Geometry of numbers

In number theory, the geometry of numbers studies convex bodies and integer vectors in n-dimensional space. The geometry of numbers was initiated by Hermann Minkowski (1910).

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 results

Main article: Minkowski's theorem

Suppose that Γ is a lattice in n-dimensional Euclidean space Rⁿ and K is a convex centrally symmetric body. Minkowski's theorem, sometimes called Minkowski's first theorem, states that if $\mathrm{vol} (K)>2^n \mathrm{vol} (\mathbb{R}^n/\Gamma)$, then K contains a nonzero vector in Γ.

Main article: Minkowski's second theorem

The successive minimum λ_k is defined to be the inf of the numbers λ such that λK contains k linearly independent vectors of Γ. Minkowski's theorem on successive minima, sometimes called Minkowski's second theorem, is a strengthening of his first theorem and states that

$\lambda_1\lambda_2\cdots\lambda_n \mathrm{vol} (K)\le 2^n \mathrm{vol} (\mathbb{R}^n/\Gamma).$

Later research in the geometry of numbers

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.

Subspace theorem of W. M. Schmidt

Main article: Subspace theorem

See also: Siegel's lemma, volume (mathematics), determinant, and Parallelepiped

In the geometry of numbers, the subspace theorem was obtained by Wolfgang M. Schmidt in 1972. It states that if n is a positive integer, and L₁,...,L_n 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

$|L_1(x)\cdots L_n(x)|<|x|^{-\varepsilon}$

lie in a finite number of proper subspaces of Qⁿ.

Influence on functional analysis

Main article: normed vector space

See also: Banach space and F-space

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.

Researchers continue to study generalizations to star-shaped sets and other non-convex sets.

Bibliography

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• (Republished in 1964 by Dover.)
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• Lovász, L.: An Algorithmic Theory of Numbers, Graphs, and Convexity, CBMS-NSF Regional Conference Series in Applied Mathematics 50, SIAM, Philadelphia, Pennsylvania, 1986
• Wolfgang M. Schmidt. Diophantine approximation. Lecture Notes in Mathematics 785. Springer. (1980 [1996 with minor corrections])
• Rolf Schneider, Convex bodies: the Brunn-Minkowski theory, Cambridge University Press, Cambridge, 1993.
• Anthony C. Thompson, Minkowski geometry, Cambridge University Press, Cambridge, 1996.