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