geometry of numbers

Etymology
The field was initiated by German mathematician (1910, Geometrie der Zahlen).

Noun

 * 1)  The subbranch of number theory which applies techniques from geometry to the study of algebraic numbers.
 * 2) * 1969,, 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.
 * 1) * 2006, Enrico Bombieri, Walter Gubler, Heights in Diophantine Geometry,, page 181,
 * The easy proof is obtained applying the pigeon-hole principle to
 * $$\{\alpha_1 x_1 + \dots + \alpha_n x_n \pmod 1 \vert x_i = 0,\dots N \}$$,
 * or by geometry of numbers by applying Minkowski's first theorem in C.2.19 to the symmetric convex body of volume $$2^{n+1}$$ given by
 * $$\vert X_0 + \alpha_1 X_1 +\dots \alpha_n X_n\vert \le N^{-n}, \vert X_i\vert\le N, i=1,\dots N$$.
 * or by geometry of numbers by applying Minkowski's first theorem in C.2.19 to the symmetric convex body of volume $$2^{n+1}$$ given by
 * $$\vert X_0 + \alpha_1 X_1 +\dots \alpha_n X_n\vert \le N^{-n}, \vert X_i\vert\le N, i=1,\dots N$$.

Translations

 * German: Geometrie der Zahlen
 * Italian: teoria dei numeri geometrica