# Ornstein-Weiss lemma

Let $G$ be a group. For a fixed $K\subseteq G$, define the $K$-boundary of $U\subseteq G$ as

 $\partial_{K}U=\left\{g\in G\mid Kg\cap U,Kg\cap(G\setminus U)\neq\emptyset% \right\}\;.$ (1)

Let $\mathcal{PF}(G)$ be the set of finite subsets of $G$. Call a Følner net for $G$ a net $\mathcal{X}=\{X_{i}\}_{i\in\mathcal{I}}\subseteq\mathcal{PF}(G),$ $\mathcal{I}$ being a directed set, such that for every finite $K\subseteq G$,

 $\lim_{i\in\mathcal{I}}\frac{|\partial_{K}X_{i}|}{|X_{i}|}=0\;,$ (2)

where the limit is taken in the sense of directed sets. Recall that $G$ has a Følner net if and only if $G$ is amenable.

###### Theorem 1 (Ornstein-Weiss lemma)

Let $G$ be an amenable group and $F:\mathcal{PF}(G)\to\mathbb{R}$ a subadditive, right-invariant function, that is:

1. 1.

For any two finite subsets $U,V$ of $G$,

 $F(U\cup V)\leq F(U)+F(V)\;.$ (3)
2. 2.

For any $g\in G$ and finite $U\subseteq G$,

 $F(Ug)=F(U)\;.$ (4)

Then for any Følner net $\mathcal{X}=\left\{X_{i}\right\}_{i\in\mathcal{I}}$ on $G$, the limit

 $L=\lim_{i\in\mathcal{I}}\frac{F(X_{i})}{|X_{i}|}$ (5)

exists, and does not depend on the choice of $\mathcal{X}$.

The Ornstein-Weiss lemma allows to prove variants of Birkhoff’s ergodic theorem for actions of amenable groups, rather than only those generated by an invertible  , measure invariant map. Moreover, it shares several similarities with Fekete’s lemma on subadditive functions over the positive integers, although it is not a complete  counterpart. In fact, putting $X_{n}=\{1,\ldots,n\}$ determines a Følner sequence on $\mathbb{Z}$; however, if $f:\mathbb{N}\to[0,\infty)$ is subadditive, then $F(U)=f(|U|)$ is right-invariant, but not necessarily subadditive. (Counterexample: $f(n)=n\,\mathrm{mod}\,2$, $U=\{1,2\}$, $V=\{2,3\}$.)

## References

Title Ornstein-Weiss lemma OrnsteinWeissLemma 2013-03-22 19:20:24 2013-03-22 19:20:24 Ziosilvio (18733) Ziosilvio (18733) 5 Ziosilvio (18733) Theorem msc 43A07