Near-optimal mean value estimates for multidimensional Weyl sums

TL;DR

This paper provides sharp estimates for multidimensional Weyl sums, advancing the understanding of Vinogradov's mean value theorem and approaching optimal variable constraints, with various applications discussed.

Contribution

It introduces near-optimal bounds for multidimensional Vinogradov-type sums, extending previous results to more general systems with translation-dilation invariance.

Findings

Achieved near-optimal variable constraints for multidimensional Weyl sums.

Extended Vinogradov's mean value theorem to arbitrary translation-dilation invariant systems.

Discussed multiple applications of the new bounds.

Abstract

We obtain sharp estimates for multidimensional generalisations of Vinogradov's mean value theorem for arbitrary translation-dilation invariant systems, achieving constraints on the number of variables approaching those conjectured to be the best possible. Several applications of our bounds are discussed.