Lattices, also known as Minkowski's theory after Hermann Minkowski, or the geometry of numbers (deprecated!) allows the usage of geometrical tools (i.e. volumes) in number theory.
The intuitive notion of a lattice (perhaps surprisingly) matches its mathematical definition. For example, lattices are formed by
points on an infinite checkerboard;
centers of a hexagonal tessellation;
integers on the real number line.
The last example should hint at how we generalize this concept to arbitrary dimensions. In general, lattices consist of discrete points which appear at "regular intervals."
is a subgroup of
are linearly independent vectors. Collectively,
form a basis of
We say a set of vectors
are linearly independent if the only solution to the equation
is when all
Taking a step back, this definition should resemble that of a vector space, with one exception: scalars are integers! The discrete nature of lattices comes from this restriction.
Some more terminology from linear algebra will be useful. The dimension of a lattice, denoted
. A lattice is complete if
. Note that we can always choose a subspace of
such that the lattice is complete, namely the subspace generated by
is known as the fundamental mesh.
In the image above, we see the points of a lattice in
. The red vectors are one set of basis vectors and the shaded region is the corresponding fundamental mesh. The green vectors also form another set of basis vectors with its corresponding fundamental mesh. We see here that the basis vectors and fundamental mesh is not unique to a lattice.
Although the fundamental mesh is not unique, it turns out that the (
dimensional) volume of the fundamental mesh is constant for any given lattice. Hence we can define the volume of a lattice as the volume of a fundamental mesh. However this definition can be hard to handle hence we provide an equivalent definition via determinants:
matrix whose rows are given by the basis vectors. Then the volume of a fundamental mesh is given by
is known as centrally symmetric if
. It is convex if for any
, the line joining
is contained in
. Finally we can introduce the most important theorem about lattices, the Minkowski's Lattice Point Theorem:
be a complete lattice of dimension
be a centrally symmetric convex set. Suppose
contains at least one nonzero point of
. This result is primarily used to prove the existence of lattice vectors.
Throughout this section,
denotes the inner product.
Proof sketch of Minkowski's theorem
This proof is by some sort of a pigeonhole argument on volumes. Consider the set
, hence the inclusion
cannot be injective, thus we can find some
is a nontrivial lattice point.
be the lattice generated by
(take the rows as basis vectors).
Compute the volume of this lattice
generates the same lattice
Show that each row in
is in the lattice but
does not generate the lattice. This is one key difference from the case of linear algebra (over fields).
matrices whose row vectors are basis for lattices
. Both lattices are the same iff there exists some
for problem 1. Note that
is the group of invertible matrices with integer coefficients, meaning
have integer coefficients.
3) Show that the condition in Minkowski's lattice point theorem is strict, i.e. for any complete lattice