On S\'ark\"ozy's theorem for shifted primes

TL;DR

This paper investigates the structure of subsets of integers avoiding differences of the form p-1, where p is prime, and establishes an upper bound on their size, extending understanding of additive combinatorics related to primes.

Contribution

It provides a new upper bound on the size of sets avoiding differences of the form p-1, advancing the understanding of shifted prime differences in additive number theory.

Findings

Sets avoiding differences p-1 are small, with size at most N^{1 - c}.

The result extends Sárközy's theorem to shifted primes.

The bound improves previous results on difference sets involving primes.

Abstract

Suppose that $A \subset \{1,\dots, N\}$ has no two elements differing by $p-1$, $p$ prime. Then $|A| \ll N^{1 - c}$.