Lattice Packing

Let \(V\) be a real vector space of \(d\) dimensions with a given inner product \(\langle \ , \ \rangle\). When we say \(\Lambda\) is a lattice in \(V\), we mean that \(\Lambda\) is a discrete closed subgroup \(\Lambda \subseteq V\) such that \(V / \Lambda\) has a finite volume from the induced measure. The volume of \(V / \Lambda\) is also called the covolume of \(\Lambda\).

Given a lattice \(\Lambda\), take a real number \(r > 0\) and consider the collection of open balls \(\{ B_{r}\left( v \right)\}_{v \in \Lambda}\). Such a collection of balls are said to be a sphere packing in \(V\) if no non-trivial pairs of these balls intersect. That is, for any \(v_1, v_2 \in \Lambda\), \(B_{r}(v_1) \cap B_{r}(v_2) \neq \emptyset \Rightarrow v_1 = v_2\). Such an arrangement is called a lattice sphere packing, or simply lattice packing.

Covering efficiency

The following is the covering efficiency of a lattice. \[\begin{align} \lim_{R \rightarrow \infty} \frac{\mu\left( B_{R}(0) \cap \left( \bigsqcup_{v \in \Lambda} B_{r}(v) \right) \right)}{\mu\left( B_{R}(0) \right)} , \end{align}\] where \(\mu\) is the Lebesgue measure on \(V\) induced by the inner product. This limit always exists for any given lattice packing and is equal to \[\begin{align} \frac{\mu( B_{r}(0) )}{ \mu(V / \Lambda)}. \end{align}\] This is always a real number in the open interval \([0,1]\). See sphere packing constant