Multigrade efficient congruencing and Vinogradov's mean value theorem
نویسندگان
چکیده
منابع مشابه
Multigrade Efficient Congruencing and Vinogradov’s Mean Value Theorem
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 8k . 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 number...
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We obtain estimates for Vinogradov’s integral which for the first time approach those conjectured to be the best possible. Several applications of these new bounds are provided. In particular, the conjectured asymptotic formula in Waring’s problem holds for sums of s kth powers of natural numbers whenever s > 2k + 2k − 3.
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We apply the efficient congruencing method to estimate Vinogradov’s integral for moments of order 2s, with 1 6 s 6 k − 1. Thereby, we show that quasi-diagonal behaviour holds when s = o(k), we obtain near-optimal estimates for 1 6 s 6 1 4k 2 + k, and optimal estimates for s > k − 1. In this way we come half way to proving the main conjecture in two different directions. There are consequences f...
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We enhance the efficient congruencing method for estimating Vinogradov’s integral for moments of order 2s, with 1 6 s 6 k− 1. In this way, we prove the main conjecture for such even moments when 1 6 s 6 1 4 (k+1) , 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...
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We apply multigrade efficient congruencing to estimate Vinogradov’s integral of degree k for moments of order 2s, establishing strongly diagonal behaviour for 1 6 s 6 1 2 k(k + 1) − 1 3 k + o(k). In particular, as k → ∞, we confirm the main conjecture in Vinogradov’s mean value theorem for 100% of the critical interval 1 6 s 6 1 2 k(k + 1).
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ژورنال
عنوان ژورنال: Proceedings of the London Mathematical Society
سال: 2015
ISSN: 0024-6115,1460-244X
DOI: 10.1112/plms/pdv034