Geometry of numbers

Typically, a ring of algebraic integers is viewed as a lattice in

and the study of these lattices provides fundamental information on algebraic numbers.

[1] Hermann Minkowski (1896) initiated this line of research at the age of 26 in his work The Geometry of Numbers.

[2] The geometry of numbers has a close relationship with other fields of mathematics, especially functional analysis and Diophantine approximation, the problem of finding rational numbers that approximate an irrational quantity.

Minkowski's theorem on successive minima, sometimes called Minkowski's second theorem, is a strengthening of his first theorem and states that[4] In 1930–1960 research on the geometry of numbers was conducted by many number theorists (including Louis Mordell, Harold Davenport and Carl Ludwig Siegel).

In recent years, Lenstra, Brion, and Barvinok have developed combinatorial theories that enumerate the lattice points in some convex bodies.

[5] In the geometry of numbers, the subspace theorem was obtained by Wolfgang M. Schmidt in 1972.

[6] It states that if n is a positive integer, and L1,...,Ln are linearly independent linear forms in n variables with algebraic coefficients and if ε>0 is any given real number, then the non-zero integer points x in n coordinates with lie in a finite number of proper subspaces of Qn.

Minkowski's geometry of numbers had a profound influence on functional analysis.

Minkowski proved that symmetric convex bodies induce norms in finite-dimensional vector spaces.

Minkowski's theorem was generalized to topological vector spaces by Kolmogorov, whose theorem states that the symmetric convex sets that are closed and bounded generate the topology of a Banach space.