We obtain estimates for Vinogradov's integral that 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 2 + 2k − 3.

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

We apply a nested variant of multigrade efficient congruencing to estimate mean values related to that of Vinogradov. We show that when φj∈Zfalse[tfalse] (1⩽j⩽k) is a system of polynomials with non‐vanishing Wronskian, and s⩽k(k+1)/2, then for all complex sequences (frakturan), and for each ε>0, one has 0true∫[0,1)k0true∑|n|⩽Xfrakturanefalse(α1φ1(n)+⋯+αkφk(n)false)2sdα≪Xε0true∑|n|⩽X|frakturan|2s.As a special case of this result, we confirm the main conjecture in Vinogradov's mean value theorem for all exponents k, recovering the recent conclusions of the author (for k=3) and Bourgain, Demeter and Guth (for k⩾4). In contrast with the l2‐decoupling method of the latter authors, we make no use of multilinear Kakeya estimates, and thus our methods are of sufficient flexibility to be applicable in algebraic number fields, and in function fields. We outline such extensions.

No abstract

We prove that any finite set of real numbers can be split into two parts, one part being highly non-additive and the other highly nonmultiplicative.Similarly, the multiplicativity of a set is measured by its multiplicative energy E × (A) = card{a ∈ A 4 : a 1 a 2 = a 3 a 4 }.