A Blichfeldt-type inequality for centrally symmetric convex bodies

TL;DR

This paper establishes a sharp asymptotic upper bound on lattice points within centrally symmetric convex bodies, extending Davenport's results through volume projection techniques.

Contribution

It introduces a generalized inequality for lattice points in symmetric convex bodies, improving bounds using projection-based volume analysis.

Findings

Derived an asymptotically sharp upper bound on lattice points

Generalized Davenport's result for symmetric convex bodies

Utilized volume projections to improve bounds

Abstract

In this note, we derive an asymptotically sharp upper bound on the number of lattice points in terms of the volume of centrally symmetric convex bodies. Our main tool is a generalization of a result of Davenport that bounds the number of lattice points in terms of volumes of suitable projections.