{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,4,22]],"date-time":"2026-04-22T05:57:38Z","timestamp":1776837458321,"version":"3.51.2"},"reference-count":27,"publisher":"American Mathematical Society (AMS)","issue":"345","license":[{"start":{"date-parts":[[2024,7,26]],"date-time":"2024-07-26T00:00:00Z","timestamp":1721952000000},"content-version":"am","delay-in-days":366,"URL":"https:\/\/www.ams.org\/publications\/copyright-and-permissions"}],"funder":[{"DOI":"10.13039\/501100006769","name":"Russian Science Foundation","doi-asserted-by":"publisher","award":["22-41-02028"],"award-info":[{"award-number":["22-41-02028"]}],"id":[{"id":"10.13039\/501100006769","id-type":"DOI","asserted-by":"publisher"}]},{"DOI":"10.13039\/501100006769","name":"Russian Science Foundation","doi-asserted-by":"publisher","award":["DMS-1802119"],"award-info":[{"award-number":["DMS-1802119"]}],"id":[{"id":"10.13039\/501100006769","id-type":"DOI","asserted-by":"publisher"}]},{"DOI":"10.13039\/100000001","name":"National Science Foundation","doi-asserted-by":"publisher","award":["22-41-02028"],"award-info":[{"award-number":["22-41-02028"]}],"id":[{"id":"10.13039\/100000001","id-type":"DOI","asserted-by":"publisher"}]},{"DOI":"10.13039\/100000001","name":"National Science Foundation","doi-asserted-by":"publisher","award":["DMS-1802119"],"award-info":[{"award-number":["DMS-1802119"]}],"id":[{"id":"10.13039\/100000001","id-type":"DOI","asserted-by":"publisher"}]}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["Math. Comp."],"abstract":"<p>In this paper we study crystallographic sphere packings and Kleinian sphere packings, introduced first by Kontorovich and Nakamura in 2017 and then studied further by Kapovich and Kontorovich in 2021. In particular, we solve the problem of existence of crystallographic sphere packings in certain higher dimensions posed by Kontorovich and Nakamura. In addition, we present a geometric doubling procedure allowing to obtain sphere packings from some Coxeter polyhedra without isolated roots, and study \u201cproperly integral\u201d packings (that is, ones which are integral but not superintegral). Our techniques rely extensively on computations with Lorentzian quadratic forms, their orthogonal groups, and associated higher\u2013dimensional hyperbolic polyhedra.<\/p>","DOI":"10.1090\/mcom\/3858","type":"journal-article","created":{"date-parts":[[2023,7,26]],"date-time":"2023-07-26T10:57:08Z","timestamp":1690369028000},"page":"505-521","source":"Crossref","is-referenced-by-count":3,"title":["Kleinian sphere packings, reflection groups, and arithmeticity"],"prefix":"10.1090","volume":"93","author":[{"given":"Nikolay","family":"Bogachev","sequence":"first","affiliation":[],"role":[{"role":"author","vocabulary":"crossref"}]},{"given":"Alexander","family":"Kolpakov","sequence":"additional","affiliation":[],"role":[{"role":"author","vocabulary":"crossref"}]},{"given":"Alex","family":"Kontorovich","sequence":"additional","affiliation":[],"role":[{"role":"author","vocabulary":"crossref"}]}],"member":"14","published-online":{"date-parts":[[2023,7,26]]},"reference":[{"key":"1","doi-asserted-by":"publisher","first-page":"737","DOI":"10.2140\/gt.2006.10.737","article-title":"Infinitely many hyperbolic Coxeter groups through dimension 19","volume":"10","author":"Allcock, Daniel","year":"2006","journal-title":"Geom. Topol.","ISSN":"https:\/\/id.crossref.org\/issn\/1465-3060","issn-type":"print"},{"issue":"1","key":"2","doi-asserted-by":"publisher","first-page":"65","DOI":"10.1137\/141000671","article-title":"Julia: a fresh approach to numerical computing","volume":"59","author":"Bezanson, Jeff","year":"2017","journal-title":"SIAM Rev.","ISSN":"https:\/\/id.crossref.org\/issn\/1095-7200","issn-type":"print"},{"issue":"4","key":"3","doi-asserted-by":"publisher","first-page":"3078","DOI":"10.1093\/imrn\/rnaa278","article-title":"On faces of quasi-arithmetic Coxeter polytopes","author":"Bogachev, Nikolay","year":"2021","journal-title":"Int. Math. Res. Not. IMRN","ISSN":"https:\/\/id.crossref.org\/issn\/1073-7928","issn-type":"print"},{"key":"4","unstructured":"[BK22] N. Bogachev and A. Kolpakov, Thin hyperbolic reflection groups, Preprint,  arXiv:2112.14642."