# On a theorem of S\'ark\"ozy for difference sets and shifted primes

## TL;DR

This paper proves an upper bound on the size of sets within [N] that avoid differences equal to one less than a prime, extending understanding of difference sets and prime-related constraints.

## Contribution

It establishes a new exponential decay bound on the size of such sets, advancing the theory of difference sets and prime-related combinatorial restrictions.

## Key findings

- Sets avoiding differences of one less than a prime are very small, with size bounded by N times an exponential decay.
- The bound improves understanding of the structure of difference sets related to primes.
- Provides a quantitative measure of how prime-related difference restrictions limit set size.

## Abstract

We show that if the difference of two elements of a set $A \subseteq [N]$ is never one less than a prime number, then $|A| = O (N \exp (-c (\log N)^{1/3}))$ for some absolute constant $c>0$.