Fourier uniformity of bounded multiplicative functions in short intervals on average

Abstract

Let $\lambda$ denote the Liouville function. We show that as $X\to \infty$, $${\int }_X^{2X}\underset{\alpha }{\sup }\left|\sum _{x<n\le x+H}\lambda (n)e(-\alpha n)\right|dx=o(XH)$$ for all $H\ge X^{\theta }$ with $\theta >0$ fixed but arbitrarily small. Previously, this was only known for $\theta >5/8$. For smaller values of $\theta$ this is the first `non-trivial' case of local Fourier uniformity on average at this scale. We also obtain the analogous statement for (non-pretentious) $1$-bounded multiplicative functions. We illustrate the strength of the result by obtaining cancellations in the sum of $\lambda (n)\Lambda (n+h)\Lambda (n+2h)$ over the ranges $h<X^{\theta }$ and $n<X$, and where $\Lambda$ is the von Mangoldt function.

52 pages, 10 figures