Lattice-point enumerators of ellipsoids

TL;DR

This paper establishes an upper bound on the number of lattice points inside ellipsoids, paralleling Minkowski's successive minima volume bound, and discusses potential extensions to general convex bodies.

Contribution

It proves a lattice point count bound for ellipsoids that mirrors Minkowski's volume bound, suggesting a possible approach for broader classes of convex bodies.

Findings

Number of lattice points in ellipsoids bounded similarly to volume bounds

Potential extension of the method to general convex bodies

Open problem regarding unconditional bounds for all convex bodies

Abstract

Minkowski's second theorem on successive minima asserts that the volume of a 0-symmetric convex body K over the covolume of a lattice \Lambda can be bounded above by a quantity involving all the successive minima of K with respect to \Lambda. We will prove here that the number of lattice points inside K can also accept an upper bound of roughly the same size, in the special case where K is an ellipsoid. Whether this is also true for all K unconditionally is an open problem, but there is reasonable hope that the inductive approach used for ellipsoids could be extended to all cases.