A quantitative bound on Furstenberg-S\'ark\"ozy patterns with shifted prime power common differences in primes

Abstract

Let $k\geq1$ be a fixed integer, and $\mathcal P_N$ be the set of primes no more than $N$. We prove that if a set $\mathcal A\subset\mathcal P_N$ contains no patterns $p_1,p_1+(p_2-1)^k$, where $p_1,p_2$ are prime numbers, then \[ \frac{|\mathcal A|}{|\mathcal P_N|}\ll(\log\log N)^{-\frac{1}{4k^3+23k^2}}. \]