{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2025,6,21]],"date-time":"2025-06-21T09:40:02Z","timestamp":1750498802527,"version":"3.41.0"},"reference-count":9,"publisher":"Cambridge University Press (CUP)","issue":"5","license":[{"start":{"date-parts":[[2007,9,1]],"date-time":"2007-09-01T00:00:00Z","timestamp":1188604800000},"content-version":"unspecified","delay-in-days":0,"URL":"https:\/\/www.cambridge.org\/core\/terms"}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["Combinator. Probab. Comp."],"published-print":{"date-parts":[[2007,9]]},"abstract":"<jats:p>Let <jats:italic>S<\/jats:italic> be a finite set of integers. We consider a problem of finding <jats:bold>\n\t      <jats:italic>D<\/jats:italic>\n\t    <\/jats:bold>(<jats:italic>S<\/jats:italic>), the minimum size of a set <jats:italic>A<\/jats:italic>, such that <jats:italic>S<\/jats:italic>\u2286 <jats:italic>A<\/jats:italic>\u2212<jats:italic>A<\/jats:italic>. We give a characterization for \u2018extremal\u2019 sets and prove lower and upper bounds on <jats:italic>D<\/jats:italic>(<jats:italic>S<\/jats:italic>) in terms of additive properties of <jats:italic>S<\/jats:italic>.<\/jats:p>","DOI":"10.1017\/s0963548307008401","type":"journal-article","created":{"date-parts":[[2007,2,28]],"date-time":"2007-02-28T10:11:16Z","timestamp":1172657476000},"page":"775-787","source":"Crossref","is-referenced-by-count":0,"title":["Difference Covers"],"prefix":"10.1017","volume":"16","author":[{"given":"TOMASZ","family":"SCHOEN","sequence":"first","affiliation":[]}],"member":"56","published-online":{"date-parts":[[2007,9,1]]},"reference":[{"key":"S0963548307008401_manual_ref-2","doi-asserted-by":"publisher","DOI":"10.1007\/BF01212974"},{"key":"S0963548307008401_manual_ref-4","doi-asserted-by":"publisher","DOI":"10.1016\/S0020-0190(00)00080-6"},{"key":"S0963548307008401_manual_ref-5","unstructured":"[5] Freiman, G. (1973) Foundations of a Structural Theory of Set Addition, Vol. 37 of Translations of Math. Monographs, AMS, Providence, RI."},{"volume-title":"Extremal Graph Theory","year":"1978","author":"Bollob\u00e1s","key":"S0963548307008401_manual_ref-3"},{"key":"S0963548307008401_manual_ref-8","first-page":"385","article-title":"Uber die Darstellung der Zahlen 1,2,\u00a8,N mittels Differenzen. (In Russian.)","volume":"24","author":"Redei","year":"1949","journal-title":"Mat. Sb., N. Ser."},{"key":"S0963548307008401_manual_ref-6","doi-asserted-by":"publisher","DOI":"10.1007\/s00039-001-0332-9"},{"key":"S0963548307008401_manual_ref-7","first-page":"173","article-title":"On the number of sums and differences.","volume":"258","author":"Hennecart","year":"1999","journal-title":"Ast\u00e9risque"},{"key":"S0963548307008401_manual_ref-9","first-page":"933","article-title":"On the cardinality of A\u00b1A and A\u2212A","volume":"18","author":"Ruzsa","year":"1978","journal-title":"Combinatorics: Proc. Fifth Hungarian Colloq., Keszthely 1976"},{"key":"S0963548307008401_manual_ref-1","doi-asserted-by":"publisher","DOI":"10.1002\/0471722154"}],"container-title":["Combinatorics, Probability and Computing"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/www.cambridge.org\/core\/services\/aop-cambridge-core\/content\/view\/S0963548307008401","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2025,6,21]],"date-time":"2025-06-21T09:07:37Z","timestamp":1750496857000},"score":1,"resource":{"primary":{"URL":"https:\/\/www.cambridge.org\/core\/product\/identifier\/S0963548307008401\/type\/journal_article"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2007,9]]},"references-count":9,"journal-issue":{"issue":"5","published-print":{"date-parts":[[2007,1]]}},"alternative-id":["S0963548307008401"],"URL":"https:\/\/doi.org\/10.1017\/s0963548307008401","relation":{},"ISSN":["0963-5483","1469-2163"],"issn-type":[{"type":"print","value":"0963-5483"},{"type":"electronic","value":"1469-2163"}],"subject":[],"published":{"date-parts":[[2007,9]]}}}