Solutions to certain linear equations in Piatetski-Shapiro sequences
Volume 177 / 2017
Abstract
Denote by $\text{PS}(\alpha)$ the image of the Piatetski-Shapiro sequence $n \mapsto \lfloor {n^\alpha} \rfloor$, where $\alpha \gt 1$ 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. $\alpha \gt 1$, it has infinitely many or at most finitely many solutions in $\text{PS}(\alpha)$ according as $\alpha \lt 2$ (and $0 \leq b \lt a$) or $\alpha \gt 2$ (and $(a,b) \neq (1,0)$). We collect a number of interesting open questions related to further results along these lines.
