Diophantine Approximation with Piatetski-Shapiro Primes

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