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The cubic case of the main conjecture in Vinogradov's mean value theorem

Trevor D. Wooley

Abstract

We apply a variant of the multigrade efficient congruencing method to estimate Vinogradov's integral of degree $3$ for moments of order $2s$, establishing strongly diagonal behaviour for $1\le s\le 6$. Consequently, the main conjecture is now known to hold for the first time in a case of degree exceeding $2$.

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 for moments of order , establishing strongly diagonal behaviour for . Consequently, the main conjecture is now known to hold for the first time in a case of degree exceeding .

Paper Structure

This paper contains 8 sections, 17 theorems, 147 equations.

Table of Contents

  1. Introduction
  2. The basic infrastructure
  3. Auxiliary systems of congruences
  4. The conditioning and pre-congruencing processes
  5. Efficient congruencing and the multigrade combination
  6. The latent monograde process
  7. The iterative process
  8. Applications

Key Result

Theorem 1.1

For each $\varepsilon>0$, one has $J_{s,3}(X)\ll X^\varepsilon (X^s+X^{2s-6})$.

Theorems & Definitions (17)

  • Theorem 1.1
  • Theorem 1.2
  • Lemma 2.1
  • Lemma 3.1
  • Lemma 3.2
  • Lemma 4.1
  • Lemma 4.2
  • Lemma 5.1
  • Lemma 5.2
  • Lemma 5.3
  • ...and 7 more