Moment estimates for cubic Weyl sums connect Fourier restriction theory with the counting of integer solutions to polynomial equations. I will discuss work in preparation that combines methods from harmonic analysis and analytic number theory to study these sums with general complex coefficients.

Wooley established a sharp ninth-moment bound and sharp major-arc eighth-moment estimates when all coefficients equal one. I will present a sharp (\ell^\infty, L^9) estimate for arbitrary coefficients, together with sharp eighth-moment bounds in the high-amplitude regime, all up to arbitrarily small power losses. The high-amplitude estimates admit refinements involving both the (\ell^2) and (\ell^\infty) norms of the coefficients.

The approach combines high–low stopping-time analysis with the circle method’s decomposition into major and minor arcs. On the minor arcs, the auxiliary square functions arising from the stopping-time argument can be lifted to a three-dimensional setting where cone decoupling applies. A second high-frequency dominance argument, applied to a mixed norm, is key to obtaining the high-amplitude estimates.