Følner sequence

Remarks

Example: in $G = \Z$, take $F_n = \set{0, 1, \ldots, n-1}$. For $k \in \Z$ and $n \ge \abs{k}$, the sets $k + F_n$ and $F_n$ differ in exactly $2\abs{k}$ points. So the ratio is $2\abs{k}/n$, which tends to $0$, and $(F_n)$ is a Følner sequence. In the same way the cubes $\set{0, \ldots, n-1}^d$ form a Følner sequence in $\Z^d$.

Non-example: in $G = \Z$, the singletons $F_n = \set{n}$ do not form a Følner sequence. For $g = 1$, the sets $1 + F_n = \set{n+1}$ and $F_n$ are disjoint, so the ratio is $2$ for every $n$.

A countable group has a Følner sequence if and only if it satisfies the Følner condition. For a locally compact group $G$ with left Haar measure $m$, one uses compact sets $F_n$ with $m(F_n) > 0$. The condition is then $m(gF_n \triangle F_n)/m(F_n) \to 0$ for every $g \in G$. Often the convergence is asked to be uniform for $g$ in compact sets.