Mean value estimates for odd cubic Weyl sums

Abstract

We establish an essentially optimal estimate for the ninth moment of the exponential sum having argument $\alpha x^3+\beta x$. The first substantial advance in this topic for over 60 years, this leads to improvements in Heath-Brown's variant of Weyl's inequality, and other applications of Diophantine type.

References in corpus (2)

• The cubic case of the main conjecture in Vinogradov's mean value theorem
• Rational solutions of pairs of diagonal equations, one cubic and one quadratic

Cited by in corpus (6)

• The cubic case of the main conjecture in Vinogradov's mean value theorem
• Nested efficient congruencing and relatives of Vinogradov's mean value theorem
• Translation invariance, exponential sums, and Waring's problem
• On integer solutions of Parsell-Vinogradov systems
• An instance where the major and minor arc integrals meet
• Discrete restriction for $(x,x^3)$ and related topics