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I work in probability and I am looking for an important example of action of an amenable countable group in other areas of math for which the (pointwise) ergodic theorem is actually quite important. ...

I prefer to proceed with a concrete example if I may. I appreciate that the answer might well be better explained with group theory, geometry and/or notions from probability theory, which I welcome. ...

Definition: Let $h$ be a polynomial in $n$ variables, then : $\gamma(h,r,R):=\{ v \in \mathbb{Z}^{n} : \vert h(v) \vert \leq r, \Vert v \Vert < R \}$ Let $\omega : \mathbb{Z}^{n} \to \{ 0 , 1\}$...

This question is related to the one in https://mathoverflow.net/questions/65322/the-structure-of-certain-maximal-sets-of-means-into-amenable-semigroups. I open a different topic because they can be ...

Let $(S,\cdot)$ be a semigroup and $W\subseteq S$ be a subset. Let me call $W$ "tile" if the following property is satisfied: there exist $s_1,...s_k\in S$ such that the sets $s_i\cdot W$ are pairwise ...

The Cohen-Lenstra measure on the set of abelian p-groups assigns $\mathbb{P}(G) = \prod_{i \geq 1} \left( 1 - \frac{1}{p^i}\right) \cdot |\mathrm{Aut}(G)|^{-1} $. Apparently, this is equivalent to ...