1.2. LATTICES AND THE SPHERE PACKING PROBLEM 5

Sd

0

bij.

←→

Od(R)\GLd(R)

PQFs bases up to orth. trans.

space of lattices

Sd

0

/GLd(Z)

bij.

←→

Od(R)\ GLd(R)/GLd(Z)

PQFs up to arith. equiv. lattices up to isometries

Figure 3. Connections between lattices and PQFs.

Setting A =

√

DS ∈ GLd(R) we get a Cholesky decomposition

(1.10) Q =

AtA,

which is uniquely determined by the PQF Q up to orthogonal transformations.

That is, if (1.10) holds as well as Q = BtB for some B ∈ GLd(R), then A = OB

with O in

Od(R) = O ∈

Rd×d

:

OtO

= idd .

1.2. Lattices and the sphere packing problem

1.2.1. Lattices. Gauß [114] interpreted

Q[x] =

xtQx

=

xtAtAx

= Ax

2

as the squared (Euclidean) length of the vector Ax. So the (point) lattice

L =

AZd

= Za1 + ··· + Zad

(with column vectors a1, . . . , ad of A) came into the focus of number theorists.

We say L is generated by the basis (a1, . . . , ad), which we simply identify with the

matrix A ∈ GLd(R). Note that lattices are precisely the discrete subgroups of Rd

of rank d (see [124]).

The bases of the lattice L are of the form A = AU with U ∈ GLd(Z). Thus

they are in one-to-one correspondence with arithmetical equivalent forms Q =

(A

)tA

= U

tAtAU

= U

tQU

of Q. Concluding, the revealed connections are shown

in Figure 3.

The matrix A ∈ GLd(R) maps the standard lattice

Zd

to the lattice L =

AZd,

and the ellipsoid E(Q, λ(Q)) in (1.6) is transformed to AE(Q, λ(Q)) =

λ(Q)Bd,

where

Bd

= {x ∈

Rd

: x ≤ 1}

denotes the (Euclidean) unit ball in Rd with respect to the Euclidean norm x =

√

xtx. The arithmetical minimum of Q has an interpretation as the squared length

of the shortest non-zero lattice vector.

The determinant of Q may be geometrically interpreted via the volume | det A|

of the parallelotope

(1.11) {λ1a1 + ··· + λdad : λ1, . . . , λd ∈ [0, 1)}.