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

Abstract

We study Piatetski-Shapiro sequences [n^c] modulo m, for non-integers c > 1 an 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 < 2we 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 [n^c] modulo m is asymptotically orthogonal to multiplicative functions bounded by 1 and with mean value 0.