Finiteness of solutions to linear Diophantine equations on Piatetski-Shapiro sequences

Abstract

A sequence of integers of the form $\lfloor n^{\alpha}\rfloor$ $(n=1,2,\ldots)$ for some fixed non-integral $\alpha>1$ is called a Piatetski-Shapiro sequence, where $\lfloor x\rfloor$ denotes the integer part of $x$. Let $\mathrm{PS}(\alpha)$ denote the set of all those terms. In this article, we show that $x+y=z$ has only finitely many solutions $(x,y,z)\in \mathrm{PS}(\alpha)^3$ for almost every $\alpha>3$. Furthermore, we show that $\mathrm{PS}(\alpha)$ has only finitely many arithmetic progressions of length $3$ for almost every $\alpha>10$. In addition, we estimate upper bounds for the Hausdorff dimension of the set of $\alpha\in [s,t]$ such that $y=a_1x_1+\cdots +a_nx_n$ has infinitely many solutions on $\mathrm{PS}(\alpha)$.