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RANDOMNESS AND NON-RANDOMNESS PROPERTIES OF PIATETSKI-SHAPIRO SEQUENCES MODULO $m$

Part of: Multiplicative number theory Sequences and sets Exponential sums and character sums Probabilistic theory: distribution modulo $1$; metric theory of algorithms

Published online by Cambridge University Press:  14 August 2019

Jean-Marc Deshouillers ,
Michael Drmota ,
Clemens Müllner  and
Lukas Spiegelhofer
Jean-Marc Deshouillers
Affiliation:
Université de Bordeaux, CNRS, Bordeaux INP, IMB, UMR 5251, F-33400, Talence, France email [email protected]
Michael Drmota
Affiliation:
Institut für Diskrete Mathematik und Geometrie, TU Wien, Wiedner Hauptstr. 8–10, 1040 Wien, Austria email [email protected]
Clemens Müllner
Affiliation:
Institut für Diskrete Mathematik und Geometrie, TU Wien, Wiedner Hauptstr. 8–10, 1040 Wien, Austria CNRS, Université de Lyon, Université Lyon 1, Institut Camille Jordan, 43 boulevard du 11 novembre 1918, 69622 Villeurbanne Cedex, France email [email protected]
Lukas Spiegelhofer
Affiliation:
Institut für Diskrete Mathematik und Geometrie, TU Wien, Wiedner Hauptstr. 8–10, 1040 Wien, Austria email [email protected]
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Abstract

We study Piatetski-Shapiro sequences $(\lfloor n^{c}\rfloor )_{n}$ modulo $m$, for non-integer $c>1$ and positive $m$, and we are particularly interested in subword occurrences in those sequences. We prove that each block $\in \{0,1\}^{k}$ of length $k<c+1$ occurs as a subword with the frequency $2^{-k}$, while there are always blocks that do not occur. In particular, those sequences are not normal. For $1<c<2$, we estimate the number of subwords from above and below, yielding the fact that our sequences are deterministic and not morphic. Finally, using the Daboussi–Kátai criterion, we prove that the sequence $\lfloor n^{c}\rfloor$ modulo $m$ is asymptotically orthogonal to multiplicative functions bounded by 1 and with mean value 0.

MSC classification

Primary: 11B50: Sequences (mod $m$) 11B83: Special sequences and polynomials
Secondary: 11K16: Normal numbers, radix expansions, Pisot numbers, Salem numbers, good lattice points, etc. 11L07: Estimates on exponential sums 11N37: Asymptotic results on arithmetic functions
Type
Research Article
Information
Mathematika , Volume 65 , Issue 4 , 2019 , pp. 1051 - 1073
Copyright
Copyright © University College London 2019 

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Footnotes

This work was supported by the Austrian Science Foundation FWF, SFB F5502-N26 “Subsequences of Automatic Sequences and Uniform Distribution”, which is a part of the Special Research Program “Quasi Monte Carlo Methods: Theory and Applications”, by the joint ANR-FWF project ANR-14-CE34-0009, I-1751 MuDeRa, Ciência sem Fronteiras (project PVE 407308/2013-0) and by the European Research Council (ERC) under the European Union’s Horizon 2020 research and innovation programme under the Grant Agreement No. 648132.

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