Density
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Definition 1 For \(A\subseteq\Group\) and Følner sequence \(\Folner=(\Folner[N])_{n=1}^\infty\), we write \[\begin{align*} \UpperDensity{A}&=\limsup_{N\rightarrow\infty}\frac{\CountingMeasure{A\cap\Folner[N]}}{\CountingMeasure{\Folner[N]}} \\\LowerDensity{A}&=\liminf_{N\rightarrow\infty}\frac{\CountingMeasure{A\cap\Folner[N]}}{\CountingMeasure{\Folner[N]}} \end{align*}\] to be the upper and lower densities of \(A\) with respect to \(\Folner\), respectively. If these agree, then we can write \[ \Density{A}=\lim_{N\rightarrow\infty}\frac{\CountingMeasure{A\cap\Folner[N]}}{\CountingMeasure{\Folner[N]}} \] to be the density of \(A\) with respect to \(\Folner\). We also define the upper Banach density of \(A\) by \[ \UpperBanachDensity{A}=\sup\{d_{\Folner}(A):\text{for Følner sequences }\Folner\text{ where }d_{\Folner}(A)\text{ exists}\}. \]
Proposition 1 (Monotonicity) Let \(A,B\subseteq\Monoid\) be subsets. For \(A\subseteq B\), we have\[\UpperDensity{A}\leq\UpperDensity{B}, \]and\[\LowerDensity{A}\leq\LowerDensity{B}. \]
Proposition 2 (Translation Invariance) Let \(B\subseteq\Monoid\) be a subset. For all \(\MonoidElement\in\Monoid\), we have that \[\UpperDensity{B}=\UpperDensity[\MonoidOperation{\MonoidElement}{\Folner}]{B}=\UpperDensity{\MonoidOperation{\MonoidElement}{B}}\] and \[\LowerDensity{B}=\LowerDensity[\MonoidOperation{\MonoidElement}{\Folner}]{B}=\LowerDensity{\MonoidOperation{\MonoidElement}{B}}.\]
Lemma 1 (Upper/Lower Density Pairwise Additive) Let \(A,B\subseteq\Monoid\) be subsets such that \(\Density{A}\) and \(\Density{B}\) both exist, then we have \[\begin{align*} \UpperDensity{A\cup B}+\LowerDensity{A\cap B}&=\Density{A}+\Density{B}, \\\LowerDensity{A\cup B}+\UpperDensity{A\cap B}&=\Density{A}+\Density{B}. \end{align*}\] Further, we have \(\Density{A\cap B}\) exists if and only if \(\Density{A\cup B}\) exists.