Metric theory of Weyl sums

TL;DR

This paper establishes that for almost all points in the unit cube, the Weyl sums exhibit large oscillations of order N^{1/2} infinitely often, improving measure-theoretic results over previous Hausdorff dimension estimates.

Contribution

It proves that the set of points with large Weyl sums has full Lebesgue measure, significantly advancing the understanding of the distribution of exponential sums.

Findings

Weyl sums are large for almost all points in [0,1)^d infinitely often.

Improved measure-theoretic bounds over previous Hausdorff dimension results.

Established bounds for exponential sums with monomials, excluding d=4.

Abstract

We prove that there exist positive constants $C$ and $c$ such that for any integer $d \ge 2$ the set of ${\mathbf x}\in [0,1)^d$ satisfying $$ cN^{1/2}\le \left|\sum^N_{n=1}\exp\left (2 \pi i \left (x_1n+\ldots+x_d n^d\right)\right) \right|\le C N^{1/2}$$ for infinitely many natural numbers $N$ is of full Lebesque measure. This substantially improves the previous results where similar sets have been measured in terms of the Hausdorff dimension. We also obtain similar bounds for exponential sums with monomials $xn^d$ when $d\neq 4$. Finally, we obtain lower bounds for the Hausdorff dimension of large values of general exponential polynomials.