Linear equations with two variables in Piatetski-Shapiro sequences

For every non-integral $α>1$, the sequence of the integer parts of $n^α$ $(n=1,2,\ldots)$ is called the Piatetski-Shapiro sequence with exponent $α$, and let $\mathrm{PS}(α)$ 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 $a\neq 1$ and $0\leq b<a$, and suppose that the equation $y=ax+b$ is solvable in $\mathbb{N}$. We show that for all $1<α<2$ the equation $y=ax+b$ is solvable in $\mathrm{PS}(α)$. Further, we investigate the set of $α\in (s,t)$ so that the equation $y=ax+b$ is solvable in $\mathrm{PS}(α)$ where $2< s <t$. Finally, we show that the Hausdorff dimension of the set is coincident with $2/s$.

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