The family of bi-Lipschitz classes of Delone sets in Euclidean space has the cardinality of the continuum (Q741166)

For a metric space \(M\), we call a set \(A \subset M\) a Delone set if there exist positive integers \(R\) and \(r\) such that (1) \(d_M(x,y) \geq r\) for any \(x,y \in A\), and (2) \(\displaystyle{\cup_{x \in A}B_R(x)=M}\). If there is a constant \(\lambda \geq 1\) and a bijection \(F:A \rightarrow B\) for two Delone sets \(A \subset M_1\) and \(B \subset M_2\), satisfying \(\frac{1}{\lambda}d_{M_1}(x,y) \leq d_{M_2}(F(x),F(y)) \leq \lambda d_{M_1}(x,y)\) for all \(x,y \in A\), we say \(A\) and \(B\) are bi-Lipschitz equivalent. In this paper, the author proves that the family of bi-Lipschitz equivalence classes of Delone sets in \(\mathbb{E}^d(d \geq 2)\) has the cardinalty of the continuum.