The Method of Trigonometrical Sums in the Theory of Numbers

By I. Vinogradov, K. F. Roth, A. Davenport. 2004

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Nested efficient congruencing and relatives of Vinogradov's mean value theorem (2017, background): “g method that we outline in §2 now also remedies this NESTED EFFICIENT CONGRUENCING 5 defect. Work prior to 2010 preceding the eﬃcient congruencing methods was, meanwhile, far weaker (see for example [23, 41, 43]). We have already expended considerable space on recording our main conclusions and des”
Vinogradov's mean value theorem via efficient congruencing, II (2011, methodology): “has J s,k(X) ≪ X2s− 1 2 k(k+1)+ε. Prior to the introduction of the eﬃcient congruencing method, conclusions of the type supplied by Theorem 1.1 were available only for s> (1+o(1))k2 logk (see [1], [15], [16], [18] and earlier work of Hua [8]). In [19, Theorem 1.1], meanwhile, we showed that Js,k(”
Strichartz estimates for Schrödinger equations on irrational tori (2013, methodology): “Then, by Van der Corput’s method [41] with”
Bounds for the maximal function associated to periodic solutions of one‐dimensional dispersive equations (2008, unknown): “For x0 = p/q, (p, q) = 1, and t = 1/q we have (see [12]) N ∑”
On the Vinogradov mean value (2016, background): “nds on Weyl sums and zero-free regions for the Riemann zeta function. The introduction of the mean value (1.2) and its signiﬁcance to number theory go back to the seminal work of I.M. Vinogradov (cf. [Vi2]). This approach is referred to as ‘Vinogradov’s Method’. Following T. Wooley, we call ‘Main Co”