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Journal of the American Mathematical Society

Published by the American Mathematical Society, the Journal of the American Mathematical Society (JAMS) is devoted to research articles of the highest quality in all areas of pure and applied mathematics.

ISSN 1088-6834 (online) ISSN 0894-0347 (print)

The 2020 MCQ for Journal of the American Mathematical Society is 4.79.

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Minkowski’s conjecture, well-rounded lattices and topological dimension
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by Curtis T. McMullen PDF
J. Amer. Math. Soc. 18 (2005), 711-734 Request permission


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{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 $n=6$ 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
  • Received by editor(s): August 27, 2004
  • Published electronically: March 24, 2005
  • Additional Notes: Research partially supported by the NSF and the Guggenheim Foundation.
  • © Copyright 2005 American Mathematical Society
    The copyright for this article reverts to public domain 28 years after publication.
  • Journal: J. Amer. Math. Soc. 18 (2005), 711-734
  • MSC (2000): Primary 11H31; Secondary 11E57, 11J83, 55M10, 55N30
  • DOI:
  • MathSciNet review: 2138142