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

Let $[\, \cdot\,]$ be the floor function and $\|x\|$ 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 $1<c<12/11$ 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 $p=[n^c]$.