A new proof of Roth's theorem on arithmetic progressions

TL;DR

This paper introduces a novel proof of Roth's theorem on arithmetic progressions that avoids traditional iterative methods and uses a density increment approach via a quantitative Varnavides's theorem, offering a different perspective on the theorem.

Contribution

The paper provides a new proof of Roth's theorem that does not rely on the usual Fourier coefficient-based iteration, instead employing a density increment through a quantitative Varnavides's theorem.

Findings

Proof avoids iteration common in previous proofs

Uses a density increment via a quantitative Varnavides's theorem

Offers a new perspective on Roth's theorem

Abstract

We present a proof of Roth's theorem that follows a slightly different structure to the usual proofs, in that there is not much iteration. Although our proof works using a type of density increment argument (which is typical of most proofs of Roth's theorem), we do not pass to a progression related to the large Fourier coefficients of our set (as most other proofs of Roth do). Furthermore, in our proof, the density increment is achieved through an application of a quantitative version of Varnavides's theorem, which is perhaps unexpected.