Proof of the main conjecture in Vinogradov's mean value theorem for degrees higher than three

  title={Proof of the main conjecture in Vinogradov's mean value theorem for degrees higher than three},
  author={Jean Bourgain and Ciprian Demeter and Larry Guth},
  journal={arXiv: Number Theory},
We prove the main conjecture in Vinogradov's Mean Value Theorem for degrees higher than three. This will be a consequence of a sharp decoupling inequality for curves 

Figures from this paper

The Cubic Case of Vinogradov's Mean Value Theorem --- A Simplified Approach to Wooley's "Efficient Congruencing"
This is an expository paper, giving a simplified proof of the cubic case of the main conjecture for Vinogradov's mean value theorem.
On the Vinogradov mean value
We discuss the recent work of C. Demeter, L. Guth and the author on the proof of the Vinogradov Main Conjecture using the decoupling theory for curves.
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). InExpand
Arithmetic combinatorics on Vinogradov systems
In this paper, we present a variant of the Balog-Szemeredi-Gowers theorem for the Vinogradov system. We then use our result to deduce a higher degree analogue of the sum-product phenomenon.
Effective Vinogradov's mean value theorem via efficient boxing
Abstract We combine Wooley's efficient congruencing method with earlier work of Vinogradov and Hua to get effective bounds on Vinogradov's mean value theorem.
Vinogradov’s Mean Value Theorem as an Ingredient in Polynomial Large Sieve Inequalities and Some Consequences
We discuss the role of Vinogradov’s mean value theorem in polynomial large sieve inequalities. We present an application to the distribution of fractions with k-th power denominators. Moreover,Expand
On a binary system of Prendiville: The cubic case
We prove sharp decoupling inequalities for a class of two dimensional non-degenerate surfaces in R^5, introduced by Prendiville. As a consequence, we obtain sharp bounds on the number of integerExpand
A large sieve inequality for power moduli
In this note we give a new bound for large sieve with characters to power moduli which improves in some range of the parameters the previous bounds of Baier/Zhao and Halupczok.
Small fractional parts of polynomials
Using the recent result of Bourgain, Demeter and Guth on Vinogradov's mean value, a number of new results about small fractional parts of polynomials and fractional parts of additive forms areExpand
On integer solutions of Parsell–Vinogradov systems
We prove a sharp upper bound on the number of integer solutions of the Parsell–Vinogradov system in every dimension $$d\ge 2$$d≥2.


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 2 s , establishing strongly diagonal behaviour for 1 ⩽ sExpand
The Asymptotic Formula in Waring's Problem
We derive a new minor arc bound, suitable for applications associated with Waring’s problem, from Vinogradov’s mean value theorem. In this way, the conjectured asymptotic formula in Waring’s problemExpand
Decoupling inequalities and some mean-value theorems
The purpose of this paper is to present some further applications of the general decoupling theory from [B-D] and [B-D2] to certain diophantine issues. In particular, we consider mean value estimatesExpand
A short proof of the multilinear Kakeya inequality
  • L. Guth
  • Mathematics
  • Mathematical Proceedings of the Cambridge Philosophical Society
  • 2014
Abstract We give a short proof of a slightly weaker version of the multilinear Kakeya inequality proven by Bennett, Carbery and Tao.
On the multilinear restriction and Kakeya conjectures
We prove d-linear analogues of the classical restriction and Kakeya conjectures in Rd. Our approach involves obtaining monotonicity formulae pertaining to a certain evolution of families ofExpand
Finite bounds for Hölder-Brascamp-Lieb multilinear inequalities
A criterion is established for the validity of multilinear inequalities of a class considered by Brascamp and Lieb, generalizing well-known inequalties of Rogers and H\"older, Young, andExpand
Decouplings for surfaces in R^4
We prove a sharp decoupling for non degenerate surfaces in $\R^4$. This puts the recent progress on the Lindel\"of hypothesis into a more general perspective.
Decoupling, exponential sums and the Riemann zeta function
We establish a new decoupling inequality for curves in the spirit of [B-D1], [B-D2] which implies a new mean value theorem for certain exponential sums crucial to the Bombieri-Iwaniec method asExpand
L p regularity of averages over curves and bounds for associated maximal operators
We prove that for a finite type curve in ℝ3 the maximal operator generated by dilations is bounded on Lp for sufficiently large p. We also show the endpoint Lp → Lp1/p regularity result for theExpand
The asymptotic formula in Waring’s problem: Higher order expansions
  • R. Vaughan, T. Wooley
  • Mathematics
  • Journal für die reine und angewandte Mathematik (Crelles Journal)
  • 2018
When {k>1} and s is sufficiently large in terms of k, we derive an explicit multi-term asymptotic expansion for the number of representations of a large natural number as the sum of sExpand