Duke Mathematical Journal

Vinogradov’s mean value theorem via efficient congruencing, II

Trevor D. Wooley

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We apply the efficient congruencing method to estimate Vinogradov’s integral for moments of order 2s, with 1sk21. Thereby, we show that quasi-diagonal behavior holds when s=o(k2), and we obtain near-optimal estimates for 1s14k2+k and optimal estimates for sk21. In this way we come halfway to proving the main conjecture in two different directions. There are consequences for estimates of Weyl type and in several allied applications. Thus, for example, the anticipated asymptotic formula in Waring’s problem is established for sums of s kth powers of natural numbers whenever s2k22k8 (k6).

Article information

Duke Math. J., Volume 162, Number 4 (2013), 673-730.

First available in Project Euclid: 15 March 2013

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Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

Primary: 11L15: Weyl sums
Secondary: 11L07: Estimates on exponential sums 11P05: Waring's problem and variants 11P55: Applications of the Hardy-Littlewood method [See also 11D85]


Wooley, Trevor D. Vinogradov’s mean value theorem via efficient congruencing, II. Duke Math. J. 162 (2013), no. 4, 673--730. doi:10.1215/00127094-2079905. https://projecteuclid.org/euclid.dmj/1363355691

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