Lattice of points

point lattice, in $ \mathbf R ^ {n} $,  with basis $ [ \mathbf e _ {1} \dots \mathbf e _ {n} ] $

The set $ \Lambda = \mathbf Z {\mathbf e _ {1} } + \dots + \mathbf Z {\mathbf e _ {n} } $ of points $ \mathbf a = g _ {1} \mathbf e _ {1} + \dots + g _ {n} \mathbf e _ {n} $, where $ g _ {1} \dots g _ {n} $ are integers.

The lattice $ \Lambda $ can be regarded as the free Abelian group with $ n $ generators. A lattice $ \Lambda $ has an infinite number of bases; their general form is $ ( \mathbf e _ {1} \dots \mathbf e _ {n} ) U $, where $ U $ runs through all integral matrices of determinant $ \pm 1 $. The quantity

$$ d ( \Lambda ) = | \mathop{\rm det} ( \mathbf e _ {1} \dots e _ {n} ) | > 0 $$

is the volume of the parallelopipedon formed by the vectors $ \mathbf e _ {1} \dots \mathbf e _ {n} $. It does not depend on the choice of a basis and is called the determinant of the lattice $ \Lambda $.

The partition of point lattices into Voronoi lattice types plays an important role in the geometry of quadratic forms (cf. Quadratic form).

#### Comments[edit]

The idea of lattices and lattice points links geometry to arithmetic (integers). Therefore it plays a central role in the geometry of numbers; integer programming (lattice point theorems); Diophantine approximations; reduction theory; analytic number theory; numerical analysis; crystallography (cf. Crystallography, mathematical); coding and decoding; combinatorics; geometric algorithms, and other areas.

#### References[edit]

[a1] |  J.W.S. Cassels, "An introduction to the geometry of numbers" , Springer (1972) MR1434478 MR0306130 MR0181613 MR0157947 Zbl 0866.11041 Zbl 0209.34401 Zbl 0131.29003 Zbl 0086.26203

|

[a2] |  P. Erdös, P.M. Gruber, J. Hammer, "Lattice points" , Longman (1989) MR1003606 Zbl 0683.10025
[a3] |  P.M. Gruber, C.G. Lekkerkerker, "Geometry of numbers" , North-Holland (1987) pp. Sect. (iv) (Updated reprint) MR0893813 Zbl 0611.10017
[a4] |  P.M. Gruber (ed.) J.M. Wills (ed.) , Handbook of convex geometry , North-Holland (1992) MR1242973 Zbl 0777.52002 Zbl 0777.52001
[a5] |  R. Kannan, L. Lovasz, "Covering minima and lattice-point-free convex bodies" Ann. of Math. , 128 (1988) pp. 577–602 MR0970611 Zbl 0659.52004