# Fourier uniformity of bounded multiplicative functions in short intervals on average

## TL;DR

This paper proves a new form of Fourier uniformity for the Liouville function and other multiplicative functions in short intervals on average, extending previous results to smaller interval lengths and demonstrating significant cancellations.

## Contribution

It establishes the first non-trivial local Fourier uniformity results for the Liouville function in very short intervals, improving upon prior bounds and applying to non-pretentious multiplicative functions.

## Key findings

- Proves Fourier uniformity for $ heta > 0$ arbitrarily small.
- Shows cancellations in sums involving $ ext{Liouville}$ and von Mangoldt functions.
- Extends previous results from $ heta > 5/8$ to smaller scales.

## Abstract

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.