theory. Nowak, Werner Georg (Vienna) <A href="mailto:werner_georg.nowak@boku.ac.at">werner_georg.nowak@boku.ac.at</A> Lattice points in large regions, analytic theory of arithmetic functions. Ohkubo, Yukio (Kagoshima) <A href="mailto:ohkubo@eco.iuk.ac.jp">ohkubo@eco.iuk.ac.jp</A> Distribution of one dimensional and multidimensional sequences, continuous uniform distribution, discrepancies. • Paštéka, Milan (Bratislava) <A href="mailto:pasteka@mat.savba.sk">pasteka@mat.savba.sk</A> Theory of densities, uniform distribution in groups and rings. Pethő, Attila (Debrecen) <A href="mailto:petho.attila@inf.unideb.hu">petho.attila@inf.unideb.hu</A> Diophantine equations, distribution of polynomials and algebraic numbers, radix representations, cryptography. Pintz, János (Budapest) <A href="mailto:pintz.janos@renyi.mta.hu">pintz.janos@renyi.mta.hu</A> Analytic number theory, distribution of prime numbers. Porubský, Štefan (Prague) <A href="mailto:sporubsky@hotmail.com">sporubsky@hotmail.com</A> Distribution functions of sequences, combinatorial number theory, arithmetic densities, arithmetic functions, summation methods, pseudorandom number generators, cryptography. Sárközy, András (Budapest) <A href="mailto:sarkozy@cs.elte.hu">sarkozy@cs.elte.hu</A> Distribution of binary sequences, pseudorandom number generators, combinatorial number theory, character sums, number theoretic ciphers. Shkredov, Ilya (Moscow) <A href="mailto:ilya.shkredov@gmail.com">ilya.shkredov@gmail.com</A> Combinatorial number theory, continued fractions. Sós, Vera T. (Budapest) <A href="mailto:sos@renyi.hu">sos@renyi.hu</A> Uniform distribution of sequences and discrepancies. Strauch, Oto (Bratislava) <A href="mailto:oto.strauch@mat.savba.sk">oto.strauch@mat.savba.sk</A> The theory of distribution functions of sequences, discrepancies, metric theory of diophantine approximations. Tezuka, Shu (Kyushu) <A href="mailto:shu_tezuka@yahoo.co.jp">shu_tezuka@yahoo.co.jp</A> Quasi-Monte Carlo integration, quasi-Monte Carlo methods in financial mathematics, pseudorandom number generators. Tichy, Robert F. (Graz) <A href="mailto:tichy@tugraz.at">tichy@tugraz.at</A> Discrepancies, quasi - Monte Carlo integration, quasi - Monte Carlo methods in financial mathematics, diophantine approximations and diophantine equations. Tóth, János T. (Ostrava) <A href="mailto:tothj@ujs.sk">tothj@ujs.sk</A> Distribution of block sequences, various kinds of dense sequences. Ustinov, Alexey. (Khabarovsk) <A href="mailto:ustinov@iam.khv.ru">ustinov@iam.khv.ru</A> Trigonometric (exponential) sums, continued fractions, geometry of numbers. Verger-Gaugry, Jean-Louis. (Le Bourget-du-Lac) <A href="mailto:Jean-Louis.Verger-Gaugry@univ-smb.fr">Jean-Louis.Verger-Gaugry@univ-smb.fr</A> Number theory, dynamics. Weber, Michel (Strasbourg) <A href="mailto:michel.weber@math.unistra.fr">michel.weber@math.unistra.fr</A> Ergotic theory, pointwise convergence, probability theory. Winkler, Reinhard (Vienna) <A href="mailto:reinhard.winkler@tuwien.ac.at">reinhard.winkler@tuwien.ac.at</A> Distribution of integer sequences and sequences from groups and generalized spaces, theory of distribution functions of sequences (limit measures), distribution of binary sequences, dynamic emerging from sequences. Winterhof, Arne (Linz) <A href="mailto:arne.winterhof@oeaw.ac.at">arne.winterhof@oeaw.ac.at</A> Pseudorandom numbers, measures of pseudorandomness, exponential sums, finite fields. Woźniakowski, Henryk (New York, Warsaw) <A href="mailto:henryk@cs.columbia.edu">henryk@cs.columbia.edu</A> Discrepancies, Quasi-Monte Carlo algorithms, Monte-Carlo algorithms, tractability of multivariate problems, computational complexity of continous problems. Publisher De Gruyter Poland Bogumiła Zuga 32A Str. 01-811 Warsaw, Poland

<DIV align=left> Open Access Statement The journal is an Open Access journal that allows a free unlimited access to all its contents without any restrictions upon publication to all users.</DIV>

<DIV align=left> 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: <UL> <LI>Distribution of one dimensional and multidimensional sequences. </LI> <LI>Distribution of binary sequences. </LI> <LI>Distribution of integer sequences and sequences from groups and </LI> <LI>Distribution problems in finite abstract sets. </LI> <LI>Continuously uniform distributions. </LI> <LI>Discrepancies. </LI> <LI>Pseudo-random number generators. </LI> <LI>Quasi-Monte Carlo integration. </LI> <LI>Quasi-Monte Carlo methods in financial mathematics. </LI> <LI>Theory of densities. </LI> <LI>Theory of distribution functions of sequences. </LI> <LI>Distribution of integer points in large domains. </LI> <LI>Spectral properties of sequences. </LI> <LI>Dynamics emerging from sequences. </LI> <LI>Number theoretic ciphers and codes. </LI> <LI>Combinatorial number theory. </LI> <LI>Trigonometric sums. </LI> <LI>Diophantine approximations and diophantine equations. </LI> <LI>Continued fractions. </LI> <LI>Quantum chaos. </LI></UL> 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; <A href="http://math.boku.ac.at/udt">math.boku.ac.at/udt</A> Archiving Sciendo archives the contents of this journal in <A href="https://www.portico.org/">Portico</A> - digital long-term preservation service of scholarly books, journals and collections. Plagiarism Policy The editorial board is participating in a growing community of <A href="https://www.crossref.org/services/similarity-check/">Similarity Check System's</A> 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. </DIV>

Uniform distribution theory

Mahler’s Conjecture on ξ(3/2)ⁿmod 1

Oto Strauch
Oto Strauch

Published Online: 02 Feb 2022

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

Page range: 49 - 70

Received: 27 Feb 2019

Accepted: 29 Aug 2021

Journal & Issue Details

Uniform distribution theory

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References

Recent Articles

Uniform distribution theory

Mahler’s Conjecture on ξ(3/2)nmod 1

Oto StrauchOto Strauch

Published Online: 02 Feb 2022

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

Page range: 49 - 70

Received: 27 Feb 2019

Accepted: 29 Aug 2021

Journal & Issue Details

Uniform distribution theory

PDF Preview

References

Recent Articles

Mahler’s Conjecture on ξ(3/2)ⁿmod 1

Oto Strauch
Oto Strauch