Improved bounds for the Fourier uniformity conjecture

Abstract

Let $\hat{I}»$ denote the Liouville function. We prove that $$\sum _{X\le x<2X}\underset{\hat{I}\pm \in \mathbb{R}/\mathbb{Z}}{\sup }|\sum _{x\le n<x+H}\hat{I}»(n)e(n\hat{I}\pm )|=o(HX)$$ as $X\to \infty$, in the regime $H=H(X)\ge \exp ((\log X{)}^{2/5+\epsilon })$. This improves upon a result of Walsh towards the Fourier uniformity conjecture.

59 pages