Squares in Piatetski--Shapiro Sequences

Abstract

We study the distribution of squares in a Piatetski-Shapiro sequence ${\left(\lfloor n^c\rfloor \right)}_{n\in \mathbb{N}}$ with $c>1$ and $c\notin \mathbb{N}$. We also study more general equations $\lfloor n^c\rfloor =s{m}^2$, $n,m\in \mathbb{N}$, $1\le n\le N$ for an integer $s$ and obtain several bounds on the number of solutions for a fixed $s$ and on average over $s$ in an interval. These results are based on various techniques chosen depending on the range of the parameters.