Covering Radii in Positive Characteristic

Abstract: A fascinating question in geometry of number pertains to the covering radius of lattice with respect to an interesting function. For example, given a convex body C and a lattice L in R^d, it is interesting to ask what is the infimal r ≥ 0 such that L + rC = R^d. Another interesting covering radius is the multiplicative covering radius, which connects to dynamics due to its invariance under the diagonal group. It was conjectured by Minkowski that the multiplicative covering radius is bounded above by 2^{-d} and that this upper bound is obtained only on AZ^d. In this talk I will discuss surprising results pertaining to covering radii in the positive characteristic setting and discover several surprising results. Some of my results include explicitly connecting between the covering radii with respect to convex bodies and successive minima and proving a positive characteristic analogue of Minkowski’s function.

dynamical systemsnumber theory

Audience: general audience

New England Dynamics and Number Theory Seminar