Mean value estimates for odd cubic Weyl sums

TL;DR

This paper provides an essentially optimal estimate for the ninth moment of a specific exponential sum involving cubic and linear terms, marking a significant advancement in the field after over 60 years.

Contribution

It introduces a new optimal estimate for the ninth moment of cubic Weyl sums, improving upon longstanding bounds and enhancing applications in Diophantine analysis.

Findings

Established an essentially optimal ninth moment estimate

Improved Heath-Brown's Weyl inequality

Enhanced applications in Diophantine problems

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.