Linear equations with two variables in Piatetski-Shapiro sequences
Volume 202 / 2022
Abstract
For every non-integral $\alpha \gt 1$, the sequence of the integer parts of $n^{\alpha }$ $(n=1,2,\ldots )$ is called the Piatetski-Shapiro sequence with exponent $\alpha $ and let $\mathrm {PS}(\alpha )$ denote the set of all terms of this sequence. 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 $(x,y)\in X^2$. Let $a,b\in \mathbb {R}$ with $a\neq 1$ and $0\leq b \lt a$, and suppose that the equation $y=ax+b$ is solvable in $\mathbb {N}$. We show that for all $1 \lt \alpha \lt 2$ the equation $y=ax+b$ is solvable in $\mathrm {PS}(\alpha )$. Further, we investigate the set of $\alpha \in (s,t)$ such that the equation $y=ax+b$ is solvable in $\mathrm {PS}(\alpha )$ where $2 \lt s \lt t$. Finally, we show that the Hausdorff dimension of the set is $2/s$.
