On the Local Fourier Uniformity Problem for Small Sets

We consider vanishing properties of exponential sums of the Liouville function λ of the form lim lim sup log1X mΣ≤X m1 sup α∈C|||| H1 Σ λ(m + h)e2πihα |||| = 0, H→∞ X→∞ h≤H 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 e2πihα is replaced by a polynomial phase e2πihtα 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.