Improved bounds for the Fourier uniformity conjecture
Fuente:
arXiv
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| Autore principale: | |
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| Natura: | Preprint |
| Pubblicazione: |
2026
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| _version_ | 1866914516680310784 |
|---|---|
| author | Pilatte, Cédric |
| author_facet | Pilatte, Cédric |
| contents | Let $λ$ denote the Liouville function. We prove that $$\sum_{X \leq x < 2X} \sup_{α\in \mathbb{R}/\mathbb{Z}} \bigg\lvert\!\sum_{x \leq n < x+H} λ(n) e(nα)\bigg\rvert = o(HX)$$ 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. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2604_26564 |
| institution | arXiv |
| publishDate | 2026 |
| record_format | arxiv |
| spellingShingle | Improved bounds for the Fourier uniformity conjecture Pilatte, Cédric Number Theory Combinatorics Let $λ$ denote the Liouville function. We prove that $$\sum_{X \leq x < 2X} \sup_{α\in \mathbb{R}/\mathbb{Z}} \bigg\lvert\!\sum_{x \leq n < x+H} λ(n) e(nα)\bigg\rvert = o(HX)$$ 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. |
| title | Improved bounds for the Fourier uniformity conjecture |
| topic | Number Theory Combinatorics |
| url | https://arxiv.org/abs/2604.26564 |