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

Jean Bourgain, Ciprian Demeter, Larry Guth

Abstract

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

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

Abstract

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

Paper Structure

This paper contains 13 sections, 31 theorems, 299 equations.

Table of Contents

  1. Introduction
  2. Notation
  3. Overview of the method
  4. From decouplings to exponential sums
  5. The $\epsilon$ removal argument
  6. Transition to larger balls
  7. A few basic tools
  8. The iteration scheme
  9. The case $n=3$
  10. The case $n=4$
  11. The case of arbitrary $n$
  12. The proof of Theorem \ref{['ntmain']} for $n\ge 3$
  13. Appendix

Key Result

Theorem 1.1

For each $s\ge 1$ and $n,N\ge 2$ we have the upper bound

Theorems & Definitions (50)

  • Theorem 1.1
  • Theorem 1.2
  • Theorem 4.1
  • proof
  • Corollary 4.2
  • proof
  • Theorem 6.1
  • Proposition 6.2
  • proof
  • Lemma 6.3
  • ...and 40 more