On the local Fourier uniformity problem for small sets

We consider vanishing properties of exponential sums of the Liouville function $\lambda$ of the form $$\underset{H\to \mathrm{\infty }}{\lim }\underset{X\to \mathrm{\infty }}{\mathrm{lim\hspace{0.17em}sup}}\frac{1}{\log X}\sum _{m\le X}\frac{1}{m}\underset{\alpha \in C}{\sup }\mid \frac{1}{H}\sum _{h\le H}\lambda (m+h)e^{2\pi ih\alpha }\mid =0,$$ 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 $e^{2\pi i{h}^t\alpha }$ for $t\ge 2$ then the statement remains true for any set $C$ of upper box-counting dimension $<1\mathrm{/}t$. 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.