geometry of numbers

A typical approach in the geometry of numbers is to view a ring of algebraic integers as a lattice in #92;textstyle#92;mathbb#123;R#125;ⁿ; the study of these lattices provides fundamental information on algebraic numbers.

1969, C. G. Lekkerkerker, Geometry of Numbers, Wolters-Noordoff, North-Holland, page 1, The geometry of numbers to which this book is devoted deals with arbitrary bodies and arbitrary lattices in the n-dimensional euclidean space. Its aim is to study various quantities describing the behaviour of a body with respect to a lattice.

2006, Enrico Bombieri, Walter Gubler, Heights in Diophantine Geometry, Cambridge University Press, page 181, The easy proof is obtained applying the pigeon-hole principle to α₁x_1+…+αₙx_n(mod 1)|x_i=0,…N, or by geometry of numbers by applying Minkowski's first theorem in C.2.19 to the symmetric convex body of volume 2ⁿ⁺¹ given by |X_0+α₁X_1+…αₙX_n|<N⁻ⁿ,|X_i|<N,i=1,…N.