Exercise 27, § 1

Let $G$ be a group operating on a set $X$. A subset $P$ (resp. $C$) of $X$ is called a $G$-packing (resp. $G$-covering) if, for every $s \neq e$ in $G$, one has $sP \cap P = \varnothing$ (resp. if $X = \bigcup_{s \in G} sC$).

A $G$-tiling is called a subset $P$ which is at the same time a $G$-packing and a $G$-covering.

a) Suppose that $X$ is locally compact, that $G$ is countable, operates continuously in $X$, and that there exists a non-zero positive measure $\mu$ on $X$ invariant under $G$. Let $P$ and $C$ be a $G$-packing and a $G$-covering which are $\mu$-integrable. Now prove that $\mu(C) \geq \mu(P)$. (Remark that $\mu(C) \geq \sum_{s \in G} \mu(C \cap sP)$).

b) Suppose in addition that there exists on $X$ a uniform structure compatible with the topology of $X$ and admitting a fundamental system $\mathscr{G}$ of open entourages invariant under $G$. Let $\Delta(G)$ denote on the other hand the greatest lower bound of the numbers $\mu(C)$ for all integrable $G$-coverings C of $G$. Let V be an entourage belonging to $\mathscr{G}$, and let $a \in X$ be such that $\mu(V(a)) > \Delta(G)$; now prove that there exists $s \in G$ such that $s \neq e$ and $(a, sa) \in V^{-1} \circ V$.

c) Suppose that $X$ is a locally compact group, $\mu$ a left Haar measure, $G$ a countable subgroup of $X$ operating by left translation. With the same definition of $\Delta(G)$ as in b), now prove that, if A is an integrable subset of X such that $\mu(A) > \Delta(G)$, there exists $s \in G \cap AA^{-1}$ such that $s \neq e$. Particular case where $X = \mathbf{R}^n$, and where G is a discrete subgroup of rank n in $\mathbf{R}^n$: $\Delta(G)$ is then equal to the absolute value of the determinant of a basis of G over $\mathbf{Z}$ with respect to the canonical basis of $\mathbf{R}^n$; if A is a symmetric convex field in $\mathbf{R}^n$ such that $\mu(A) \geq 2^n \Delta(G)$, now prove that there exists a point of $A \cap G$ distinct from 0 (« Minkowski theorem »).

d) The hypotheses being those of a), let f be a function $\geq 0$ and $\mu$-integrable in X. Now prove that there exist two points $a, b$ of X such that $\mu(C) \sum_{s \in G} f(sa) \geq \int_X f(x) d\mu(x)$ and $\mu(P) \sum_{s \in G} f(sb) \leq \int_X f(x) d\mu(x)$. (Observe that if g is an integrable function $\geq 0$, E an $\mu$-integrable set in X, there exists $c \in E$ such that $\int_E g(x) d\mu(x) \leq g(c) \mu(E)$, and $c' \in E$ such that $\int_E g(x) d\mu(x) \geq g(c') \mu(E)$).