Geometry of numbers

Geometry of numbers is the part of number theory which uses geometry for the study of algebraic numbers. Typically, a ring of algebraic integers is viewed as a lattice in ${\displaystyle \mathbb {R} ^{n},}$ and the study of these lattices provides fundamental information on algebraic numbers.Hermann Minkowski (1896) initiated this line of research at the age of 26 in his work The Geometry of Numbers.

Best rational approximants for π (green circle), e (blue diamond), ϕ (pink oblong), (√3)/2 (grey hexagon), 1/√2 (red octagon) and 1/√3 (orange triangle) calculated from their continued fraction expansions, plotted as slopes y/x with errors from their true values (black dashes)  

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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

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

The successive minimum ${\displaystyle \lambda _{k}}$ is defined to be the inf of the numbers ${\displaystyle \lambda }$ such that ${\displaystyle \lambda K}$ contains ${\displaystyle k}$ linearly independent vectors of ${\displaystyle \Gamma }$. Minkowski's theorem on successive minima, sometimes called Minkowski's second theorem, is a strengthening of his first theorem and states that

${\displaystyle \lambda _{1}\lambda _{2}\cdots \lambda _{n}\operatorname {vol} (K)\leq 2^{n}\operatorname {vol} (\mathbb {R} ^{n}/\Gamma ).}$

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

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

${\displaystyle |L_{1}(x)\cdots L_{n}(x)|<|x|^{-\varepsilon }}$

lie in a finite number of proper subspaces of Qⁿ.

Influence on functional analysis

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.

References

1. MSC classification, 2010, available at http://www.ams.org/msc/msc2010.html, Classification 11HXX.
2. Minkowski, Hermann (2013-08-27). Space and Time: Minkowski's papers on relativity. Minkowski Institute Press. ISBN 978-0-9879871-1-2.
3. Schmidt's books. Grötschel, Martin; Lovász, László; Schrijver, Alexander (1993), Geometric algorithms and combinatorial optimization, Algorithms and Combinatorics, vol. 2 (2nd ed.), Springer-Verlag, Berlin, doi:10.1007/978-3-642-78240-4, ISBN 978-3-642-78242-8, MR 1261419
4. Cassels (1971) p. 203
5. Grötschel et al., Lovász et al., Lovász, and Beck and Robins.
6. Schmidt, Wolfgang M. Norm form equations. Ann. Math. (2) 96 (1972), pp. 526–551. See also Schmidt's books; compare Bombieri and Vaaler and also Bombieri and Gubler.
7. For Kolmogorov's normability theorem, see Walter Rudin's Functional Analysis. For more results, see Schneider, and Thompson and see Kalton et al.
8. Kalton et al. Gardner

Bibliography

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