  An integral sequence of the form $\lfloor n^{\alpha}\rfloor$ $(n=1,2,\ldots)$
for some non-integral $\alpha>1$ is called a Piatetski-Shapiro sequence, where
$\lfloor x\rfloor$ denotes the integer part of $x$. Let $\mathrm{PS}(\alpha)$
denote the set of all those terms. In this article, we reveal that $x+y=z$ has
at most finitely many solutions $(x,y,z)\in \mathrm{PS}(\alpha)^3$ for almost
every $\alpha>3$ in the sense of one-dimensional Lebesgue measure. Furthermore,
we show that $\mathrm{PS}(\alpha)$ has at most finitely many arithmetic
progressions of length $3$ for almost every $\alpha>10$. In addition, we obtain
results on the Hausdorff dimension and a more general equation such as
$y=a_1x_1+\cdots +a_nx_n$.
