Distributions of Arithmetic Progressions in Piatetski-Shapiro Sequence

Kota Saito, Yuuya Yoshida

Published 2020-06-24Version 1

It is known that for all $\alpha\in(1,2)$ and all integers $k\ge3$ and $r\ge1$, there exist infinitely many $n\in\mathbb{N}$ such that the sequence $(\lfloor{(n+rj)^\alpha}\rfloor)_{j=0}^{k-1}$ is an arithmetic progression of length $k$. In this paper, we show that the asymptotic density of all the above $n$ is equal to $1/(k-1)$. Although the common difference $r$ is arbitrarily fixed in the above result, we also examine the case when $r$ is not fixed. Furthermore, we also examine the number of the above $n$ that are contained in a short interval.