Bounds for discrete moments of Weyl sums and applications

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 $k \geq 4$.