Squares in Piatetski--Shapiro Sequences

TL;DR

This paper investigates the distribution of perfect squares within Piatetski-Shapiro sequences and analyzes related equations, providing bounds on solutions using diverse techniques tailored to parameter ranges.

Contribution

It offers new bounds on the occurrence of squares in Piatetski-Shapiro sequences and solutions to related equations, extending understanding of their distribution.

Findings

Bounds on the number of squares in Piatetski-Shapiro sequences

Estimates for solutions to equations involving these sequences

Analysis of solutions on average over parameters

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 $c\not\in\mathbb N$. 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.