$L^p$ maximal estimates for Weyl sums with $k\ge3$ on $\mathbb{T}$

TL;DR

This paper investigates the $L^p$ maximal bounds for higher-order Weyl sums on the torus, providing sharp results for the case $k=3$ through analysis of large value sets using rational approximation and exponential sum estimates.

Contribution

It establishes new $L^p$ maximal estimates for Weyl sums with $k extgreater 2$, including sharp results for the cubic case, advancing understanding of their size and distribution.

Findings

Sharp $L^p$ estimates for $k=3$ Weyl sums.

Positive and negative bounds for higher $k$.

Structural analysis of large value sets using rational approximation.

Abstract

In this paper, we study the $L^p$ maximal estimates for the Weyl sums $\sum_{n=1}^{N}e^{2\pi i(nx + n^{k}t)}$ with higher-order $k\ge3$ on $\mathbb{T}$, and obtain the positive and negative results. Especially for the case $k=3$, our result is sharp up to the endpoint. The main idea is to investigate the structure of the set where large values of Weyl sums are achieved by making use of the rational approximation and the refined estimate for the exponential sums.