Strong Bounds for 3-Progressions

TL;DR

This paper establishes new bounds showing that sufficiently large subsets of integers necessarily contain non-trivial three-term arithmetic progressions, improving previous size thresholds using novel analytic techniques.

Contribution

The authors develop new analytic methods for finite fields and adapt them to prove stronger bounds for 3-term arithmetic progressions in integers.

Findings

Sets of size at least $2^{-O((\log N)^eta)} imes N$ contain 3-term APs.

Improved bounds over previous results for the existence of 3-term APs.

New analytic techniques for finite fields and integers.

Abstract

We show that for some constant $\beta > 0$, any subset $A$ of integers $\{1,\ldots,N\}$ of size at least $2^{-O((\log N)^\beta)} \cdot N$ contains a non-trivial three-term arithmetic progression. Previously, three-term arithmetic progressions were known to exist only for sets of size at least $N/(\log N)^{1 + c}$ for a constant $c > 0$. Our approach is first to develop new analytic techniques for addressing some related questions in the finite-field setting and then to apply some analogous variants of these same techniques, suitably adapted for the more complicated setting of integers.