},{"key":"5","unstructured":"[BKK] N. Bogachev, A. Kolpakov, and A. Kontorovich, Crystallographic packings, \\url{https:\/\/github.com\/sashakolpakov\/crystallographic-packings}."},{"issue":"5","key":"6","doi-asserted-by":"publisher","first-page":"769","DOI":"10.4213\/mzm11889","article-title":"Vinberg\u2019s algorithm for hyperbolic lattices","volume":"103","author":"Bogachev, N. V.","year":"2018","journal-title":"Mat. Zametki","ISSN":"https:\/\/id.crossref.org\/issn\/0025-567X","issn-type":"print"},{"issue":"1","key":"7","doi-asserted-by":"publisher","first-page":"133","DOI":"10.1016\/0021-8693(87)90245-6","article-title":"Automorphism groups of Lorentzian lattices","volume":"111","author":"Borcherds, Richard","year":"1987","journal-title":"J. Algebra","ISSN":"https:\/\/id.crossref.org\/issn\/0021-8693","issn-type":"print"},{"issue":"2","key":"8","doi-asserted-by":"publisher","first-page":"319","DOI":"10.1215\/S0012-7094-00-10424-3","article-title":"Reflection groups of Lorentzian lattices","volume":"104","author":"Borcherds, Richard E.","year":"2000","journal-title":"Duke Math. J.","ISSN":"https:\/\/id.crossref.org\/issn\/0012-7094","issn-type":"print"},{"key":"9","unstructured":"[Bot1] R. Bottinelli, On computational aspects of hyperbolic reflection groups & two homology theories for graphs, Ph.D. Thesis, 2021."},{"key":"10","unstructured":"[Bot2] R. Bottinelli, Vinny the Coxboy: Vinberg\u2019s algorithm over number fields, \\url{https:\/\/github.com\/bottine\/VinbergsAlgorithmNF}."},{"key":"11","isbn-type":"print","doi-asserted-by":"publisher","first-page":"33","DOI":"10.1007\/bf00883263","article-title":"Arithmetic crystallographic groups generated by reflections, and reflective hyperbolic lattices","author":"Bugaenko, V. O.","year":"1992","ISBN":"https:\/\/id.crossref.org\/isbn\/0821841076"},{"key":"12","unstructured":"[Col] G. Collinet, Personal communication to A. Kolpakov, 10 November 2014."},{"issue":"1","key":"13","doi-asserted-by":"publisher","first-page":"103","DOI":"10.1006\/jnth.1996.0141","article-title":"\u00dcber die maximale Dimension von Lorentz-Gittern mit coendlicher Spiegelungsgruppe","volume":"61","author":"Esselmann, Frank","year":"1996","journal-title":"J. Number Theory","ISSN":"https:\/\/id.crossref.org\/issn\/0022-314X","issn-type":"print"},{"key":"14","unstructured":"[Gug] R. Guglielmetti, Hyperbolic isometries in (in\u2013)finite dimensions and discrete reflection groups: theory and computations, Ph.D. Thesis, Universit\u00e9 de Fribourg, 2017."},{"key":"15","doi-asserted-by":"publisher","first-page":"105","DOI":"10.1515\/crelle-2023-0004","article-title":"On superintegral Kleinian sphere packings, bugs, and arithmetic groups","volume":"798","author":"Kapovich, Michael","year":"2023","journal-title":"J. Reine Angew. Math.","ISSN":"https:\/\/id.crossref.org\/issn\/0075-4102","issn-type":"print"},{"issue":"2","key":"16","doi-asserted-by":"publisher","first-page":"436","DOI":"10.1073\/pnas.1721104116","article-title":"Geometry and arithmetic of crystallographic sphere packings","volume":"116","author":"Kontorovich, Alex","year":"2019","journal-title":"Proc. Natl. Acad. Sci. USA","ISSN":"https:\/\/id.crossref.org\/issn\/0027-8424","issn-type":"print"},{"key":"17","unstructured":"[KS] M. Kirschmer and R. Scharlau, Classification of reflective Lorentzian lattices in higher dimensions, Work in progress, 2022."