  We prove that for every irrational number $\alpha$, real number $c$
satisfying $1<c<9/8$ and positive real number $\theta$ satisfying
$\theta<(9/c-8)/10$, there exist infinitely many primes of the form
$p=\left[n^c\right]$ with $n\in \mathbb{N}$ such that $||\alpha
p||<p^{-\theta}$.
