The cubic case of the main conjecture in Vinogradov's mean value theorem

Abstract

We apply a variant of the multigrade efficient congruencing method to estimate Vinogradov's integral of degree 3 for moments of order 2s, establishing strongly diagonal behaviour for 1 s 6. Consequently, the main conjecture is now known to hold for the first time in a case of degree exceeding 2. s (1 j k),(1.1)with 1 x i , y i X (1 i s). The main conjecture in Vinogradov's mean value theorem asserts that for each ε > 0, one hasan estimate that, but for the presence of the factor X ε , would be best possible (see [5, equation (7.4)]). Despite eighty years of intense investigation, such an estimate has been established only in two cases, namely the (trivial) linear case with k = 1, and the quadratic case with k = 2 in which the elementary theory of quadratic forms can be brought to bear. Our goal in this paper is the first proof of the main conjecture (1.2) in a case with k > 2.Theorem 1.1. For each ε > 0, one has J s,3 (X) ≪ X ε (X s + X 2s−6 ).The estimate for J s,3 (X) recorded in this theorem, which establishes the main conjecture in Vinogradov's mean value theorem in the cubic case k = 3, goes substantially beyond the estimates available hitherto. By means of Newton's formulae concerning the roots of polynomials, it is apparent that J s,3 (X) = s!X s + O(X s−1 ) for 1 s 3, since the solutions of (1.1) are then simply the diagonal ones with {x 1 , . . . , x s } = {y 1 , . . . , y s }. Moreover, from [6, Theorem 1.5] one has J 4,3 (X) = 4!X 4 + O(X 10/3 (log 2X) 35 ).These estimates confirm (1.2) for 1 s 4 in a particularly strong form when k = 3, though in the latter range the estimate (1.2) has been known since at

Discrete Fourier restriction via Efficient Congruencing

“…The arguments required for the proof of Theorem 9.1 are straightforward analogues of those required in the case d = 1 central to this paper, and involve none of the complications demanded by the multigrade efficient congruencing methods of [23,24,25]. We consequently propose to expand no further on this subject, leaving the reader to complete the routine exercises needed for its proof, and to apply [16] as the necessary framework.…”

“…Indeed, the basic [7,22] and multigrade variants of efficient congruencing introduced in [23,24,25] may all be modified to accommodate the weighted setting appropriate for restriction theory. Although this task is not especially easy, by adapting these methods one may establish the Main Conjecture (recorded above as Conjecture 1.4) for 1 s Moreover, one may take ε = 0 when s > k(k−1).…”

“…Very recently, with the arrival of the efficient congruencing method, such conclusions have become available for all k 2 with s 0 (k) k(k + 1) (see Wooley [21,Theorem 1.1]). The very latest refinements of such work [25,Theorem 1.2] show that one may even take s 0 (k) k(k − 1) whenever k 3.…”

Mean value estimates for odd cubic Weyl sums

“…Our starting point for the proof of Theorems 1.1 and 1.2 is the mean value estimate supplied by the cubic case of the main conjecture in Vinogradov's mean value theorem, established in our very recent work [10…”

Rational solutions of pairs of diagonal equations, one cubic and one quadratic

“…In this memoir we attain this technical limit in the previously inaccessible case of two diagonal equations, one cubic and one quadratic, in 11 variables. It transpires that progress is possible here owing to the author's recent proof [21] of the cubic case of the main conjecture in Vinogradov's mean value theorem, though wielding the latter to achieve our present purpose entails further innovations beyond the conventional repertoire of Hardy-Littlewood artisans.…”