A generalization of the discrete version of Minkowski's fundamental theorem

TL;DR

This paper extends Minkowski's discrete lattice point theorem to include bodies with multiple interior lattice points, establishing an optimal relation between boundary and interior lattice points using additive combinatorics.

Contribution

It provides the first optimal discrete relation for convex bodies with multiple interior lattice points, generalizing Minkowski's original theorem.

Findings

Established an optimal boundary-internal lattice point relation.

Used additive combinatorics techniques in the proof.

Extended classical lattice point bounds to multiple interior points.

Abstract

One of the most fruitful results from Minkowski's geometric viewpoint on number theory is his so called 1st Fundamental Theorem. It provides an optimal upper bound for the volume of an o-symmetric convex body whose only interior lattice point is the origin. Minkowski also obtained a discrete analog by proving optimal upper bounds on the number of lattice points in the boundary of such convex bodies. Whereas the volume inequality has been generalized to any number of interior lattice points already by van der Corput in the 1930s, a corresponding result for the discrete case remained to be proven. Our main contribution is a corresponding optimal relation between the number of boundary and interior lattice points of an o-symmetric convex body. The proof relies on a congruence argument and a difference set estimate from additive combinatorics.