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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.

##### MSC:
 52C22 Tilings in $$n$$ dimensions (aspects of discrete geometry)
##### Keywords:
linearly repetitive Delone set; substitution tilings
Full Text:
##### References:
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