Fourier uniformity of bounded multiplicative functions in short intervals on average

Kaisa Matomäki, Maksym Radziwiłł, Terence Tao

Published 2018-12-04Version 1

Let $\lambda$ denote the Liouville function. We show that as $X \rightarrow \infty$, $$ \int_{X}^{2X} \sup_{\alpha} \left | \sum_{x < n \leq x + H} \lambda(n) e(-\alpha n) \right | dx = o ( X H) $$ for all $H \geq X^{\theta}$ 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.