{"paper":{"title":"On the Origin of Crystallinity: a Lower Bound for the Regularity Radius of Delone Sets","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.MG","authors_text":"Alexey Garber, Egon Schulte, Igor A. Baburin, Mikhail Bouniaev, Nikolay Dolbilin, Nikolay Yu. Erokhovets, Sergey V. Krivovichev","submitted_at":"2018-04-13T17:05:56Z","abstract_excerpt":"The local theory of regular or multi-regular systems aims at finding sufficient local conditions for a Delone set $X$ to be a regular or multi-regular system. One of the main goals is to estimate the regularity radius $\\hat{\\rho}_d$ for Delone sets $X$ in terms of the radius $R$ of the largest \"empty ball\" for $X$.\n  The present paper establishes the lower bound $\\hat{\\rho_d}\\geq 2dR$ for all $d$, which is linear in $d$. The best previously known lower bound had been $\\hat{\\rho}_d\\geq 4R$ for $d\\geq 2$. The proof of the new lower bound is accomplished through explicit constructions of Delone s"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1804.05035","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"}