Geometrynumbers

In number theory,geometrynumbers refers totopicmethod arising fromworkHermann Minkowski, onrelationship between convex setslatticesn-dimensional space. It has frequently been usedan auxiliary roleproofs, particularlydiophantine approximation. The subject was givengreat dealattention inperiod 1930-1960 by some leading number theorists (including Louis Mordell, Harold DavenportCarl Ludwig Siegel).

To begin with, Minkowski's theorem establishesrelation between symmetric convex setsinteger points; we might as well say, between any latticeany Banach space normn dimensions. The topic therefore belongs properly tosortaffine geometry simplification oftheoryquadratic forms (Hilbert space normsrelationlattices). To relaxconvexity technique innon-trivial way may be technically difficult.

The theoretical foundations can be considered as dealing withspacelatticesn dimensions, whicha prioricoset space GL_n(R)/GL_n(Z). This isn't very easydealdirectly (itan example fortheory ratherarithmetic groups). One foundational resultMahler's compactness theorem describingrelatively compact subsets (the coset spacenon-compact, as can be seen already incase n = 2, where therecusps).

One can say thatgeometrynumbers takes on some ofwork that continued fractions do,diophantine approximation questionstwo or more dimensions - thereno straightforward generalisation.