Følner sequence
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1 Følner Sequences
Definition 1 We define a right-Følner sequence in \(\Group\) as a sequence \(\Folner =(\Folner[N])_{N\in\mathbb{N}}\) of finite subsets of \(\Group\) satisfying \[\lim_{N\rightarrow\infty}\frac{\CountingMeasure{\GroupOperation{(\GroupOperation{\Folner[N]}{\Inverse{\GroupElement}})}{\Folner[N]}}}{\CountingMeasure{\Folner[N]}}=1,\] for all \(\GroupElement\in\Group\).
Definition 2 Similarly, we define a left-Følner sequence in \(\Group\) as a sequence \(\Folner =(\Folner[N])_{N\in\mathbb{N}}\) of finite subsets of \(\Group\) satisfying \[\lim_{N\rightarrow\infty}\frac{\CountingMeasure{(\Inverse{\GroupElement}\cdot\Folner[N])\cap\Folner[N]}}{\CountingMeasure{\Folner[N]}}=1,\] for all \(\GroupElement\in\Group\).
Definition 3 We call a sequence in \(\Group\) a Følner sequence if it is both a left and right Følner sequence.
1.1 Alternative definitions for Monoids
Definition 4 Let \(\Monoid\) be a countably-infinite left-cancellative monoid with discrete topology. We define a left-Følner sequence in \(\Monoid\) as a sequence of finite subsets \(\Folner =(\Folner[N])_{N\in\mathbb{N}}\) satisfying \[\lim_{N\rightarrow\infty}\frac{\CountingMeasure{\MonoidOperation{(\MonoidOperation{\MonoidElement}{\Folner[N]})}{\Folner[N]}}}{\CountingMeasure{\Folner[N]}}=1\] for all \(g\in M\).
Definition 5 Similarly, for a countably-infinite right-cancellative monoid with discrete topology \(\Monoid\), we define a right-Følner sequence in \(\Monoid\) as a sequence of finite subsets \(\Folner =(\Folner[N])_{N\in\mathbb{N}}\) satisfying \[\lim_{N\rightarrow\infty}\frac{\CountingMeasure{\MonoidOperation{(\MonoidOperation{\Folner[N]}{\MonoidElement})}{\Folner[N]}}}{\CountingMeasure{\Folner[N]}}=1\] for all \(g\in M\).
1.2 Equivalent definitions using Set Differences
Equivalent definitions can be constructed by using set differences instead of intersections.
For example, the equivalent definition of a left-Følner sequence, \(\Folner\), in \(\Monoid\) requires \[\lim_{N\rightarrow\infty}\frac{\CountingMeasure{(\MonoidOperation{\Folner[N]}{\MonoidElement})\triangle\Folner[N]}}{\CountingMeasure{\Folner[N]}}=0, \] to be satisfied for all \(\MonoidElement\in\Monoid\).
This alternative definition will be useful when looking at proving some of the properties of density.
2 Tempered Følner Sequences
Definition 6 (Lindenstrauss (2001), Definition 1.1) A sequence of sets \(\Folner=(\Folner[N])_{N\in\mathbb{N}}\) will be said to be tempered if, for some \(b>0\) and all \(n\in\mathbb{N}\),
\[ \CountingMeasure{\bigcup_{1\leq k<N}\Folner[k]^{-1}\Folner[N]}\leq b\CountingMeasure{\Folner[N]}. \tag{1}\]
is referred to as the Shulman condition.
Proposition 1 (Lindenstrauss (2001), Proposition 1.4)