  For every non-integral $\alpha>1$, the sequence of the integer parts of
$n^{\alpha}$ $(n=1,2,\ldots)$ is called the Piatetski-Shapiro sequence with
exponent $\alpha$, and let $\mathrm{PS}(\alpha)$ denote the set of all those
terms. For all $X\subseteq \mathbb{N}$, we say that an equation $y=ax+b$ is
solvable in $X$ if the equation has infinitely many solutions of distinct pairs
$(x,y)\in X^2$. Let $a,b\in \mathbb{R}$ with $a\neq 1$ and $0\leq b<a$, and
suppose that the equation $y=ax+b$ is solvable in $\mathbb{N}$. We show that
for all $1<\alpha<2$ the equation $y=ax+b$ is solvable in
$\mathrm{PS}(\alpha)$. Further, we investigate the set of $\alpha \in (s,t)$ so
that the equation $y=ax+b$ is solvable in $\mathrm{PS}(\alpha)$ where $2< s
<t$. Finally, we show that the Hausdorff dimension of the set is coincident
with $2/s$.
