Counterexamples to Minkowski's conjecture and escape of mass in positive characteristic (Q6631399)

Minkowski's conjecture is the following:\N\NFor every \(d \geq 2\) and for every \( x \in \operatorname{SL}_d({\mathbb R})/\operatorname{SL}_d({\mathbb Z})\):\N\begin{itemize}\N\item[1.] \(\mu(x) \leq 2^{-d} = \mu({\mathbb Z}^d),\)\N\item[2.] \( \mu(x) = 2^{-d}\) if and only if \(x \in A {\mathbb Z}^d,\)\N\end{itemize}\Nwhere \(A\) is the group of diagonal matrices with determinant 1 and \(\mu\) is the covering radius. The author gives infinitely many counterexamples to this famous conjecture in positive characteristic by constructing a sequence of compact \(A\)-orbits, where \(\mu\) obtains its conjectured upper bound but violates the uniqueness condition. Additionally, it is shown that these orbits must exhibit complete escape of mass.