Counterexamples to Minkowski's Uniqueness Conjecture and Escape of Mass in Positive Characteristic

Abstract

We show that there are infinitely many counterexamples to Minkowski's conjecture in positive characteristic regarding uniqueness of the upper bound of the multiplicative covering radius, $\mu$, by constructing a sequence of compact $A$ orbits where $\mu$ obtains its conjectured upper bound. In addition, we show that these orbits, as well as a slightly larger sequence of orbits, must exhibit complete escape of mass.