The lattice discrepancy of certain three-dimensional bodies

TL;DR

This paper derives an asymptotic formula for counting lattice points in a three-dimensional body related to a Lame disc, accounting for boundary points with zero Gaussian curvature, advancing understanding of lattice discrepancies in geometric bodies.

Contribution

It introduces a new asymptotic formula for lattice point counts in three-dimensional bodies derived from Lame discs, with special focus on boundary curvature effects.

Findings

Established an asymptotic formula for lattice points in the body

Analyzed the impact of boundary points with zero Gaussian curvature

Provided a more precise approximation for lattice discrepancy

Abstract

Based on a fairly precise approximation to the lattice discrepancy of a Lame disc, an asymptotic formula is established for the number of lattice points in a related three-dimensional body, linearly dilated by a large real parameter x. Particular care is taken of the boundary points of Gaussian curvature zero.