On the Local Fourier Uniformity Problem for Small Sets

We consider vanishing properties of exponential sums of the Liouville function λ of the form $$ {align*} & limH→∞X→∞1/log X∑m≤ X1/mα∈ C|1/H∑h≤ Hλ(m+h)e2π ihα|=0, {align*} $$ where $C⊂{{T}}$. The case $C={{T}}$ 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 $C⊂{{T}}$ of zero Lebesgue measure. Moreover, we prove that extending this to any set $C$ with non-empty interior is equivalent to the $C={{T}}$ 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 $e2π ihα$ is replaced by a polynomial phase $e^{2π ihᵗα }$ for $t≥ 2$ 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.