Mahler's compactness theorem
Encyclopedia
In mathematics
, Mahler's compactness theorem, proved by , 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. 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
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 coset
s, 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 assertion of Mahler's theorem is equivalent to the compactness of the space of unit-covolume lattices in
whose systole
is larger than
.
Mahler's compactness theorem was generalized to semisimple Lie groups by Mumford; see Mumford's compactness theorem.
Mathematics
Mathematics is the study of quantity, space, structure, and change. Mathematicians seek out patterns and formulate new conjectures. Mathematicians resolve the truth or falsity of conjectures by mathematical proofs, which are arguments sufficient to convince other mathematicians of their validity...
, Mahler's compactness theorem, proved by , is a foundational result on lattices
Lattice (group)
In mathematics, especially in geometry and group theory, a lattice in Rn is a discrete subgroup of Rn which spans the real vector space Rn. Every lattice in Rn can be generated from a basis for the vector space by forming all linear combinations with integer coefficients...
in Euclidean space
Euclidean space
In mathematics, Euclidean space is the Euclidean plane and three-dimensional space of Euclidean geometry, as well as the generalizations of these notions to higher dimensions...
, 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
Sequence
In mathematics, a sequence is an ordered list of objects . Like a set, it contains members , and the number of terms is called the length of the sequence. Unlike a set, order matters, and exactly the same elements can appear multiple times at different positions in the sequence...
of lattices. In intuitive terms it says that this is possible in just two ways: becoming coarse-grained with a fundamental domain
Fundamental domain
In geometry, the fundamental domain of a symmetry group of an object is a part or pattern, as small or irredundant as possible, which determines the whole object based on the symmetry. More rigorously, given a topological space and a group acting on it, the images of a single point under the group...
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
- GLn(R)/GLn(Z)
that parametrises lattices in Rn, with its quotient topology. There is a well-defined
Well-defined
In mathematics, well-definition is a mathematical or logical definition of a certain concept or object which uses a set of base axioms in an entirely unambiguous way and satisfies the properties it is required to satisfy. Usually definitions are stated unambiguously, and it is clear they satisfy...
function Δ on X, which is the absolute value
Absolute value
In mathematics, the absolute value |a| of a real number a is the numerical value of a without regard to its sign. So, for example, the absolute value of 3 is 3, and the absolute value of -3 is also 3...
of the determinant
Determinant
In linear algebra, the determinant is a value associated with a square matrix. It can be computed from the entries of the matrix by a specific arithmetic expression, while other ways to determine its value exist as well...
of a matrix — this is constant on the coset
Coset
In mathematics, if G is a group, and H is a subgroup of G, and g is an element of G, thenA coset is a left or right coset of some subgroup in G...
s, 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
If and only if
In logic and related fields such as mathematics and philosophy, if and only if is a biconditional logical connective between statements....
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 assertion of Mahler's theorem is equivalent to the compactness of the space of unit-covolume lattices in
whose systoleSystole
Systole may refer to:*Systole , a term describing the contraction of the heart*Systolic array, a term used in computer architecture*Systolic geometry, a term used in mathematics...
is larger than
.Mahler's compactness theorem was generalized to semisimple Lie groups by Mumford; see Mumford's compactness theorem.