We consider vanishing properties of exponential sums of the Liouville function of the form

where . The case corresponds to the local 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 of zero Lebesgue measure. Moreover, we prove that extending this to any set with non-empty interior is equivalent to the 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 is replaced by a polynomial phase for then the statement remains true for any set of upper box-counting dimension . 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 step nilpotent Lie group. Furthermore, we discuss the unweighted version of the local 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.