{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,4,1]],"date-time":"2026-04-01T14:43:36Z","timestamp":1775054616643,"version":"3.50.1"},"reference-count":27,"publisher":"Springer Science and Business Media LLC","issue":"1","license":[{"start":{"date-parts":[[2024,6,22]],"date-time":"2024-06-22T00:00:00Z","timestamp":1719014400000},"content-version":"tdm","delay-in-days":0,"URL":"https:\/\/creativecommons.org\/licenses\/by\/4.0"},{"start":{"date-parts":[[2024,6,22]],"date-time":"2024-06-22T00:00:00Z","timestamp":1719014400000},"content-version":"vor","delay-in-days":0,"URL":"https:\/\/creativecommons.org\/licenses\/by\/4.0"}],"funder":[{"name":"Northeastern University USA"}],"content-domain":{"domain":["link.springer.com"],"crossmark-restriction":false},"short-container-title":["Discrete Comput Geom"],"published-print":{"date-parts":[[2025,7]]},"abstract":"Abstract<\/jats:title>\n Delone sets are discrete point sets X<\/jats:italic> in \n \n $${\\mathbb {R}}^d$$<\/jats:tex-math>\n \n \n \n R<\/mml:mi>\n <\/mml:mrow>\n d<\/mml:mi>\n <\/mml:msup>\n <\/mml:math>\n <\/jats:alternatives>\n <\/jats:inline-formula> characterized by parameters (r<\/jats:italic>,\u00a0R<\/jats:italic>), where (usually) 2r<\/jats:italic> is the smallest inter-point distance of X<\/jats:italic>, and R<\/jats:italic> is the radius of a largest \u201cempty ball\u201d that can be inserted into the interstices of X<\/jats:italic>. The regularity radius \n \n $${\\hat{\\rho }}_d$$<\/jats:tex-math>\n \n \n \n \u03c1<\/mml:mi>\n ^<\/mml:mo>\n <\/mml:mover>\n d<\/mml:mi>\n <\/mml:msub>\n <\/mml:math>\n <\/jats:alternatives>\n <\/jats:inline-formula> is defined as the smallest positive number \n \n $$\\rho $$<\/jats:tex-math>\n \n \u03c1<\/mml:mi>\n <\/mml:math>\n <\/jats:alternatives>\n <\/jats:inline-formula> such that each Delone set with congruent clusters of radius \n \n $$\\rho $$<\/jats:tex-math>\n \n \u03c1<\/mml:mi>\n <\/mml:math>\n <\/jats:alternatives>\n <\/jats:inline-formula> is a regular system, that is, a point orbit under a crystallographic group. We discuss two conjectures on the growth behavior of the regularity radius. Our \u201cWeak Conjecture\u201d states that \n \n $${\\hat{\\rho }}_{d}={\\textrm{O}(d^2\\log _2 d)}R$$<\/jats:tex-math>\n \n \n \n \n \u03c1<\/mml:mi>\n ^<\/mml:mo>\n <\/mml:mover>\n d<\/mml:mi>\n <\/mml:msub>\n =<\/mml:mo>\n \n O<\/mml:mtext>\n (<\/mml:mo>\n \n d<\/mml:mi>\n 2<\/mml:mn>\n <\/mml:msup>\n \n log<\/mml:mo>\n 2<\/mml:mn>\n <\/mml:msub>\n d<\/mml:mi>\n )<\/mml:mo>\n <\/mml:mrow>\n R<\/mml:mi>\n <\/mml:mrow>\n <\/mml:math>\n <\/jats:alternatives>\n <\/jats:inline-formula> as \n \n $$d\\rightarrow \\infty $$<\/jats:tex-math>\n \n \n d<\/mml:mi>\n \u2192<\/mml:mo>\n \u221e<\/mml:mi>\n <\/mml:mrow>\n <\/mml:math>\n <\/jats:alternatives>\n <\/jats:inline-formula>, independent of\u00a0r<\/jats:italic>. This is verified in the paper for two important subfamilies of Delone sets: those with full-dimensional clusters of radius 2r<\/jats:italic> and those with full-dimensional sets of d<\/jats:italic>-reachable points. We also offer support for the plausibility of a \u201cStrong Conjecture\u201d, stating that \n \n $${\\hat{\\rho }}_{d}={\\textrm{O}(d\\log _2 d)}R$$<\/jats:tex-math>\n \n \n \n \n \u03c1<\/mml:mi>\n ^<\/mml:mo>\n <\/mml:mover>\n d<\/mml:mi>\n <\/mml:msub>\n =<\/mml:mo>\n \n O<\/mml:mtext>\n (<\/mml:mo>\n d<\/mml:mi>\n \n log<\/mml:mo>\n 2<\/mml:mn>\n <\/mml:msub>\n d<\/mml:mi>\n )<\/mml:mo>\n <\/mml:mrow>\n R<\/mml:mi>\n <\/mml:mrow>\n <\/mml:math>\n <\/jats:alternatives>\n <\/jats:inline-formula> as \n \n $$d\\rightarrow \\infty $$<\/jats:tex-math>\n \n \n d<\/mml:mi>\n \u2192<\/mml:mo>\n \u221e<\/mml:mi>\n <\/mml:mrow>\n <\/mml:math>\n <\/jats:alternatives>\n <\/jats:inline-formula>, independent of r<\/jats:italic>.<\/jats:p>","DOI":"10.1007\/s00454-024-00666-6","type":"journal-article","created":{"date-parts":[[2024,6,22]],"date-time":"2024-06-22T20:10:26Z","timestamp":1719087026000},"page":"78-94","update-policy":"https:\/\/doi.org\/10.1007\/springer_crossmark_policy","source":"Crossref","is-referenced-by-count":1,"title":["Bounds for the Regularity Radius of Delone Sets"],"prefix":"10.1007","volume":"74","author":[{"given":"Nikolay","family":"Dolbilin","sequence":"first","affiliation":[],"role":[{"role":"author","vocabulary":"crossref"}]},{"given":"Alexey","family":"Garber","sequence":"additional","affiliation":[],"role":[{"role":"author","vocabulary":"crossref"}]},{"ORCID":"https:\/\/orcid.org\/0000-0001-9725-3589","authenticated-orcid":false,"given":"Egon","family":"Schulte","sequence":"additional","affiliation":[],"role":[{"role":"author","vocabulary":"crossref"}]},{"given":"Marjorie","family":"Senechal","sequence":"additional","affiliation":[],"role":[{"role":"author","vocabulary":"crossref"}]}],"member":"297","published-online":{"date-parts":[[2024,6,22]]},"reference":[{"key":"666_CR1","series-title":"Lecture Notes in Computer Science","doi-asserted-by":"publisher","first-page":"229","DOI":"10.1007\/978-3-030-76657-3_16","volume-title":"Discrete Geometry and Mathematical Morphology, DGMM 2021","author":"O Anosova","year":"2022","unstructured":"Anosova, O., Kurlin, V.: An isometry classification of periodic point sets. 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