Popular differences in primes along fractional powers

Abstract

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

22 pages, 1 figure