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Solutions to certain linear equations in Piatetski-Shapiro sequences

Volume 177 / 2017

Daniel Glasscock Acta Arithmetica 177 (2017), 39-52 MSC: Primary 11J83; Secondary 11B83. DOI: 10.4064/aa8355-10-2016 Published online: 22 December 2016

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.

Authors

  • Daniel GlasscockDepartment of Mathematics
    The Ohio State University
    231 W. 18th Ave.
    Columbus, OH 43210, U.S.A.
    e-mail

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