Popular differences in primes along fractional powers

We prove that ${}_{m\le M}{}_{n\le N}\mathrm{\Lambda }(n)\mathrm{\Lambda }(n+\lfloor m^c\rfloor )=1+\mathrm{O}({\log }^{2-Bc}N)$, where $c>2$ is a non-integer, $B\ge 3\mathrm{/}c$, and $M$ is of order $N^{1\mathrm{/}c}{\log }^{-B}N$. As a combinatorial consequence, we obtain that the primes contain infinitely many pairs whose difference belongs to the Piatetski-Shapiro sequence $\{\lfloor m^c\rfloor \text{\hspace{0.17em}⁣}:m\in \mathbb{N}\}$ for any non-integer $c>2$.