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)}.
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