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

For a fixed non-integral $\alpha \gt 1$, let $\mathrm{PS}(\alpha ) = \{\lfloor n^\alpha \rfloor \colon n =1,2,\ldots \}$. We show that $x+y=z$ has only finitely many solutions $(x,y,z)\in \mathrm{PS}(\alpha )^3$ for almost every $\alpha \gt 3$. Furthermore, we show that $\mathrm{PS}(\alpha )$ contains only finitely many arithmetic progressions of length $3$ for almost every $\alpha \gt 10$. In addition, we give upper bounds for the Hausdorff dimension of the set of $\alpha \in [s,t]$ such that $y=a_1x_1+\cdots +a_nx_n$ has infinitely many solutions in $\mathrm {PS}(\alpha )$.