mathematics Mathematics is an area of knowledge that includes the topics of numbers, formulas and related structures, shapes and the spaces in which they are contained, and quantities and their changes. These topics are represented in modern mathematics ...

, Mahler's compactness theorem, proved by , is a foundational result on lattices in

Euclidean space Euclidean space is the fundamental space of geometry, intended to represent physical space. Originally, that is, in Euclid's Elements, Euclid's ''Elements'', it was the three-dimensional space of Euclidean geometry, but in modern mathematics ther ...

, 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 Degeneracy, degenerate, or degeneration may refer to: Arts and entertainment * Degenerate (album), ''Degenerate'' (album), a 2010 album by the British band Trigger the Bloodshed * Degenerate art, a term adopted in the 1920s by the Nazi Party i ...

(''go off to infinity'') in a

sequence In mathematics, a sequence is an enumerated collection of objects in which repetitions are allowed and order matters. Like a set, it contains members (also called ''elements'', or ''terms''). The number of elements (possibly infinite) is calle ...

of lattices. In intuitive terms it says that this is possible in just two ways: becoming ''coarse-grained'' with a

fundamental domain Given a topological space and a group acting on it, the images of a single point under the group action form an orbit of the action. A fundamental domain or fundamental region is a subset of the space which contains exactly one point from each o ...

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). Let ''X'' be the space :

\mathrm_n(\mathbb)/\mathrm_n(\mathbb)

that parametrises lattices in

\mathbb^n

, with its

quotient topology In topology and related areas of mathematics, the quotient space of a topological space under a given equivalence relation is a new topological space constructed by endowing the quotient set of the original topological space with the quotient t ...

. There is a

well-defined In mathematics, a well-defined expression or unambiguous expression is an expression whose definition assigns it a unique interpretation or value. Otherwise, the expression is said to be ''not well defined'', ill defined or ''ambiguous''. A funct ...

function Δ on ''X'', which is the

absolute value In mathematics, the absolute value or modulus of a real number x, is the non-negative value without regard to its sign. Namely, , x, =x if is a positive number, and , x, =-x if x is negative (in which case negating x makes -x positive), an ...

of the

determinant In mathematics, the determinant is a scalar value that is a function of the entries of a square matrix. It characterizes some properties of the matrix and the linear map represented by the matrix. In particular, the determinant is nonzero if and ...

of a matrix – this is constant on the

coset In mathematics, specifically group theory, a subgroup of a group may be used to decompose the underlying set of into disjoint, equal-size subsets called cosets. There are ''left cosets'' and ''right cosets''. Cosets (both left and right) ...

s, since an

invertible In mathematics, the concept of an inverse element generalises the concepts of opposite () and reciprocal () of numbers. Given an operation denoted here , and an identity element denoted , if , one says that is a left inverse of , and that is ...

integer matrix has

1 or −1. Mahler's compactness theorem states that a subset ''Y'' of ''X'' is

relatively compact In mathematics, a relatively compact subspace (or relatively compact subset, or precompact subset) of a topological space is a subset whose closure is compact. Properties Every subset of a compact topological space is relatively compact (sin ...

if and only if In logic and related fields such as mathematics and philosophy, "if and only if" (shortened as "iff") is a biconditional logical connective between statements, where either both statements are true or both are false. The connective is bicondi ...

Δ is bounded on ''Y'', and there is a neighbourhood ''N'' of 0 in

\mathbb^n

such that for all Λ in ''Y'', the only lattice point of Λ in ''N'' is 0 itself. The assertion of Mahler's theorem is equivalent to the compactness of the space of unit-covolume lattices in

\mathbb^n

whose

systole Systole ( ) is the part of the cardiac cycle during which some chambers of the heart contract after refilling with blood. The term originates, via New Latin, from Ancient Greek (''sustolē''), from (''sustéllein'' 'to contract'; from ''sun ...

is larger or equal than any fixed

\varepsilon>0

. Mahler's compactness theorem was generalized to

semisimple Lie group In mathematics, a Lie algebra is semisimple if it is a direct sum of modules, direct sum of simple Lie algebras. (A simple Lie algebra is a non-abelian Lie algebra without any non-zero proper Lie algebra#Subalgebras.2C ideals and homomorphisms, i ...

s by

David Mumford David Bryant Mumford (born 11 June 1937) is an American mathematician known for his work in algebraic geometry and then for research into vision and pattern theory. He won the Fields Medal and was a MacArthur Fellow. In 2010 he was awarded t ...

; see

Mumford's compactness theorem In mathematics, Mumford's compactness theorem states that the space of compact Riemann surfaces of fixed genus ''g'' > 1 with no closed geodesics of length less than some fixed ''ε'' > 0 in the Poincaré metric is compact. It w ...

References

*William Andrew Coppel (2006), ''Number theory'', p. 418. *{{Citation , last1=Mahler , first1=Kurt , authorlink=Kurt Mahler, title=On lattice points in n-dimensional star bodies. I. Existence theorems , jstor=97965 , mr=0017753 , year=1946 , journal= Proceedings of the Royal Society of London. Series A: Mathematical, Physical and Engineering Sciences , issn=0962-8444 , volume=187 , pages=151–187, doi=10.1098/rspa.1946.0072, doi-access=free Geometry of numbers Discrete groups Compactness theorems Theorems in number theory