$L^p$ maximal estimates for Weyl sums with $k\ge 3$ on $\mathbb{T}$

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 $k\ge 3$ 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.