Linear equations with two variables in Piatetski-Shapiro sequences

Abstract

For every non-integral $\alpha >1$, the sequence of the integer parts of $n^{\alpha }$ $(n=1,2,\dots )$ is called the Piatetski-Shapiro sequence with exponent $\alpha$, and let $\mathrm{PS}(\alpha )$ denote the set of all those terms. For all $X\subseteq \mathbb{N}$, we say that an equation $y=ax+b$ is solvable in $X$ if the equation has infinitely many solutions of distinct pairs $(x,y)\in X^2$. Let $a,b\in \mathbb{R}$ with $a\ne 1$ and $0\le b<a$, and suppose that the equation $y=ax+b$ is solvable in $\mathbb{N}$. We show that for all $1<\alpha <2$ the equation $y=ax+b$ is solvable in $\mathrm{PS}(\alpha )$. Further, we investigate the set of $\alpha \in (s,t)$ so that the equation $y=ax+b$ is solvable in $\mathrm{PS}(\alpha )$ where $2<s<t$. Finally, we show that the Hausdorff dimension of the set is coincident with $2/s$.

12 pages