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

A sequence of integers of the form $\lfloor n^{\alpha }\rfloor$ $(n=1,2,\dots )$ for some fixed non-integral $\alpha >1$ is called a Piatetski-Shapiro sequence, where $\lfloor x\rfloor$ denotes the integer part of $x$. Let $\mathrm{P}\mathrm{S}(\alpha )$ denote the set of all those terms. In this article, we show that $x+y=z$ has only finitely many solutions $(x,y,z)\in \mathrm{P}\mathrm{S}(\alpha {)}^3$ for almost every $\alpha >3$. Furthermore, we show that $\mathrm{P}\mathrm{S}(\alpha )$ 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 $y=a_1{x}_1+\cdots +a_n{x}_n$ has infinitely many solutions on $\mathrm{P}\mathrm{S}(\alpha )$.