},{"issue":"1","key":"18","doi-asserted-by":"publisher","first-page":"63","DOI":"10.4171\/CMH\/4","article-title":"On right-angled reflection groups in hyperbolic spaces","volume":"80","author":"Potyagailo, Leonid","year":"2005","journal-title":"Comment. Math. Helv.","ISSN":"https:\/\/id.crossref.org\/issn\/0010-2571","issn-type":"print"},{"key":"19","unstructured":"[Sage] SageMath, The Sage Mathematics Software System. Version 9.4, The Sage Developers, 2021, \\url{https:\/\/www.sagemath.org}."},{"issue":"209","key":"20","first-page":"15","article-title":"Integral lattices and hyperbolic reflection groups","author":"Scharlau, Rudolf","year":"1992","journal-title":"Ast\\'{e}risque","ISSN":"https:\/\/id.crossref.org\/issn\/0303-1179","issn-type":"print"},{"key":"21","doi-asserted-by":"publisher","first-page":"516","DOI":"10.1016\/j.jalgebra.2018.06.013","article-title":"Reflective Lorentzian lattices of signature (5,1)","volume":"513","author":"Turkalj, Ivica","year":"2018","journal-title":"J. Algebra","ISSN":"https:\/\/id.crossref.org\/issn\/0021-8693","issn-type":"print"},{"key":"22","first-page":"471","article-title":"Discrete groups generated by reflections in Loba\u010devski\u012d spaces","volume":"72(114)","author":"Vinberg, \u00c8. B.","year":"1967","journal-title":"Mat. Sb. (N.S.)","ISSN":"https:\/\/id.crossref.org\/issn\/0368-8666","issn-type":"print"},{"key":"23","doi-asserted-by":"crossref","unstructured":"[Vin72] \u00c8. B. Vinberg, On groups of unit elements of certain quadratic forms, Mathematics of the USSR \u2013 Sbornik, Vol. 16. No. 1, pp. 17\u201335, 1972.","DOI":"10.1070\/SM1972v016n01ABEH001346"},{"key":"24","unstructured":"[Vin85] \u00c8. B. Vinberg, The absence of crystallographic groups of reflections in Lobachevsky spaces of large dimension, In: Trans. Moscow Math. Soc., 1985, pp. 75\u2013112."},{"issue":"3","key":"25","doi-asserted-by":"publisher","first-page":"593","DOI":"10.17323\/1609-4514-2015-15-3-593-602","article-title":"Non-arithmetic hyperbolic reflection groups in higher dimensions","volume":"15","author":"Vinberg, E. B.","year":"2015","journal-title":"Mosc. Math. J.","ISSN":"https:\/\/id.crossref.org\/issn\/1609-3321","issn-type":"print"},{"issue":"6","key":"26","first-page":"1273","article-title":"The groups \ud835\udc42_{18,1}(\ud835\udc4d) and \ud835\udc42_{19,1}(\ud835\udc4d)","volume":"238","author":"Vinberg, \u00c8. B.","year":"1978","journal-title":"Dokl. Akad. Nauk SSSR","ISSN":"https:\/\/id.crossref.org\/issn\/0002-3264","issn-type":"print"},{"key":"27","isbn-type":"print","doi-asserted-by":"publisher","first-page":"139","DOI":"10.1007\/978-3-662-02901-5_2","article-title":"Discrete groups of motions of spaces of constant curvature","author":"Vinberg, \u00c8. B.","year":"1993","ISBN":"https:\/\/id.crossref.org\/isbn\/3540520007"}],"container-title":["Mathematics of Computation"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/www.ams.org\/mcom\/2024-93-345\/S0025-5718-2023-03858-3\/S0025-5718-2023-03858-3.pdf","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2026,4,22]],"date-time":"2026-04-22T05:10:12Z","timestamp":1776834612000},"score":1,"resource":{"primary":{"URL":"https:\/\/www.ams.org\/mcom\/2024-93-345\/S0025-5718-2023-03858-3\/"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2023,7,26]]},"references-count":27,"journal-issue":{"issue":"345","published-print":{"date-parts":[[2024,1]]}},"alternative-id":["S0025-5718-2023-03858-3"],"URL":"https:\/\/doi.org\/10.1090\/mcom\/3858","archive":["CLOCKSS","Portico"],"relation":{},"ISSN":["1088-6842","0025-5718"],"issn-type":[{"value":"1088-6842","type":"electronic"},{"value":"0025-5718","type":"print"}],"subject":[],"published":{"date-parts":[[2023,7,26]]}}}