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Question: How does property A relate to sequence of cubes?
Let \(X\) be a uniformly discrete metric space. We say that \(X\) has
property A if for every \(\epsilon > 0\) and \(R>0\) there exists a
collection of finite subest \(\{ A_{x} \}_{x \in X}, A_{x} \subseteq X
\times \mathbb{N}\) for every \(x \in X\), and a constant \(S > 0\) such
that
- \(\frac{\#(A_{x}\Delta A_{y})}{\#(A_{x} \cap A_{y})} \leq \epsilon \text{ when } d(x,y) \leq R, \text{ and}\)
- \(A_{x} \subseteq B(x,S) \times \mathbb{N}\).
For a coarse disjoint untion, the conditions can be local except that \(S\) must be independent of the cube that contains \(x,y\).
So \(A_{x}\) is a set of tuples which are a binary string and a natural number.