Maximal estimates for the Weyl sums on $\mathbb{T}^{d}$ (with an appendix by Alex Barron)

TL;DR

This paper establishes sharp maximal estimates for Weyl sums on multi-dimensional tori, explores variants along rational lines and generic tori, and applies these results to bounds on Hausdorff dimensions, with an appendix providing an improved proof and Strichartz estimates.

Contribution

It provides the first sharp maximal estimates for Weyl sums on $ ext{T}^d$, including variants and applications, with an improved proof in the appendix.

Findings

Sharp maximal estimate for Weyl sums on $ ext{T}^d$

New bounds on Hausdorff dimension related to Weyl sums

Improved proof and Strichartz estimates in the appendix

Abstract

In this paper, we obtain the maximal estimate for the Weyl sums on the torus $\mathbb{T}^d$ with $d\geq 2$, which is sharp up to the endpoint. We also consider two variants of this problem which include the maximal estimate along the rational lines and on the generic torus. Applications, which include some new upper bound on the Hausdorff dimension of the sets associated to the large value of the Weyl sums, reflect the compound phenomenon between the square root cancellation and the constructive interference. In the Appendix, an alternate proof of Theorem 1.1 inspired by Baker's argument in [1] is given by Barron, which also improves the $N^{\epsilon}$ loss in Theorem 1.1, and the Strichartz-type estimates for the Weyl sums with logarithmic losses are obtained by the same argument.