## Acta Mathematica

### On Vinogradov’s mean value theorem: strongly diagonal behaviour via efficient congruencing

#### Abstract

We enhance the efficient congruencing method for estimating Vinogradov’s integral for moments of order 2s, with ${1\leqslant s\leqslant k^{2}-1}$. In this way, we prove the main conjecture for such even moments when ${1\leqslant s\leqslant \tfrac{1}{4}(k+1)^{2}}$, showing that the moments exhibit strongly diagonal behaviour in this range. There are improvements also for larger values of s, these finding application to the asymptotic formula in Waring’s problem.

#### Note

The first author was supported in part by National Science Foundation grants DMS-0901339 and DMS-1201442.

#### Note

The second author was supported in part by a Royal Society Wolfson Research Merit Award.

#### Article information

Source
Acta Math., Volume 213, Number 2 (2014), 199-236.

Dates
Revised: 5 November 2013
First available in Project Euclid: 30 January 2017

https://projecteuclid.org/euclid.acta/1485801866

Digital Object Identifier
doi:10.1007/s11511-014-0119-0

Mathematical Reviews number (MathSciNet)
MR3286035

Zentralblatt MATH identifier
1307.11102

Rights

#### Citation

Ford, Kevin; Wooley, Trevor D. On Vinogradov’s mean value theorem: strongly diagonal behaviour via efficient congruencing. Acta Math. 213 (2014), no. 2, 199--236. doi:10.1007/s11511-014-0119-0. https://projecteuclid.org/euclid.acta/1485801866

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