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

## TL;DR

This paper investigates the properties of Piatetski-Shapiro sequences modulo m, revealing their non-normality, deterministic nature, and orthogonality to certain multiplicative functions, with detailed analysis of subword occurrences.

## Contribution

It provides new results on the distribution, subword frequency, and orthogonality properties of Piatetski-Shapiro sequences modulo m for non-integer c > 1.

## Key findings

- Sequences are not normal, with some subwords never occurring.
- For 1<c<2, sequences are deterministic and not morphic.
- Sequences are asymptotically orthogonal to bounded multiplicative functions.

## 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-K\'{a}tai 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 $0$.