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Linearly repetitive Delone sets are rectifiable. (English) Zbl 1288.52011
Summary: We show that every linearly repetitive Delone set in the Euclidean \(d\)-space \(\mathbb R^d\), with \(d\geqslant 2\), is equivalent, up to a bi-Lipschitz homeomorphism, to the integer lattice \(\mathbb Z^d\). In the particular case when the Delone set \(X\) in \(\mathbb R^d\) comes from a primitive substitution tiling of \(\mathbb R^d\), we give a condition on the eigenvalues of the substitution matrix which ensures the existence of a homeomorphism with bounded displacement from \(X\) to the lattice \(\beta\mathbb Z^d\) for some positive \(\beta\). This condition includes primitive Pisot substitution tilings but also concerns a much broader set of substitution tilings.

52C22 Tilings in \(n\) dimensions (aspects of discrete geometry)
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