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Journal of the American Mathematical Society

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


Author: Curtis T. McMullen
Journal: J. Amer. Math. Soc. 18 (2005), 711-734
MSC (2000): Primary 11H31; Secondary 11E57, 11J83, 55M10, 55N30
Published electronically: March 24, 2005
MathSciNet review: 2138142
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Abstract: Let $A \subset {\operatorname{SL}}_n({\mathbb R})$ be the diagonal subgroup, and identify ${\operatorname{SL}}_n({\mathbb R})/ {\operatorname{SL}}_n({\mathbb Z})$ with the space of unimodular lattices in ${\mathbb R}^n$. In this paper we show that the closure of any bounded orbit

\begin{displaymath}A \cdot L \subset {\operatorname{SL}}_n({\mathbb R})/{\operatorname{SL}}_n({\mathbb Z}) \end{displaymath}

meets the set of well-rounded lattices. This assertion implies Minkowski's conjecture for $n=6$ 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 ${\mathbb R}^n$ and $T^n$.


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Additional Information

Curtis T. McMullen
Affiliation: Department of Mathematics, Harvard University, 1 Oxford St, Cambridge, Massachusetts 02138-2901

DOI: http://dx.doi.org/10.1090/S0894-0347-05-00483-2
Received by editor(s): August 27, 2004
Published electronically: March 24, 2005
Additional Notes: Research partially supported by the NSF and the Guggenheim Foundation.
Article copyright: © Copyright 2005 American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication.