Mahler's compactness theorem, proved by Kurt Mahler in 1946, characterizes when a set of lattices in Euclidean space is relatively compact, in the sense that it is bounded in the space that parametrizes lattices. It shows that a set of lattices fails to stay bounded in exactly two ways: either the volumes of their fundamental domains grow without bound, or the lattices contain vectors that shrink toward zero length. The theorem is foundational to the geometry of numbers, and David Mumford later generalized it to semisimple Lie groups.
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StatementA set of lattices in Euclidean space can fail to be bounded in only two ways: fundamental domains of ever larger volume, or ever shorter vectors. 1 Classification
Statement FormCharacterization Theorem 1 Mahler's Compactness Theorem
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Source Mahler's compactness theorem (Wikipedia)
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Source Mahler's compactness theorem (Wikipedia)
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1. Mahler's compactness theorem (Wikipedia)
Introduction, first paragraph
it says that this is possible in just two ways: becoming coarse-grained with a fundamental domain that has ever larger volume; or containing shorter and shorter vectors
Introduction, first sentence
Kurt Mahler (1946)
Principal Result Of: Kurt Mahler, Lead sentence of the article on the result
In mathematics, Mahler's compactness theorem, proved by Kurt Mahler (1946), is a foundational result on lattices in Euclidean space, characterising sets of lattices that are 'bounded' in a certain definite sense.
Proved By: Kurt Mahler, Lead paragraph
In mathematics, Mahler's compactness theorem, proved by Kurt Mahler (1946), is a foundational result on lattices in Euclidean space, characterising
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