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

```@article{Bourgain2015ProofOT,
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},
year={2015}
}```
• Published 4 December 2015
• Mathematics
• 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
190 Citations

#### Paper Mentions

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#### References

SHOWING 1-10 OF 22 REFERENCES
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
• Mathematics
• 2005
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
• Mathematics
• 2005
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
• Mathematics
• 2015
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
• Mathematics
• 2005
We prove that for a finite type curve in â3 the maximal operator generated by dilations is bounded on Lp for sufficiently large p. We also show the endpoint Lp â Lp1/p regularity result for theExpand
The asymptotic formula in Waring’s problem: Higher order expansions
• 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