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