{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,2,28]],"date-time":"2026-02-28T09:18:38Z","timestamp":1772270318158,"version":"3.50.1"},"reference-count":9,"publisher":"Cambridge University Press (CUP)","issue":"2","license":[{"start":{"date-parts":[[2010,12,6]],"date-time":"2010-12-06T00:00:00Z","timestamp":1291593600000},"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":[[2011,3]]},"abstract":"<jats:p>Let <jats:italic>A<\/jats:italic> be a finite non-empty set of integers. An asymptotic estimate of the size of the sum of several dilates was obtained by Bukh. The unique known exact bound concerns the sum |<jats:italic>A<\/jats:italic> + <jats:italic>k<\/jats:italic>\u22c5<jats:italic>A<\/jats:italic>|, where <jats:italic>k<\/jats:italic> is a prime and |<jats:italic>A<\/jats:italic>| is large. In its full generality, this bound is due to Cilleruelo, Serra and the first author.<\/jats:p><jats:p>Let <jats:italic>k<\/jats:italic> be an odd prime and assume that |<jats:italic>A<\/jats:italic>| &gt; 8<jats:italic>k<\/jats:italic><jats:sup><jats:italic>k<\/jats:italic><\/jats:sup>. A corollary to our main result states that |2\u22c5<jats:italic>A<\/jats:italic> + <jats:italic>k<\/jats:italic>\u22c5<jats:italic>A<\/jats:italic>|\u2265(<jats:italic>k<\/jats:italic>+2)|<jats:italic>A<\/jats:italic>|\u2212<jats:italic>k<\/jats:italic><jats:sup>2<\/jats:sup>\u2212<jats:italic>k<\/jats:italic>+2. Notice that |2\u22c5<jats:italic>P<\/jats:italic>+<jats:italic>k<\/jats:italic>\u22c5<jats:italic>P<\/jats:italic>|=(<jats:italic>k<\/jats:italic>+2)|<jats:italic>P<\/jats:italic>|\u22122<jats:italic>k<\/jats:italic>, if <jats:italic>P<\/jats:italic> is an arithmetic progression.<\/jats:p>","DOI":"10.1017\/s0963548310000520","type":"journal-article","created":{"date-parts":[[2010,12,6]],"date-time":"2010-12-06T10:47:13Z","timestamp":1291632433000},"page":"249-256","source":"Crossref","is-referenced-by-count":14,"title":["A Lower Bound for the Size of a Minkowski Sum of Dilates"],"prefix":"10.1017","volume":"20","author":[{"given":"Y. O.","family":"HAMIDOUNE","sequence":"first","affiliation":[],"role":[{"role":"author","vocabulary":"crossref"}]},{"given":"J.","family":"RU\u00c9","sequence":"additional","affiliation":[],"role":[{"role":"author","vocabulary":"crossref"}]}],"member":"56","published-online":{"date-parts":[[2010,12,6]]},"reference":[{"key":"S0963548310000520_ref5","unstructured":"[5] Garaev M. Z. (2007) An explicit sum-product estimate in p . Internat. Math. Res. Notices 2007 #rnm035."},{"key":"S0963548310000520_ref8","unstructured":"[8] Nathanson M. B. Inverse problems for linear forms over finite sets of integers. Available online at: arXiv: 0708.2304v2."},{"key":"S0963548310000520_ref6","doi-asserted-by":"publisher","DOI":"10.1090\/S0002-9939-08-09385-4"},{"key":"S0963548310000520_ref4","article-title":"A sumset problem","volume":"2","author":"Cilleruelo","year":"2010","journal-title":"J. Combin. Number Theory"},{"key":"S0963548310000520_ref7","doi-asserted-by":"crossref","first-page":"157","DOI":"10.1007\/BF02771981","article-title":"Distance sets of well-distributed planar sets for polygonal norms","volume":"152","author":"\u0141aba","year":"2006","journal-title":"Israel J. Math."},{"key":"S0963548310000520_ref2","doi-asserted-by":"publisher","DOI":"10.1017\/S096354830800919X"},{"key":"S0963548310000520_ref9","doi-asserted-by":"publisher","DOI":"10.4064\/aa129-4-5"},{"key":"S0963548310000520_ref3","doi-asserted-by":"publisher","DOI":"10.1017\/S0963548309990307"},{"key":"S0963548310000520_ref1","doi-asserted-by":"publisher","DOI":"10.4064\/aa131-1-4"}],"container-title":["Combinatorics, Probability and Computing"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/www.cambridge.org\/core\/services\/aop-cambridge-core\/content\/view\/S0963548310000520","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2019,4,27]],"date-time":"2019-04-27T19:17:44Z","timestamp":1556392664000},"score":1,"resource":{"primary":{"URL":"https:\/\/www.cambridge.org\/core\/product\/identifier\/S0963548310000520\/type\/journal_article"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2010,12,6]]},"references-count":9,"journal-issue":{"issue":"2","published-print":{"date-parts":[[2011,3]]}},"alternative-id":["S0963548310000520"],"URL":"https:\/\/doi.org\/10.1017\/s0963548310000520","relation":{},"ISSN":["0963-5483","1469-2163"],"issn-type":[{"value":"0963-5483","type":"print"},{"value":"1469-2163","type":"electronic"}],"subject":[],"published":{"date-parts":[[2010,12,6]]}}}