On the distribution of $\alpha p^2$ modulo one over primes of the form $[n^c]$

Let $[\cdot ]$ be the floor function and $\mathrm{\parallel }x\mathrm{\parallel }$ denote 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 $\frac{13}{14}<\gamma <1$, there exist infinitely many prime numbers $p$ satisfying the inequality \begin{equation*} |αp^2+β|< p^{\frac{13-14γ}{29}+\varepsilon} \end{equation*} and such that $p=[n^{1\mathrm{/}\gamma }]$.