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

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