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

Kevin Ford; Trevor D. Wooley

doi:10.1007/s11511-014-0119-0

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

Kevin Ford, Trevor D. Wooley

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

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Abstract

We enhance the efficient congruencing method for estimating Vinogradov’s integral for moments of order 2s, with 1⩽s⩽k2−1. In this way, we prove the main conjecture for such even moments when 1⩽s⩽14(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.

Original language	English
Pages (from-to)	199-236
Number of pages	38
Journal	Acta Mathematica
Volume	213
Issue number	2
Early online date	25 Nov 2014
DOIs	https://doi.org/10.1007/s11511-014-0119-0
Publication status	Published - 1 Dec 2014

Keywords

math.NT
11L15, 11L07, 11P05, 11P55

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10.1007/s11511-014-0119-0

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abstract = "We enhance the efficient congruencing method for estimating Vinogradov{\textquoteright}s integral for moments of order 2s, with 1⩽s⩽k2−1. In this way, we prove the main conjecture for such even moments when 1⩽s⩽14(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{\textquoteright}s problem.",

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AU - Wooley, Trevor D.

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N2 - We enhance the efficient congruencing method for estimating Vinogradov’s integral for moments of order 2s, with 1⩽s⩽k2−1. In this way, we prove the main conjecture for such even moments when 1⩽s⩽14(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.

AB - We enhance the efficient congruencing method for estimating Vinogradov’s integral for moments of order 2s, with 1⩽s⩽k2−1. In this way, we prove the main conjecture for such even moments when 1⩽s⩽14(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.

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JF - Acta Mathematica

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

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