Minkowski’s Conjecture, Well-Rounded Lattices and Topological Dimension

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Minkowski’s Conjecture, Well-Rounded Lattices and Topological Dimension

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Title: Minkowski’s Conjecture, Well-Rounded Lattices and Topological Dimension
Author: McMullen, Curtis T.
Citation: McMullen, Curtis T. 2005. Minkowski’s conjecture, well-rounded lattices and topological dimension. Journal of the American Mathematical Society 18: 711-734. Revised 2007.
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Abstract: Let A ⊂ SLn(R) be the diagonal subgroup, and identify SLn(R)/SLn(Z)with the space of unimodular lattices in Rn. In this paper we show that the closure of any bounded orbit A • L ⊂ SLn(R)/SLn(Z)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 Rn and Tn.
Published Version: doi:10.1090/S0894-0347-05-00483-2
Terms of Use: This article is made available under the terms and conditions applicable to Other Posted Material, as set forth at http://nrs.harvard.edu/urn-3:HUL.InstRepos:dash.current.terms-of-use#LAA
Citable link to this page: http://nrs.harvard.edu/urn-3:HUL.InstRepos:3446013
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