Mahler's compactness theorem
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In mathematics, Mahler's compactness theorem is a foundational result on 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 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.
Let X be the space
- GLn(R)/GLn(Z)
that parametrises lattices in Rn, 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 cosets, since an invertible integer matrix has determinant 1 or −1.
Mahler's compactness theorem states that a subset Y of X is relatively compact if and only if Δ is bounded on Y, and there is a neighbourhood N of {0} in Rn such that for all Λ in Y, the only lattice point of Λ in N is {0} itself.
The theorem is due to Kurt Mahler (1903-1988).
