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

@article{Wooley2018NestedEC,
  title={Nested efficient congruencing and relatives of Vinogradov's mean value theorem},
  author={Trevor D. Wooley},
  journal={Proceedings of the London Mathematical Society},
  year={2018},
  volume={118}
}
  • T. Wooley
  • Published 3 August 2017
  • Mathematics
  • Proceedings of the London Mathematical Society
We apply a nested variant of multigrade efficient congruencing to estimate mean values related to that of Vinogradov. We show that when φj∈Z[t] (1⩽j⩽k) is a system of polynomials with non‐vanishing Wronskian, and s⩽k(k+1)/2 , then for all complex sequences (an) , and for each ε>0 , one has ∫[0,1)k∑|n|⩽Xane(α1φ1(n)+⋯+αkφk(n))2sdα≪Xε∑|n|⩽X|an|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… 
Subconvexity in inhomogeneous Vinogradov systems
When k and s are natural numbers and h ∈ Z, denote by Js,k(X ;h) the number of integral solutions of the system s ∑ i=1 (xji − y j i ) = hj (1 6 j 6 k), with 1 6 xi, yi 6 X . When s < k(k + 1)/2 and
Paucity problems and some relatives of Vinogradov's mean value theorem
When k > 4 and 0 6 d 6 (k− 2)/4, we consider the system of Diophantine equations x 1 + . . .+ xjk = y j 1 + . . .+ y k (1 6 j 6 k, j 6= k − d). We show that in this cousin of a Vinogradov system,
Counterexamples for high-degree generalizations of the Schr\"odinger maximal operator
. In 1980 Carleson posed a question on the minimal regularity of an initial data function in a Sobolev space H s ( R n ) that implies pointwise convergence for the solution of the linear Schr¨odinger
Efficient congruencing in ellipsephic sets: the general case
In this paper, we bound the number of solutions to a general Vinogradov system of equations $x_1^j+\dots+x_s^j=y_1^j+\dots+y_s^j$, $(1\leq j\leq k)$, as well as other related systems, in which the
Additive energy and a large sieve inequality for sparse sequences
We consider the large sieve inequality for sparse sequences of moduli and give a general result depending on the additive energy (both symmetric and asymmetric) of the sequence of moduli. For
Simultaneous equations and inequalities
Let λi, μj be non-zero real numbers not all of the same sign and let ai, bk be non-zero integers not all of the same sign. We investigate a mixed Diophantine system of the shape $ ’& ’% ˇ̌ λ1x θ 1 `
Energy estimates in sum-product and convexity problems
We prove a new class of low-energy decompositions which, amongst other consequences, imply that any finite set A of integers may be written as A = B ∪ C, where B and C are disjoint sets satisfying
Reversing a Philosophy: From Counting to Square Functions and Decoupling
Breakthrough work of Bourgain, Demeter, and Guth recently established that decoupling inequalities can prove powerful results on counting integral solutions to systems of Diophantine equations. In
A stationary set method for estimating oscillatory integrals
We propose a new method of estimating oscillatory integrals, which we call a “stationary set” method. We use it to obtain the sharp convergence exponents of Tarry’s problems in dimension two for
DIOPHANTINE INEQUALITIES OF FRACTIONAL DEGREE
This paper is concerned with the study of diagonal Diophantine inequalities of fractional degree θ, where θ > 2 is real and non-integral. For fixed non-zero real numbers λi not all of the same sign
...
...

References

SHOWING 1-10 OF 89 REFERENCES
A special case of Vinogradov's mean value theorem
i=1 (xji − y i ) = 0 (1 ≤ j ≤ k) with xi, yi ∈ [1, P ] ∩ Z are of great utility. This is perhaps best illustrated by the seminal works of Vinogradov from the first half of this century (see, for
Vinogradov's Integral and Bounds for the Riemann Zeta Function
The main result is an upper bound for the Riemann zeta function in the critical strip: ζ(σ+it)⩽A|t|B(1−σ)3/2log2/3⁡|t| with A = 76.2 and B = 4.45, valid for ½ ⩽ σ ⩽ 1 and |t| ⩾ 3. The previous best
Vinogradov's mean value theorem via efficient congruencing, II
We apply the efficient congruencing method to estimate Vinogradov's integral for moments of order 2s, with 1 =k^2-1. In this way we come half way to proving the main conjecture in two different
Discrete Fourier restriction via Efficient Congruencing
We show that whenever s > k(k + 1), then for any complex sequence (an)n∈Z, one has ∫ [0,1)k ∣∣∣∣ ∑ |n|6X ane(α1n+ . . .+ αkn ) ∣∣∣∣2s dα Xs−k(k+1)/2( ∑ |n|6X |an| )s . Bounds for the constant in the
ON VINOGRADOV'S MEAN VALUE THEOREM
The object of this paper is to obtain improvements in Vinogradov's mean value theorem widely applicable in additive number theory. Let J s,k (P) denote the number of solutions of the simultaneous
On Vinogradov’s mean value theorem: strongly diagonal behaviour via efficient congruencing
We enhance the efficient congruencing method for estimating Vinogradov’s integral for moments of order 2s, with $${1\leqslant s\leqslant k^{2}-1}$$1⩽s⩽k2-1. In this way, we prove the main conjecture
APPROXIMATING THE MAIN CONJECTURE IN VINOGRADOV'S MEAN VALUE THEOREM
We apply multigrade efficient congruencing to estimate Vino- gradov's integral of degree k for moments of order 2s, establishing strongly diagonal behaviour for 1 6 s 6 1 k(k + 1) − 1 k + o(k). In
Multigrade efficient congruencing and Vinogradov's mean value theorem
We develop a substantial enhancement of the efficient congruencing method to estimate Vinogradov's integral of degree k for moments of order 2s , thereby obtaining for the first time near‐optimal
The cubic case of the main conjecture in Vinogradov's mean value theorem
Vinogradov's mean value theorem via efficient congruencing
We obtain estimates for Vinogradov’s integral that for the rst time approach those conjectured to be the best possible. Several applications of these new bounds are provided. In particular, the
...
...