Difference sets and shifted primes

TL;DR

This paper establishes an upper bound on the size of subsets of {1,...,n} that avoid differences of the form p-1, where p is prime, revealing new insights into the structure of such sets.

Contribution

It provides a novel upper bound on the size of difference-avoiding sets related to primes, advancing understanding in additive combinatorics.

Findings

Sets avoiding differences p-1 are small, with size bounded by n times a slowly decreasing logarithmic factor.

The bound involves iterated logarithms, indicating very slow growth of the maximum size.

This result connects prime number properties with combinatorial set structures.

Abstract

We show that if A is a subset of {1, ..., n} such that it has no pairs of elements whose difference is equal to p-1 with p a prime number, then the size of A is O(n(loglog n)^(-clogloglogloglog n)) for some positive constant c.