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 $0 \leq b < a$) or $\alpha > 2$ (and $(a,b) \neq (1,0)$). We collect a number of interesting open questions related to further results along these lines.