Minkowski’s conjecture, well-rounded lattices and topological dimension

Abstract

Let A ⊂ SL n ( R ) A \subset {\operatorname {SL}}_n({\mathbb R}) be the diagonal subgroup, and identify SL n ( R ) / SL n ( Z ) {\operatorname {SL}}_n({\mathbb R})/ {\operatorname {SL}}_n({\mathbb Z}) with the space of unimodular lattices in R n {\mathbb R}^n . In this paper we show that the closure of any bounded orbit A ⋅ L ⊂ SL n ( R ) / SL n ( Z ) \begin{equation*} A \cdot L \subset {\operatorname {SL}}_n({\mathbb R})/{\operatorname {SL}}_n({\mathbb Z}) \end{equation*} meets the set of well-rounded lattices. This assertion implies Minkowski’s conjecture for n = 6 n=6 and yields bounds for the density of algebraic integers in totally real sextic fields. The proof is based on the theory of topological dimension, as reflected in the combinatorics of open covers of R n {\mathbb R}^n and T n T^n .