Bounds for discrete moments of Weyl sums and applications

TL;DR

This paper establishes new bounds for discrete moments of Weyl sums using recent advances in Vinogradov's mean value theorem, leading to improved results in exponential sum estimates and polynomial large sieve inequalities.

Contribution

It introduces two bounds for Weyl sums' moments, including a sharper bound in certain ranges, and applies these to enhance derivative tests and large sieve inequalities.

Findings

Sharper bounds for Weyl sums in specific ranges

Improved derivative test for integer points near curves

Enhanced polynomial large sieve inequality for degree ≥ 4

Abstract

We prove two bounds for discrete moments of Weyl sums. The first one can be obtained using a standard approach. The second one involves an observation how this method can be improved, which leads to a sharper bound in certain ranges. The proofs both build on the recently proved main conjecture for Vinogradov's mean value theorem. We present two selected applications: First, we prove a new $k$- th derivative test for the number of integer points close to a curve by an exponential sum approach. This yields a stronger bound than existing results obtained via geometric methods, but it is only applicable for specific functions. As second application we prove a new improvement of the polynomial large sieve inequality for one-variable polynomials of degree $k \geq 4$.