A quantitative improvement for Roth's theorem on arithmetic progressions

TL;DR

This paper enhances the quantitative bounds in Roth's theorem on three-term arithmetic progressions, providing sharper estimates for sets avoiding such progressions over integers and function fields, and extending to sumset problems.

Contribution

It introduces a new method that improves the bounds in Roth's theorem and related problems over integers and function fields.

Findings

Improved bound: |A| ≪ N(log log N)^4 / log N for sets with no 3-term APs.

Extended results to finite fields $ ext{F}_q[t]$ and sumset problems.

Method yields sharper quantitative estimates in additive combinatorics.

Abstract

We improve the quantitative estimate for Roth's theorem on three-term arithmetic progressions, showing that if $A\subset\{1,\ldots,N\}$ contains no non-trivial three-term arithmetic progressions then $\lvert A\rvert\ll N(\log\log N)^4/\log N$. By the same method we also improve the bounds in the analogous problem over $\mathbb{F}_q[t]$ and for the problem of finding long arithmetic progressions in a sumset.