On the Local Fourier Uniformity Problem for Small Sets.

We consider vanishing properties of exponential sums of the Liouville function |$\boldsymbol{\lambda }$| of the form $$ \begin{align*} & \lim_{H\to\infty}\limsup_{X\to\infty}\frac{1}{\log X}\sum_{m\leq X}\frac{1}{m}\sup_{\alpha\in C}\bigg|\frac{1}{H}\sum_{h\leq H}\boldsymbol{\lambda}(m+h)e^{2\pi ih\alpha}\bigg|=0, \end{align*} $$ 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. [ABSTRACT FROM AUTHOR]