{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,8,21]],"date-time":"2026-08-21T14:49:40Z","timestamp":1787323780245,"version":"3.56.0"},"reference-count":20,"publisher":"Society for Industrial & Applied Mathematics (SIAM)","issue":"3","funder":[{"DOI":"10.13039\/501100001459","name":"Ministry of Education - Singapore","doi-asserted-by":"publisher","award":["R-263-000-B61-112"],"award-info":[{"award-number":["R-263-000-B61-112"]}],"id":[{"id":"10.13039\/501100001459","id-type":"DOI","asserted-by":"publisher"}]}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["SIAM J. Discrete Math."],"published-print":{"date-parts":[[2017,1]]},"abstract":"<jats:p>A $ B_h $ set (or Sidon set of order $ h $) in an Abelian group $ G $ is any subset $ \\{b_0, b_1, \\ldots,b_{n}\\} $ of $ G $ with the property that all the sums $ b_{i_1} + \\cdots + b_{i_h} $ are different up to the order of the summands. Let $ \\phi(h,n) $ denote the order of the smallest Abelian group containing a $ B_h $ set of cardinality $ n + 1 $. It is shown that ${\\scriptstyle\\lim_{h \\to \\infty} \\frac{ \\phi(h,n) }{ h^n } = \\frac{1}{n! \\ \\delta_{\\sc l}(\\triangle^n)}},$ where $ \\delta_{\\sc l}(\\triangle^n) $ is the lattice packing density of an $ n $-simplex in Euclidean space. This determines the asymptotics exactly in cases where this density is known ($ n \\leq 3 $) and gives improved bounds on $ \\phi(h,n) $ in the remaining cases. The corresponding geometric characterization of bases of order $ h $ in finite Abelian groups in terms of lattice coverings by simplices is also given.<\/jats:p>","DOI":"10.1137\/16m1099182","type":"journal-article","created":{"date-parts":[[2017,9,28]],"date-time":"2017-09-28T11:07:08Z","timestamp":1506596828000},"page":"2269-2278","source":"Crossref","is-referenced-by-count":7,"title":["Improved Bounds on Sidon Sets via Lattice Packings of Simplices"],"prefix":"10.1137","volume":"31","author":[{"given":"Mladen","family":"Kova\u010devi\u0107","sequence":"first","affiliation":[],"role":[{"vocabulary":"crossref","role":"author"}]},{"given":"Vincent Y. F.","family":"Tan","sequence":"additional","affiliation":[],"role":[{"vocabulary":"crossref","role":"author"}]}],"member":"351","published-online":{"date-parts":[[2017,9,28]]},"reference":[{"key":"atypb1","doi-asserted-by":"crossref","unstructured":"T. Beth, D. Jungnickel, and H. Lenz,\n                      Design Theory\n                      , 2nd ed., Cambridge University Press, Cambridge, UK, 1999.","DOI":"10.1017\/CBO9781139507660"},{"key":"atypb2","doi-asserted-by":"publisher","DOI":"10.1007\/BF02566968"},{"key":"atypb3","doi-asserted-by":"publisher","DOI":"10.1090\/S0002-9939-1994-1196162-9"},{"key":"atypb4","doi-asserted-by":"publisher","DOI":"10.1109\/TIT.2004.824915"},{"key":"atypb5","first-page":"318","author":"To\u0301th G. Fejes","year":"1983","journal-title":"Basel"},{"key":"atypb6","doi-asserted-by":"publisher","DOI":"10.1137\/0118025"},{"key":"atypb7","doi-asserted-by":"publisher","DOI":"10.1007\/BF01297011"},{"key":"atypb8","unstructured":"P. M. Gruber,\n                      Convex and Discrete Geometry\n                      , Springer, New York, 2007."},{"key":"atypb9","unstructured":"P. M. Gruber and C. G. Lekkerkerker,\n                      Geometry of Numbers\n                      , 2nd ed., North-Holland, Amsterdam, 1987."},{"key":"atypb10","doi-asserted-by":"crossref","unstructured":"H. Halberstam and K. F. Roth,\n                      Sequences\n                      , Springer-Verlag, New York, 1983.","DOI":"10.1007\/978-1-4613-8227-0"},{"key":"atypb11","doi-asserted-by":"publisher","DOI":"10.1090\/S0002-9904-1970-12400-4"},{"key":"atypb12","doi-asserted-by":"publisher","DOI":"10.1006\/jnth.1993.1037"},{"key":"atypb13","unstructured":"M. Kovac\u030cevic\u0301,\n                      Codes in ${A}_n$ lattices: Geometry of $ {B}_h $ sets and difference sets\n                      , preprint,arXiv:1409.5276v4, 2016."},{"key":"atypb14","author":"O'Bryant K.","year":"2004","journal-title":"Electron. J. Combin., DS11 ("},{"key":"atypb15","unstructured":"C. A. 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Szabo\u0301,\n                      Algebra and Tiling: Homomorphisms in the Service of Geometry\n                      , The Mathematical Association of America, Washington, DC, 1994.","DOI":"10.5948\/UPO9781614440246"}],"container-title":["SIAM Journal on Discrete Mathematics"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/epubs.siam.org\/doi\/pdf\/10.1137\/16M1099182","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2026,8,21]],"date-time":"2026-08-21T14:13:31Z","timestamp":1787321611000},"score":1,"resource":{"primary":{"URL":"https:\/\/epubs.siam.org\/doi\/10.1137\/16M1099182"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2017,1]]},"references-count":20,"journal-issue":{"issue":"3","published-print":{"date-parts":[[2017,1]]}},"alternative-id":["10.1137\/16M1099182"],"URL":"https:\/\/doi.org\/10.1137\/16m1099182","relation":{},"ISSN":["0895-4801","1095-7146"],"issn-type":[{"value":"0895-4801","type":"print"},{"value":"1095-7146","type":"electronic"}],"subject":[],"published":{"date-parts":[[2017,1]]}}}