Geometry of numbers

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In number theory, the geometry of numbers is a topic and method arising from the work of Hermann Minkowski, on the relationship between convex sets and lattices in n-dimensional space. It has frequently been used in an auxiliary role in proofs, particularly in diophantine approximation. The subject was given a great deal of attention in the period 1930-1960 by some leading number theorists (including Louis Mordell, Harold Davenport and Carl Ludwig Siegel).

Minkowski's theorem establishes a relation between symmetric convex sets and integer points; we might as well say, between any lattice and any Banach space norm in n dimensions. The topic therefore belongs properly to a sort of affine geometry simplification of the theory of quadratic forms (Hilbert space norms in relation to lattices). To relax the convexity technique in a non-trivial way may be technically difficult.

The theoretical foundations can be considered as dealing with the space of lattices in n dimensions, which is a priori the coset space GL_n(R)/GL_n(Z). This is not easy to deal with directly (it is an example for the theory rather of arithmetic groups). One foundational result is Mahler's compactness theorem describing the relatively compact subsets (the coset space is non-compact, as can be seen already in the case n = 2, where there are cusps).

One can say that the geometry of numbers takes on some of the work that continued fractions do, for diophantine approximation questions in two or more dimensions - there is no straightforward generalisation.

[edit] References

• Siegel, Carl Ludwig (1989). Lectures on the Geometry of Numbers. Springer-Verlag. 

• Hancock, Harris (1939). Development of the Minkowski Geometry of Numbers. Macmillan.  (Republished in 1964 by Dover.)