On the Local Fourier Uniformity Problem for Small Sets

Abstract

<p>We consider vanishing properties of exponential sums of the Liouville function of the form<br></p><p> lim(->infinity) lim(->infinity) sup 1/log Sigma(<=) 1/ sup(is an element of)|1/ Sigma (<=) ( + )(2)| = 0, <br></p><p>where subset of . The case = 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 subset of 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 (2) is replaced by a polynomial phase (2) for >= 2 then the statement remains true for any set of upper box-counting dimension <1/. 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 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.<br></p>