Vinogradov's mean value theorem via efficient congruencing

TL;DR

This paper advances bounds on Vinogradov's integral using efficient congruencing, confirming conjectured asymptotics in Waring's problem for sufficiently large s.

Contribution

Introduces improved estimates for Vinogradov's integral through efficient congruencing, approaching the conjectured optimal bounds for the first time.

Findings

New bounds for Vinogradov's integral close to conjectured limits

Validation of the asymptotic formula in Waring's problem for s ≥ 2k^2+2k-3

Applications demonstrating the effectiveness of the new bounds

Abstract

We obtain estimates for Vinogradov's integral which for the first time approach those conjectured to be the best possible. Several applications of these new bounds are provided. In particular, the conjectured asymptotic formula in Waring's problem holds for sums of s kth powers of natural numbers whenever s is at least 2k^2+2k-3.