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arXiv · 2604.26564

Improved bounds for the Fourier uniformity conjecture

Cédric Pilatte
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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.

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Cédric Pilatte. 2026-04-29. Improved bounds for the Fourier uniformity conjecture. https://arxiv.org/abs/2604.26564

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