On the local Fourier uniformity problem for small sets

TL;DR

This paper proves the local Fourier uniformity conjecture for Liouville functions on sets of zero Lebesgue measure and explores higher-order variants, advancing understanding of multiplicative functions in number theory.

Contribution

It establishes the conjecture for zero measure sets, relates it to the full circle case, and extends results to polynomial phases and nilsequences, showing near-optimal conditions.

Findings

Proves the conjecture for zero Lebesgue measure sets.

Shows equivalence of the full circle case to sets with interior.

Extends results to polynomial phases and nilsequences.

Abstract

We consider vanishing properties of exponential sums of the Liouville function $\lambda$ of the form $$ \lim_{H\to\infty}\limsup_{X\to\infty}\frac{1}{\log X}\sum_{m\leq X}\frac{1}{m}\sup_{\alpha\in C}\bigg|\frac{1}{H}\sum_{h\leq H}\lambda(m+h)e^{2\pi ih\alpha}\bigg|=0, $$ where $C\subset\mathbb{T}$. The case $C=\mathbb{T}$ corresponds to the local $1$-Fourier uniformity conjecture of Tao, a central open problem in the study of multiplicative functions with far-reaching number-theoretic applications. We show that the above holds for any closed set $C\subset\mathbb{T}$ of zero Lebesgue measure. Moreover, we prove that extending this to any set $C$ with non-empty interior is equivalent to the $C=\mathbb{T}$ case, which shows that our results are essentially optimal without resolving the full conjecture. We also consider higher-order variants. We prove that if the linear phase $e^{2\pi ih\alpha}$ is replaced by a polynomial phase $e^{2\pi ih^t\alpha}$ for $t\geq 2$ then the statement remains true for any set $C$ of upper box-counting dimension $<1/t$. The statement also remains true if the supremum over linear phases is replaced with a supremum over all nilsequences coming form a compact countable ergodic subsets of any $t$-step nilpotent Lie group. Furthermore, we discuss the unweighted version of the local $1$-Fourier uniformity problem, showing its validity for a class of ``rigid'' sets (of full Hausdorff dimension) and proving a density result for all closed subsets of zero Lebesgue measure.