A Fourier-Free Density-Increment Proof of Roth's Theorem

TL;DR

This paper presents an elementary, Fourier-free proof of Roth's theorem, using a combinatorial approach instead of Fourier analysis, simplifying the original density-increment strategy.

Contribution

It introduces a novel combinatorial method to prove Roth's theorem, avoiding Fourier analysis and making the proof more accessible.

Findings

Provides a simpler, Fourier-free proof of Roth's theorem

Uses combinatorial arguments involving averages over sub-progressions

Maintains the effectiveness of the original density-increment approach

Abstract

We give an elementary, Fourier-free proof of Roth's theorem. The proof follows Roth's original density-increment strategy, but replaces the usual Fourier-analytic step with a direct combinatorial argument involving averages over sub-progressions.