Une remarque \`a propos de l'\'equivalence bilipschitzienne entre des ensembles de Delone. (A remark concerning bi-Lipschitz equivalence between Delone sets.)

TL;DR

This paper demonstrates that linearly repetitive Delone sets in Euclidean space can be transformed into integer lattice points through bi-Lipschitz homeomorphisms, establishing a form of rectifiability.

Contribution

It proves that linearly repetitive Delone sets are bi-Lipschitz equivalent to integer lattices, advancing understanding of their geometric structure.

Findings

Linearly repetitive Delone sets are rectifiable via bi-Lipschitz homeomorphisms.

Such sets can be mapped to integer lattice points.

The result links Delone set structure to lattice geometry.

Abstract

Linearly repetitive Delone sets are shown to be rectifiable by a bi-Lipschitz homeomorphisms of the Euclidean space that sends the Delone set to the set of points with integer coordinates.