Vinogradov's mean value theorem, averages of Weyl sums and applications

Vinogradov's mean value theorem from the 1930s has recently been affirmed by Bourgain/Demeter/Guth and Wooley. In this talk, we discuss some consequences for Weyl sum estimates and applications. When considering averages over Weyl sums, a small improvement in the upper bound exponent can be achieved in certain ranges compared to the known direct approaches.

Averages over Weyl sums occur in several problems in number theory. In the problem of the polynomial large sieve inequality, we show that the new exponent improves upon the former record directly. In the problem of counting integer points close to a curve we show
(work in progress) that new nontrivial bounds can be achieved, but these have to be compared with existing strong results that have been gained without an exponential sum technique.