Multigrade efficient congruencing and Vinogradov's mean value theorem

Trevor D. Wooley

Research output: Contribution to journalArticle (Academic Journal)peer-review

23 Citations (Scopus)
328 Downloads (Pure)


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 languageEnglish
Pages (from-to)519-560
Number of pages42
JournalProceedings of the London Mathematical Society
Issue number3
Early online date15 Jul 2015
Publication statusPublished - 3 Sep 2015


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


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