On Covering Radii in Function Fields

TL;DR

This paper explores covering radii in the geometry of numbers over function fields with positive characteristic, proving analogues of classical conjectures and establishing new results about lattice coverings and spectra.

Contribution

It introduces closed-form formulas for covering radii in positive characteristic and proves analogues of Woods' and Minkowski's conjectures in this setting.

Findings

Closed form for covering radii with respect to convex bodies

Proof of positive characteristic analogue of Woods' conjecture

Triviality of the Gruber-Mordell spectrum in all dimensions

Abstract

In this paper, we shall discuss topics in geometry of numbers in the positive characteristic setting, such as covering radii. We find a closed form for covering radii with respect to convex bodies, which will lead to a proof of the positive characteristic analogue of Woods' conjecture in this setting. Then, we will prove a positive characteristic analogue of Minkowski's conjecture about the multiplicative covering radius. To do this, we shall prove a positive characteristic analogue of Solan's result that every diagonal orbit intersects the set of well rounded lattices. This implies that the Gruber-Mordell spectrum in positive characteristic is trivial in every dimension.