$L^p$ maximal estimates for quadratic Weyl sums

TL;DR

This paper establishes sharp $L^p$ bounds for quadratic Weyl sums evaluated at their maximum points, advancing understanding of their size and distribution in harmonic analysis.

Contribution

It provides matching upper and lower bounds for the $L^p$ norm of quadratic Weyl sums at their maximum points, a novel result in the analysis of Weyl sums.

Findings

Derived precise $L^p$ bounds for quadratic Weyl sums at maxima

Established the order of magnitude for the $L^p$ norm of these sums

Enhanced understanding of the distribution of Weyl sums in harmonic analysis

Abstract

Let $S(x,t)$ denote the Weyl sum with associated polynomial $xn + tn^2$. Suppose that $|S(x,t)|$ attains its maximum for given $x$ at $t = t(x)$. We give upper and lower bounds of the same order of magnitude for the $L^p$ norm of $S(x,t(x))$.