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Bounds for discrete moments of Weyl sums and applications

Volume 194 / 2020

Karin Halupczok Acta Arithmetica 194 (2020), 1-28 MSC: Primary 11L15; Secondary 11J54. DOI: 10.4064/aa181207-23-9 Published online: 20 February 2020

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 to specific functions. As a second application we prove a new improvement of the polynomial large sieve inequality for one-variable polynomials of degree $k\geq 4$.

Authors

  • Karin HalupczokMathematisches Institut der HHU Düsseldorf
    Universitätsstraße 1
    D-40225 Düsseldorf, Germany
    e-mail

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