# 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

#### 182 Citations

The Cubic Case of Vinogradov's Mean Value Theorem --- A Simplified Approach to Wooley's "Efficient Congruencing"

- Mathematics
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This is an expository paper, giving a simplified proof of the cubic case of the main conjecture for Vinogradov's mean value theorem.

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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

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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… Expand

Arithmetic combinatorics on Vinogradov systems

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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

- Mathematics
- Journal of Number Theory
- 2019

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

- Mathematics
- 2018

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

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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… Expand

A large sieve inequality for power moduli

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- Acta Arithmetica
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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

- Mathematics
- 2016

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… Expand

On integer solutions of Parsell–Vinogradov systems

- Mathematics
- Inventiones mathematicae
- 2019

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.

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