Nested efficient congruencing and relatives of Vinogradov's mean value theorem

Abstract

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.

Bounds for discrete moments of Weyl sums and applications

“…The theorem is elementary for k = 1 and k = 2. For the highly nontrivial cases k ≥ 3 it has been proved in [7] by Bourgain, Demeter and Guth for k ≥ 4 and in [19] by Wooley for k = 3, and again in [20] by Wooley for k ≥ 3. In our analysis, we will make use of this deep estimate.…”

“…The recent breakthroughs of Bourgain, Demeter and Guth in [7] and Wooley in [19,20] has led to a full proof of the main conjecture in Vinogradov's Mean Value Theorem (VMVT for short). As one consequence among many, new estimates for Weyl sums are available.…”

ℓ² decoupling in ℝ² for curves with vanishing curvature

“…Now we finish the paper with a simple proof of Corollary 1.5 and a corresponding discretized version. The latter might be useful for future applications which gives bounds similar to those of Wooley [7].…”

New Bounds of Weyl Sums

“…We also remark that as mentioned by Bourgain [4, Section 3], for d 6 better results are known. On the other hand, the behaviour of the average value of the Weyl sums has recently been fully unveiled in works of Bourgain, Demeter and Guth [5] (for d 4) and Wooley [21] (for d = 3) (see also [23]) in the best possible form (1) , N → ∞, of the Vinogradov mean value theorem, where for q ∈ R we denote (1.4) s(q) = q(q + 1) 2 .…”