A quantitative bound on Furstenberg-Sárközy patterns with shifted prime power common differences in primes

Abstract

Let $k\ge 1$ 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 \[ \frac{|\mathcal A|}{|\mathcal P_N|}\ll(\log\log N)^{-\frac{1}{4k^3+23k^2}}. \]

the final version, incorporating the referee's numerous helpful comments and corrections

References in corpus (2)

• Difference sets and the primes
• On a theorem of Sárközy for difference sets and shifted primes