Estimates for smooth Weyl sums on major arcs

TL;DR

This paper provides improved estimates for smooth Weyl sums on major arcs, enhancing the Hardy-Littlewood method's effectiveness in analyzing exponential sums related to number theory problems.

Contribution

It introduces new mean value estimates for smooth Weyl sums of degree k, achieving near-optimal bounds for moments exceeding a specific threshold.

Findings

Derived essentially optimal bounds for moments of smooth Weyl sums

Established mean value estimates on major arcs for degree k sums

Enhanced the analytical tools for applications of the Hardy-Littlewood method

Abstract

We present estimates for smooth Weyl sums of use on sets of major arcs in applications of the Hardy-Littlewood method. In particular, we derive mean value estimates on major arcs for smooth Weyl sums of degree $k$ delivering essentially optimal bounds for moments of order $u$ whenever $u>2\lfloor k/2\rfloor +4$.