On certain other sets of integers

TL;DR

This paper establishes an upper bound on the size of subsets of the first N natural numbers that contain no non-trivial three-term arithmetic progressions, improving understanding of their density.

Contribution

It provides a new bound on the maximum size of such progression-free sets, refining previous results in additive combinatorics.

Findings

|A|=O(N/ log^{3/4-o(1)} N) for progression-free sets

Improved bounds on the density of sets avoiding 3-term arithmetic progressions

Advances understanding of structure in additive number theory

Abstract

We show that if A is a subset of {1,...,N} containing no non-trivial three-term arithmetic progressions then |A|=O(N/ log^{3/4-o(1)} N).