# Congruence Arguments in the Geometry of Numbers and a General Discrete Minkowksi-type Theorem

WHEN: 22.01.15 at 14:15

WHERE: Seminar Room, Arnimallee 2, FU Berlin

Speaker: Matthias Henze (FU Berlin)

Congruence Arguments in the Geometry of Numbers and a General

Discrete Minkowksi-type Theorem

Abstract: One of the most fruitful results from Minkowski's geometric

viewpoint on number theory is his so called 1st Fundamental Theorem.

It says that the volume of every o-symmetric n-dimensional convex body

whose only interior lattice point is the origin is bounded from above

by the volume of the orthogonal n-cube of edge length two. Minkowski

also obtained a discrete analog by identifying the n-cube as a

maximizer of 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. Using congruence arguments for lattice points and an

inequality in additive combinatorics, we determine a best possible

relation of this kind. In the talk, we will moreover highlight the

usefulness of considering congruences on lattice points in the

geometry of numbers.

This is joint work with Bernardo González Merino.

### Time & Location

Jan 22, 2015 | 02:15 PM

Seminar Room, Arnimallee 2, FU Berlin