Improved bounds for the Fourier uniformity conjecture

TL;DR

This paper advances the Fourier uniformity conjecture by establishing new bounds on the Liouville function's Fourier transform behavior over large intervals, improving previous results.

Contribution

It proves a stronger bound on the Liouville function's Fourier uniformity for larger interval lengths, refining prior work by Walsh.

Findings

Established that the sum over X to 2X of the supremum of the Fourier transform of the Liouville function is o(HX).

Achieved bounds for H(X) as large as exp((log X)^{2/5+ε}), improving previous results.

Progressed towards the Fourier uniformity conjecture with new bounds.

Abstract

Let $\lambda$ denote the Liouville function. We prove that $$\sum_{X \leq x < 2X} \sup_{\alpha \in \mathbb{R}/\mathbb{Z}} \bigg\lvert\!\sum_{x \leq n < x+H} \lambda(n) e(n\alpha)\bigg\rvert = o(HX)$$ as $X\to \infty$, in the regime $H = H(X) \geq \exp((\log X)^{2/5+\varepsilon})$. This improves upon a result of Walsh towards the Fourier uniformity conjecture.