Introduction

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."

Definitions

A lattice LL is a subgroup of Rn\mathbb{R}^ngenerated by bib_i, i.e.

L=i=1dZbi={i=1daibiaiZ}L=\sum_{i=1}^d\mathbb{Z} b_i = \left\{\left. \sum_{i=1}^d a_i b_i \right | a_i \in \mathbb{Z} \right\}

where bib_i are linearly independent vectors. Collectively, {bi}i=1d\left\{b_i\right\}_{i=1}^d form a basis ofLL.

We say a set of vectors viv_iare linearly independent if the only solution to the equation iaibi=0\sum_{i} a_i b_i = 0 is when all aia_iare zero.

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, denoteddimL\dim L, is dd. A lattice is complete if d=nd=n. Note that we can always choose a subspace of Rn\mathbb R^nsuch that the lattice is complete, namely the subspace generated by bib_i.

The region

Φ={i=1dxibi0xi<1}\Phi=\left\{\left.\sum_{i=1}^dx_ib_i\right|0\leq x_i<1\right\}

is known as the fundamental mesh.

In the image above, we see the points of a lattice in R2\mathbb R^2. 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 (mmdimensional) 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:

LetB\mathcal Bbe ad×nd\times nmatrix whose rows are given by the basis vectors. Then the volume of a fundamental mesh is given by

vol(L)=det(BBT)\text{vol}(L)=\sqrt{\left|\det\left(\mathcal B\mathcal B^T\right)\right|}

A subset XXof Rn\mathbb R^nis known as centrally symmetric if xXx\in Ximplies xX-x\in X. It is convex if for any x,yXx,y\in X, the line joining x,yx,y is contained in XX, i.e. {tx+(1t)y0t1}X\left\{tx+(1-t)y|0\leq t\leq1\right\}\subset X. Finally we can introduce the most important theorem about lattices, the Minkowski's Lattice Point Theorem:

Let LLbe a complete lattice of dimensionnn and XXbe a centrally symmetric convex set. Suppose

vol(X)>2nvol(L)\text{vol}(X)>2^n\text{vol}(L)

Then XXcontains at least one nonzero point of LL . This result is primarily used to prove the existence of lattice vectors.

Throughout this section, v=ivi2\left\lVert v\right\rVert=\sqrt{\sum_iv_i^2} denotes the 2\ell_2norm and a,b=iaibi\langle a,b\rangle=\sum_ia_ib_i 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

12X={12xxX}\frac12X=\left\{\frac12x|x\in X\right\}

We have vol(12X)>vol(L)\text{vol}\left(\frac12 X\right)>\text{vol}(L), hence the inclusion 12XRn/L\frac12X\to\mathbb R^n/Lcannot be injective, thus we can find some x1=x2+x_1=x_2+\ell, x1,x212X,L,x1x2x_1,x_2\in\frac12 X,\ell\in L,x_1\neq x_2. Hence x1x2Lx_1-x_2\in Lis a nontrivial lattice point.

Exercises

1) Let LLbe the lattice generated by B=(198187)\mathcal B=\begin{pmatrix}-1&9&8\\1&-8&-7\end{pmatrix}(take the rows as basis vectors).

  • Compute the volume of this lattice

  • Show that B=(101011)\mathcal B'=\begin{pmatrix}1&0&1\\0&1&1\end{pmatrix}generates the same lattice

  • Show that each row in C=(101022)\mathcal C=\begin{pmatrix}1&0&1\\0&2&2\end{pmatrix}is in the lattice butC\mathcal Cdoes not generate the lattice. This is one key difference from the case of linear algebra (over fields).

2) LetB,B\mathcal B,\mathcal B'be d×nd\times nmatrices whose row vectors are basis for lattices L,LL,L'. Both lattices are the same iff there exists some UGLd(Z)U\in\text{GL}_d(\mathbb Z) such that B=UB\mathcal B'=U\mathcal B. Find UUfor problem 1. Note that GLd(Z)\text{GL}_d(\mathbb Z)is the group of invertible matrices with integer coefficients, meaning UUand U1U^{-1}have integer coefficients.

3) Show that the condition in Minkowski's lattice point theorem is strict, i.e. for any complete latticeLLof dimension nn, we can find some centrally symmetric convex setXXwithvol(X)=2nvol(L)\text{vol}(X)=2^n\text{vol}(L)but the only lattice point inXXis the origin.

4*) Letvvbe the shortest nonzero vector for some lattice LLwith dimensionnn. Show that

v2πΓ(n2+1)1nvol(L)1n\left\lVert v\right\rVert\leq\frac2{\sqrt\pi}\Gamma\left(\frac n2+1\right)^{\frac1n}\text{vol}(L)^\frac1n