Lattice points in bodies of revolution II

TL;DR

This paper extends bounds on lattice point discrepancies for bodies of revolution by removing a derivative condition, incorporating Diophantine analysis of Fourier phase coefficients, thus broadening applicable convex bodies.

Contribution

It generalizes previous results by eliminating the non-vanishing third derivative condition, allowing bodies with analytic boundaries to be included.

Findings

Improved lattice point discrepancy bounds for a wider class of revolution bodies.

Inclusion of bodies with analytic boundary in discrepancy estimates.

Use of Diophantine properties of Fourier phase coefficients in analysis.

Abstract

In a previous article it was shown that when a three-dimensional smooth convex body has rotational symmetry around a coordinate axis one can find better bounds for the lattice point discrepancy than what is known for more general convex bodies. To accomplish this, however, it was necessary to assume a non-vanishing condition on the third derivative of the generatrix. In this article we drop this condition, showing that the aforementioned bound holds for a wider family of revolution bodies, which includes those with analytic boundary. A novelty in our approach is that, besides the usual analytic methods, it requires studying some Diophantine properties of the Taylor coefficients of the phase on the Fourier transform side.