Olds C.D., Lax A., Davidoff G. The Geometry of Numbers

American Mathematical Society, 1962 - 193 p.
The geometry of numbers is a branch of number theory that originated
with the publication of Minkowski’s seminal work in 1896 and ultimately
established itself as an important field of study in its own right. Its focus
is the conversion of arithmetic questions into geometric contexts, with the
result that certain difficult questions in arithmetic can be answered geometrically
by reasonably obvious constructions. One fundamental problem
is to define conditions under which a given region contains a lattice point,
that is, a point (p, q) with integer coordinates; another is to ask how to
characterize regions, <, for which any point can be moved by an integer
translation to coincide with another point in <, that is, to ask what conditions
must hold in < if for any point P there is to be a point Q such that
P − Q is a lattice point.