DESHOUILLERS, Jean-Marc; DRMOTA, Michael; MÜLLNER, Clemens; SPIEGELHOFFER, Lukas

doi:10.1112/S0025579319000287

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DESHOUILLERS, Jean-Marc
Institut de Mathématiques de Bordeaux [IMB]

DRMOTA, Michael
Vienna University of Technology = Technische Universität Wien [TU Wien]

MÜLLNER, Clemens
Combinatoire, théorie des nombres [CTN]

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Mathematika. 2019, vol. 65, n° 4, p. 1051-1073

University College London

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We study Piatetski-Shapiro sequences ([n(c)])(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 is an element of {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-Katai criterion, we prove that the sequence [n(c)] modulo m is asymptotically orthogonal to multiplicative functions bounded by 1 and with mean value 0.< Leer menos

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Randomness and non-randomness properties of Piatetski-Shapiro sequences modulo m

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