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| In [[mathematics]], '''Mahler's compactness theorem''', proved by {{harvs|txt|authorlink=Kurt Mahler|first=Kurt|last=Mahler|year=1946}}, is a foundational result on [[lattice (group)|lattices]] in [[Euclidean space]], characterising sets of lattices that are 'bounded' in a certain definite sense. Looked at another way, it explains the ways in which a lattice could [[degeneracy (mathematics)|degenerate]] (''go off to infinity'') in a [[sequence]] of lattices. In intuitive terms 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. It is also called his '''selection theorem''', following an older convention used in naming compactness theorems, because they were formulated in terms of [[sequential compactness]] (the possibility of selecting a convergent subsequence).
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| Let ''X'' be the space
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| :<math>\mathrm{GL}_n(\mathbb{R})/\mathrm{GL}_n(\mathbb{Z})</math>
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| that parametrises lattices in <math>\mathbb{R}^n</math>, with its [[quotient topology]]. There is a [[well-defined]] function Δ on ''X'', which is the [[absolute value]] of the [[determinant]] of a matrix – this is constant on the [[coset]]s, since an [[invertible]] integer matrix has [[determinant]] 1 or −1.
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| '''Mahler's compactness theorem''' states that a subset ''Y'' of ''X'' is [[relatively compact]] [[if and only if]] Δ is [[bounded set|bounded]] on ''Y'', and there is a neighbourhood ''N'' of {0} in <math>\mathbb{R}^n</math> such that for all ''Λ'' in ''Y'', the only lattice point of Λ in ''N'' is 0 itself.
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| The assertion of Mahler's theorem is equivalent to the compactness of the space of unit-covolume lattices in <math>\mathbb{R}^n</math> whose [[systolic geometry|systole]] is larger or equal than any fixed <math>\epsilon>0</math>.
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| Mahler's compactness theorem was generalized to [[semisimple Lie group]]s by Mumford; see [[Mumford's compactness theorem]].
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| ==References==
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| *William Andrew Coppel (2006), ''Number theory'', p. 418.
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| {{reflist}}
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| *{{Citation | last1=Mahler | first1=K. | title=On lattice points in <var>n</var>-dimensional star bodies. I. Existence theorems | jstor=97965 | mr=0017753 | year=1946 | journal=Proceedings of the Royal Society. London. Series A. Mathematical, Physical and Engineering Sciences | issn=0962-8444 | volume=187 | pages=151–187}}
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| [[Category:Geometry of numbers]]
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| [[Category:Discrete groups]]
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| [[Category:Compactness theorems]]
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| [[Category:Theorems in number theory]]
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