{"paper":{"title":"Linearly repetitive Delone sets are rectifiable","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":["math-ph","math.MG","math.MP"],"primary_cat":"math.DS","authors_text":"D. Coronel, J. Aliste-Prieto, J.-M. Gambaudo","submitted_at":"2011-03-28T17:27:41Z","abstract_excerpt":"In this paper we prove that, for any integer $d>0$, every linearly repetitive Delone set in the Euclidean $d$-space $\\RR^d$ is equivalent, up to a bi-Lipschitz homeomorphism, to the integer lattice $\\ZZ^d$. In the particular case when the Delone set $X$ in $\\RR^d$ comes from a primitive substitution tiling of $\\RR^d$, we give a condition on the eigenvalues of the substitution matrix which implies the existence of a homeomorphism with bounded displacement from $X$ to the lattice lattice $\\lambda\\ZZ^d$ for some positive $\\lambda$. This condition includes primitive Pisot substitution tilings but "},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1103.5423","kind":"arxiv","version":3},"verdict":{"id":null,"model_set":{},"created_at":null,"strongest_claim":"","one_line_summary":"","pipeline_version":null,"weakest_assumption":"","pith_extraction_headline":""},"references":{"count":0,"sample":[],"resolved_work":0,"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57","internal_anchors":0},"formal_canon":{"evidence_count":0,"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"author_claims":{"count":0,"strong_count":0,"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"builder_version":"pith-number-builder-2026-05-17-v1"}