Solutions to certain linear equations in Piatetski-Shapiro sequences

Abstract

Denote by $\text{PS}(\alpha )$ the image of the Piatetski-Shapiro sequence $n\mapsto \lfloor n^{\alpha }\rfloor$ where $\alpha >1$ is non-integral and $\lfloor x\rfloor$ is the integer part of $x\in \mathbb{R}$. We partially answer the question of which bivariate linear equations have infinitely many solutions in $\text{PS}(\alpha )$: if $a,b\in \mathbb{R}$ are such that the equation $y=ax+b$ has infinitely many solutions in the positive integers, then for Lebesgue-a.e. $\alpha >1$, it has infinitely many or at most finitely many solutions in $\text{PS}(\alpha )$ according as $\alpha <2$ (and $0\le b<a$) or $\alpha >2$ (and $(a,b)\ne (1,0)$). We collect a number of interesting open questions related to further results along these lines.

arXiv comment13 pages