Appendix B - Geometry of numbers

Summary

Geometry of numbers turns out to be a very useful tool in Diophantine approximation. For instance, it allows us to construct non-zero integer polynomials taking small values at prescribed points. In the course of the book, we applied several times the ‘first Theorem of Minkowski’ and the ‘second Theorem of Minkowski’, which are Theorems B.2 and B.3 below, respectively. We give a full proof of Theorem B.2, but not of Theorem B.3, which is much deeper. Throughout this Appendix. n denotes a positive integer. A set C in ℝⁿ having inner points and contained in the closure of its open kernel is called a body (or a domain).

theorem B.1. Let C be a bounded convex body in ℝⁿ, symmetric about the origin and of volume vol(C). If vol(C) > 2ⁿor if vol(C) = 2ⁿand C is compact, then C contains a point with integer coordinates, other than the origin.

proof. This proof is due to Mordell [429]. By classical arguments from elementary topology, it is enough to treat the case where vol(C) > 2ⁿ. For any positive integer m, denote by C_m the set of points of C having rational coordinates with denominator m. As m tends to infinity, the cardinality of C_m becomes equivalent to vol(C)mⁿ, and is thus strictly larger than (2m)ⁿ when m is large enough.