Difference sets and the primes

TL;DR

This paper investigates the size of subsets of {1,...,N} avoiding differences that are one less than a prime, establishing an exponential decay bound on their maximum size.

Contribution

It provides a new upper bound on the size of such subsets, connecting difference set properties with prime-related constraints.

Findings

Maximum size of A is bounded by O(N exp(-c(log N)^{1/4}))

Difference sets avoiding primes have exponentially small density

The result advances understanding of additive combinatorics involving primes.

Abstract

Suppose that A is a subset of {1,...,N} such that the difference between any two elements of A is never one less than a prime. We show that |A| = O(N exp(-c(log N)^{1/4})) for some absolute c>0.