Improved bounds for the Fourier uniformity conjecture

Let $\lambda$ denote the Liouville function. We prove that $$\sum _{X\le x<2X}\underset{\alpha \in \mathbb{R}\mathrm{/}\mathbb{Z}}{\sup }\mid \text{\hspace{0.17em}⁣}\sum _{x\le n<x+H}\lambda (n)e(n\alpha )\mid =o(HX)$$ as $X\to \mathrm{\infty }$, in the regime $H=H(X)\ge \exp ((\log X{)}^{2\mathrm{/}5+\epsilon })$. This improves upon a result of Walsh towards the Fourier uniformity conjecture.