On Roth's theorem on progressions

TL;DR

This paper proves an upper bound on the size of subsets of {1,...,N} that contain no non-trivial three-term arithmetic progressions, improving understanding of Roth's theorem.

Contribution

It introduces a different approach to bounding progression-free sets, refining previous results on Roth's theorem.

Findings

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

Provides a novel method differing from prior approaches

Enhances bounds related to Roth's theorem

Abstract

We show that if A is a subset of {1,...,N} contains no non-trivial three-term arithmetic progressions then |A|=O(N/ log^{1-o(1)} N). The approach is somewhat different from that used in arXiv:1007.5444.