{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,1,29]],"date-time":"2026-01-29T12:15:49Z","timestamp":1769688949715,"version":"3.49.0"},"reference-count":16,"publisher":"Cambridge University Press (CUP)","issue":"5","license":[{"start":{"date-parts":[[2011,7,7]],"date-time":"2011-07-07T00:00:00Z","timestamp":1309996800000},"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":[[2011,9]]},"abstract":"<jats:p>A set of reals <jats:italic>A<\/jats:italic> = {<jats:italic>a<\/jats:italic><jats:sub>1<\/jats:sub>,.\u00a0.\u00a0.,<jats:italic>a<\/jats:italic><jats:sub><jats:italic>n<\/jats:italic><\/jats:sub>} is called convex if <jats:italic>a<\/jats:italic><jats:sub><jats:italic>i<\/jats:italic>+1<\/jats:sub> \u2212 <jats:italic>a<\/jats:italic><jats:sub><jats:italic>i<\/jats:italic><\/jats:sub> &gt; <jats:italic>a<\/jats:italic><jats:sub><jats:italic>i<\/jats:italic><\/jats:sub> \u2212 <jats:italic>a<\/jats:italic><jats:sub><jats:italic>i<\/jats:italic>\u22121<\/jats:sub> for all <jats:italic>i<\/jats:italic>. We prove, among other results, that for some <jats:italic>c<\/jats:italic> &gt; 0 every convex <jats:italic>A<\/jats:italic> satisfies |<jats:italic>A<\/jats:italic>\u2212<jats:italic>A<\/jats:italic>| \u2265 <jats:italic>c<\/jats:italic>|<jats:italic>A<\/jats:italic>|<jats:sup>8\/5<\/jats:sup>log<jats:sup>\u22122\/5<\/jats:sup>|<jats:italic>A<\/jats:italic>|.<\/jats:p>","DOI":"10.1017\/s0963548311000277","type":"journal-article","created":{"date-parts":[[2011,7,7]],"date-time":"2011-07-07T10:19:19Z","timestamp":1310033959000},"page":"793-798","source":"Crossref","is-referenced-by-count":35,"title":["On Sumsets of Convex Sets"],"prefix":"10.1017","volume":"20","author":[{"given":"TOMASZ","family":"SCHOEN","sequence":"first","affiliation":[]},{"given":"ILYA D.","family":"SHKREDOV","sequence":"additional","affiliation":[]}],"member":"56","published-online":{"date-parts":[[2011,7,7]]},"reference":[{"key":"S0963548311000277_ref11","unstructured":"[11] Sanders T. On Roth's theorem on progressions. Ann. of Math., to appear."},{"key":"S0963548311000277_ref4","doi-asserted-by":"publisher","DOI":"10.1016\/j.jnt.2006.09.018"},{"key":"S0963548311000277_ref13","unstructured":"[13] Schoen T. and Shkredov I. D. Additive properties of multiplicative subgroups of . Quart. J. Math., to appear."},{"key":"S0963548311000277_ref6","doi-asserted-by":"publisher","DOI":"10.1007\/BF01949064"},{"key":"S0963548311000277_ref15","article-title":"Sumas contra productos","volume":"12","author":"Solymosi","year":"2009","journal-title":"Gaceta de la Real Sociedad Matematica Espa\u00f1ola"},{"key":"S0963548311000277_ref7","doi-asserted-by":"publisher","DOI":"10.1007\/s00454-005-1194-y"},{"key":"S0963548311000277_ref1","doi-asserted-by":"publisher","DOI":"10.1006\/jnth.1999.2386"},{"key":"S0963548311000277_ref5","doi-asserted-by":"publisher","DOI":"10.1017\/S1446788700008041"},{"key":"S0963548311000277_ref8","doi-asserted-by":"publisher","DOI":"10.1137\/080717286"},{"key":"S0963548311000277_ref9","unstructured":"[9] Konyagin V. S. (2000) An estimate of the L 1-norm of an exponential sum. In The Theory of Approximations of Functions and Operators: Abstracts of Papers of the International Conference Dedicated to Stechkin's 80th Anniversary (in Russian), pp. 88\u201389."},{"key":"S0963548311000277_ref10","doi-asserted-by":"publisher","DOI":"10.1017\/S1446788710000236"},{"key":"S0963548311000277_ref3","doi-asserted-by":"publisher","DOI":"10.1216\/rmjm\/1194275934"},{"key":"S0963548311000277_ref12","doi-asserted-by":"publisher","DOI":"10.1215\/00127094-1276283"},{"key":"S0963548311000277_ref2","doi-asserted-by":"publisher","DOI":"10.1023\/A:1026600414468"},{"key":"S0963548311000277_ref16","doi-asserted-by":"publisher","DOI":"10.1007\/BF02579194"},{"key":"S0963548311000277_ref14","unstructured":"[14] Shkredov I. D. and V'ugin I. V. Additive shifts of multiplicative subgroups. Mat. Sbornik, to appear."}],"container-title":["Combinatorics, Probability and Computing"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/www.cambridge.org\/core\/services\/aop-cambridge-core\/content\/view\/S0963548311000277","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2019,4,27]],"date-time":"2019-04-27T06:33:46Z","timestamp":1556346826000},"score":1,"resource":{"primary":{"URL":"https:\/\/www.cambridge.org\/core\/product\/identifier\/S0963548311000277\/type\/journal_article"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2011,7,7]]},"references-count":16,"journal-issue":{"issue":"5","published-print":{"date-parts":[[2011,9]]}},"alternative-id":["S0963548311000277"],"URL":"https:\/\/doi.org\/10.1017\/s0963548311000277","relation":{},"ISSN":["0963-5483","1469-2163"],"issn-type":[{"value":"0963-5483","type":"print"},{"value":"1469-2163","type":"electronic"}],"subject":[],"published":{"date-parts":[[2011,7,7]]}}}