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

TL;DR

This paper establishes a quantitative upper bound on the density of prime subsets avoiding specific polynomial difference patterns involving shifted prime powers, advancing understanding of prime pattern restrictions.

Contribution

It provides the first explicit bound on the size of prime sets avoiding Furstenberg-Sárközy type patterns with shifted prime power differences.

Findings

Sets avoiding the pattern are very sparse, with density decreasing faster than any fixed power of log log N.

The bound depends explicitly on the parameter k, the power in the pattern.

The result extends previous work on polynomial patterns in primes with a new quantitative estimate.

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