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

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

[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