{"paper":{"title":"A metric characterisation of repulsive tilings","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":["math.CO","math.OA"],"primary_cat":"math.MG","authors_text":"J. Savinien","submitted_at":"2014-10-27T14:25:18Z","abstract_excerpt":"A tiling of $\\mathbb{R}^d$ is repulsive if no $r$-patch can repeat arbitrarily close to itself, relative to $r$. This is a characteristic property of aperiodic order, for a non repulsive tiling has arbitrarily large local periodic patterns.\n  We consider an aperiodic, repetitive tiling $T$ of $\\mathbb{R}^d$, with finite local complexity. From a spectral triple built on the discrete hull $\\Xi$ of $T$, and its Connes distance, we derive two metrics $d_{sup}$ and $d_{inf}$ on $\\Xi$. We show that $T$ is repulsive if and only if $d_{sup}$ and $d_{inf}$ are Lipschitz equivalent. This generalises pre"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1410.7251","kind":"arxiv","version":1},"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"}