Vinogradov's mean value theorem via efficient congruencing

Abstract

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.

Discrete Fourier restriction via Efficient Congruencing

“…Indeed, one has K p,X ∼ A 1/2 p,X . By adapting the efficient congruencing method introduced in work [21] of the author associated with Vinogradov's mean value theorem, we obtain in § §6 and 7 the following conclusion. Theorem 1.1.…”

“…Although analogous to that of our corresponding work [21] concerning Vinogradov's mean value theorem, we are forced to deviate significantly from our earlier path.…”

“…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.…”

“…Our basic strategy in proving Theorem 1.1 is to adapt to the present setting the efficient congruencing method introduced by the author in the context of Vinogradov's mean value theorem (see [21]). Several complications must be surmounted in such a plan of attack.…”

Subconvexity for additive equations: Pairs of undenary cubic forms

“…The simplest situations to describe are those wherein one has non-trivial minor arc estimates in mean square for each of the polynomials φ, ψ, χ, ω. Such is the case, for example, when these polynomials are suitably nonsingular forms in a number of variables exceeding (d−1)2 d−1 , as a consequence of the work of Birch [2], and also when these polynomials are diagonal forms of degree d in d 2 variables (see [33,34]). In the latter case, moreover, if one restricts the variables to be smooth then one can reduce the number of variables required to 1 2 d(log d + log log d + O (1)) (see the methods of [29,30]).…”