Solutions to certain linear equations in Piatetski-Shapiro sequences

Daniel Glasscock¹

¹ Department of Mathematics The Ohio State University 231 W. 18th Ave. Columbus, OH 43210, U.S.A.

Acta Arithmetica, Tome 177 (2017) no. 1, pp. 39-52

Résumé

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.

Détail

DOI : 10.4064/aa8355-10-2016

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