On the distribution of $\alpha p$ modulo one over Piatetski-Shapiro primes

Let $[\cdot ]$ be the floor function and $\mathrm{\parallel }x\mathrm{\parallel }$ denotes the distance from $x$ to the nearest integer. In this paper we show that whenever $\alpha$ is irrational and $\beta$ is real then for any fixed $1<c<12\mathrm{/}11$ there exist infinitely many prime numbers $p$ satisfying the inequality \begin{equation*} |\alpha p+\beta|\ll p^{\frac{11c-12}{26c}}\log^6p \end{equation*} and such that $p=[n^c]$.