- Geometry of numbers
In
number theory , the geometry of numbers is a topic and method arising from the work ofHermann Minkowski , on the relationship betweenconvex set s and lattices inn-dimensional space. It has frequently been used in an auxiliary role in proofs, particularly indiophantine approximation . The subject was given a great deal of attention in the period 1930-1960 by some leadingnumber theorist s (includingLouis Mordell ,Harold Davenport andCarl Ludwig Siegel ).Minkowski's theorem establishes a relation between symmetric convex sets and integer points; we might as well say, between any lattice and anyBanach space norm in n dimensions. The topic therefore belongs properly to a sort ofaffine geometry simplification of the theory ofquadratic form s (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 group s). One foundational result isMahler'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 "cusp s").One can say that the geometry of numbers takes on some of the work that
continued fraction s do, for diophantine approximation questions in two or more dimensions — there is no straightforward generalisation.References
*cite book
author = Siegel, Carl Ludwig
authorlink = Carl Ludwig Siegel
title = Lectures on the Geometry of Numbers
year = 1989
publisher = Springer-Verlag*cite book
author = Hancock, Harris
title = Development of the Minkowski Geometry of Numbers
year = 1939
publisher = Macmillan (Republished in 1964 by Dover.)
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