Unit packings, upper density, the sphere-packing and lattice-packing constants

This file introduces five definitions, all parametrised by a natural number $d \in \{0,1,2,\dots\}$. Throughout, $\mathbb{R}^d$ denotes the Euclidean space of real $d$-tuples $x=(x_0,\dots,x_{d-1})$ with the Euclidean ($\ell^2$) distance $\operatorname{dist}(x,y)=\sqrt{\sum_i (x_i-y_i)^2}$; for $d=0$ this space consists of the single point $0$. The measure $\operatorname{vol}$ on $\mathbb{R}^d$ is the default volume of a finite-dimensional real inner-product space: the translation-invariant (Haar) measure normalised so that the parallelepiped spanned by an orthonormal basis has measure $1$, i.e. standard Lebesgue measure; in dimension $0$ it gives the one-point space total mass $1$. Values of $\operatorname{vol}$ lie in the extended non-negative reals $[0,\infty]$, and all suprema, quotients and limsups below are taken in $[0,\infty]$.

Point sets. A point set in dimension $d$ is an arbitrary subset $C\subseteq\mathbb{R}^d$. No measurability, discreteness, countability or finiteness is required.

Unit packing. A point set $C\subseteq\mathbb{R}^d$ is a unit packing iff

The inequality is non-strict (distinct centres at distance exactly $2$ are allowed). The empty set and every singleton are unit packings, since the condition is then vacuous; for $d=0$ every subset of the one-point space is a unit packing.

Upper density. For a point set $C\subseteq\mathbb{R}^d$ its upper density $\overline{\delta}(C)\in[0,\infty]$ is

where $R$ ranges over all real numbers, $B(x,r)=\{y\in\mathbb{R}^d:\operatorname{dist}(y,x)<r\}$ is the open ball, and $0$ is the origin. The following conventions are part of the formula:

• The quotient is division in $[0,\infty]$, $a/b=a\cdot b^{-1}$ with $0^{-1}=\infty$ and $\infty^{-1}=0$. Hence $a/0=\infty$ for $a\neq0$, $0/0=0$, and $a/\infty=0$ for every $a$ (including $a=\infty$).
• The limsup is the limsup in the complete lattice $[0,\infty]$ along $R\to+\infty$: the infimum of all $a\in[0,\infty]$ such that the ratio is $\le a$ for all sufficiently large $R$; equivalently $\inf_{R_0\in\mathbb{R}}\ \sup_{R\ge R_0}(\text{ratio at }R)$. It always exists (possibly $\infty$) and ignores the values of the ratio for small $R$.
• For $R\le0$, $B(0,R)=\emptyset$, so numerator and denominator are both $0$ and the ratio is $0/0=0$; this has no effect on the limsup.
• $\operatorname{vol}$ is applied as an outer measure; no measurability of $C$ is assumed (the union of open balls is open in any case).
• For $d\ge1$ and $R>0$ the denominator $\operatorname{vol}(B(0,R))$ is positive and finite and the numerator is at most the denominator, so the ratio lies in $[0,1]$ and $\overline{\delta}(C)\in[0,1]$. If $C=\emptyset$ the union is empty and $\overline{\delta}(\emptyset)=0$.
• For $d=0$: $B(0,R)=\{0\}$ for every $R>0$ and $\operatorname{vol}(\{0\})=1$, so $\overline{\delta}(\{0\})=1$ and $\overline{\delta}(\emptyset)=0$.

Sphere-packing constant. For $d\in\mathbb{N}$,

Formally this is an indexed supremum over all point sets $C$ and then over all proofs that $C$ is a unit packing; when $C$ is not a unit packing the inner supremum has an empty index set and contributes the bottom element $0$, so the value is exactly the supremum of $\overline{\delta}$ over unit packings. The index set is never empty, since $\emptyset$ is a unit packing. There is no requirement that $C$ be discrete, countable, measurable, non-empty or of any particular structure. Edge case $d=0$: the unit packings are $\emptyset$ and $\{0\}$, so $\Delta_0=\max(0,1)=1$.

Lattice packing constant. For $d\in\mathbb{N}$,

Here $L$ ranges over all additive subgroups of $\mathbb{R}^d$ (subsets containing $0$ and closed under addition and negation; nothing else is assumed, in particular no discreteness, closedness, finite generation or $\mathbb{Z}$-basis). "$L$ is a unit packing" means that the underlying set of $L$ satisfies the unit-packing condition above, i.e. $2\le\operatorname{dist}(x,y)$ for all distinct $x,y\in L$; $\overline{\delta}(L)$ is the upper density of that underlying set. $\operatorname{span}_{\mathbb{R}}(L)$ is the smallest real linear subspace of $\mathbb{R}^d$ containing the set $L$, and it is required to equal the whole space. As before the supremum is formally over $L$ and over proofs of the two side conditions, with each failed condition contributing $0$, so $\Delta^L_d$ is the supremum of $\overline{\delta}(L)$ over the subgroups satisfying both conditions, and it would equal $0$ if no subgroup satisfied both. Edge case $d=0$: the only subgroup is $\{0\}$; it is a unit packing, its span is the whole (trivial) space, and $\overline{\delta}(\{0\})=1$, so $\Delta^L_0=1$.