Linear Diophantine equations in Piatetski-Shapiro sequences

A Piatetski-Shapiro sequence with exponent $\alpha$ is a of integer parts $n^\alpha$ $(n = 1,2,\ldots)$ non-integral $\alpha > 0$. We let $\mathrm{PS}(\alpha)$ denote the set those terms. In this article, we study so that equation $ax + by cz$ has infinitely many pairwise distinct solutions $(x,y,z) \in \mathrm{PS}(\alpha)^3$, and give lower bound for its Hausdorff dimension. As corollary, find uncountably 2$ such contains arithmetic progressions length $3$.