Approximating the main conjecture in Vinogradov's mean value theorem

Abstract

We apply multigrade efficient congruencing to estimate Vinogradov's integral of degree $k$ for moments of order $2s$, establishing strongly diagonal behaviour for $1\le s\le \frac{1}{2}k(k+1)-\frac{1}{3}k+o(k)$. In particular, as $k\to \infty$, we confirm the main conjecture in Vinogradov's mean value theorem for 100% of the critical interval $1\le s\le \frac{1}{2}k(k+1)$.

arXiv admin note: text overlap with arXiv:1310.8447

References in corpus (1)

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

Cited by in corpus (2)

• Nested efficient congruencing and relatives of Vinogradov's mean value theorem
• On integer solutions of Parsell-Vinogradov systems