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

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.

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

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