Multigrade efficient congruencing and Vinogradov's mean value theorem

Trevor D. Wooley

doi:10.1112/plms/pdv034

Multigrade efficient congruencing and Vinogradov's mean value theorem

Trevor D. Wooley

Research output: Contribution to journal › Article (Academic Journal) › peer-review

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Abstract

We develop a substantial enhancement of the efficient congruencing method to estimate Vinogradov's integral of degree k for moments of order 2s, thereby obtaining for the first time near-optimal estimates for s>(5/8)k^2. There are numerous applications. In particular, when k is large, the anticipated asymptotic formula in Waring's problem is established for sums of s kth powers of natural numbers whenever s⩾1.543k^2.

Original language	English
Pages (from-to)	519-560
Number of pages	42
Journal	Proceedings of the London Mathematical Society
Volume	111
Issue number	3
Early online date	15 Jul 2015
DOIs	https://doi.org/10.1112/plms/pdv034
Publication status	Published - 3 Sep 2015

Keywords

math.NT
11L15
11L07
11P05
11P55

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10.1112/plms/pdv034

20150526mgecvin
This is the accepted version of the paper accepted to appear for publication in Proceedings of the London Mathematical Society.
Accepted author manuscript, 519 KB

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Multigrade efficient congruencing and Vinogradov's mean value theorem

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Keywords

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