Squares in Piatetski-Shapiro sequences
Volume 181 / 2017
Acta Arithmetica 181 (2017), 239-252
MSC: 11B83, 11K65, 11L07, 11L40.
DOI: 10.4064/aa8644-8-2017
Published online: 20 November 2017
Abstract
We study the distribution of squares in a Piatetski-Shapiro sequence $(\lfloor n^c\rfloor)_{n\in\mathbb N}$ with $c \gt 1$ and $c\not\in\mathbb N$. We also study more general equations $\lfloor n^c\rfloor = sm^2$, $n,m\in \mathbb N$, $1\le n \le N$, 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.
