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}
}
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.
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
Arithmetic combinatorics on Vinogradov systems
TLDR
A variant of the Balog-Szemeredi-Gowers theorem for the Vinogradov system is presented and a higher degree analogue of the sum-product phenomenon is deduced.
Effective Vinogradov's mean value theorem via efficient boxing
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,
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 integer
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 are
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.
...
...

References

SHOWING 1-10 OF 22 REFERENCES
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 problem
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 estimates
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 of
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, and
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 as
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 the
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 s
...
...