{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2022,4,2]],"date-time":"2022-04-02T16:03:26Z","timestamp":1648915406884},"reference-count":12,"publisher":"Cambridge University Press (CUP)","issue":"6","license":[{"start":{"date-parts":[[2011,10,6]],"date-time":"2011-10-06T00:00:00Z","timestamp":1317859200000},"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":[[2011,11]]},"abstract":"<jats:p>We obtain a first non-trivial estimate for the sum of dilates problem in the case of groups of prime order, by showing that if <jats:italic>t<\/jats:italic> is an integer different from 0, 1 or \u22121 and if <jats:private-char><jats:inline-graphic xmlns:xlink=\"http:\/\/www.w3.org\/1999\/xlink\" mime-subtype=\"gif\" mimetype=\"image\" xlink:type=\"simple\" xlink:href=\"S0963548311000447_char1\" \/><\/jats:private-char> \u2282 \u2124\/<jats:italic>p<\/jats:italic>\u2124 is not too large (with respect to <jats:italic>p<\/jats:italic>), then |<jats:private-char><jats:inline-graphic xmlns:xlink=\"http:\/\/www.w3.org\/1999\/xlink\" mime-subtype=\"gif\" mimetype=\"image\" xlink:type=\"simple\" xlink:href=\"S0963548311000447_char1\" \/><\/jats:private-char> + <jats:italic>t<\/jats:italic> \u22c5 <jats:private-char><jats:inline-graphic xmlns:xlink=\"http:\/\/www.w3.org\/1999\/xlink\" mime-subtype=\"gif\" mimetype=\"image\" xlink:type=\"simple\" xlink:href=\"S0963548311000447_char1\" \/><\/jats:private-char>| is significantly larger than 2|<jats:private-char><jats:inline-graphic xmlns:xlink=\"http:\/\/www.w3.org\/1999\/xlink\" mime-subtype=\"gif\" mimetype=\"image\" xlink:type=\"simple\" xlink:href=\"S0963548311000447_char1\" \/><\/jats:private-char>| (unless |<jats:italic>t<\/jats:italic>| = 3). In the important case |<jats:italic>t<\/jats:italic>| = 2, we obtain for instance |<jats:private-char><jats:inline-graphic xmlns:xlink=\"http:\/\/www.w3.org\/1999\/xlink\" mime-subtype=\"gif\" mimetype=\"image\" xlink:type=\"simple\" xlink:href=\"S0963548311000447_char1\" \/><\/jats:private-char> + <jats:italic>t<\/jats:italic> \u22c5 <jats:private-char><jats:inline-graphic xmlns:xlink=\"http:\/\/www.w3.org\/1999\/xlink\" mime-subtype=\"gif\" mimetype=\"image\" xlink:type=\"simple\" xlink:href=\"S0963548311000447_char1\" \/><\/jats:private-char>| \u2265 2.08 |<jats:private-char><jats:inline-graphic xmlns:xlink=\"http:\/\/www.w3.org\/1999\/xlink\" mime-subtype=\"gif\" mimetype=\"image\" xlink:type=\"simple\" xlink:href=\"S0963548311000447_char1\" \/><\/jats:private-char>|\u22122.<\/jats:p>","DOI":"10.1017\/s0963548311000447","type":"journal-article","created":{"date-parts":[[2011,10,6]],"date-time":"2011-10-06T10:30:40Z","timestamp":1317897040000},"page":"867-873","source":"Crossref","is-referenced-by-count":5,"title":["Sums of Dilates in Groups of Prime Order"],"prefix":"10.1017","volume":"20","author":[{"given":"ALAIN","family":"PLAGNE","sequence":"first","affiliation":[],"role":[{"role":"author","vocabulary":"crossref"}]}],"member":"56","published-online":{"date-parts":[[2011,10,6]]},"reference":[{"key":"S0963548311000447_ref6","unstructured":"[6] Freiman G. A. (1973) Foundations of a Structural Theory of Set Addition, Vol. 37 of Translations of Mathematical Monographs, AMS."},{"key":"S0963548311000447_ref10","article-title":"More on points and arcs","volume":"7","author":"Lev","year":"2007","journal-title":"Integers: Electron. J. Combin. Number Theory"},{"key":"S0963548311000447_ref8","doi-asserted-by":"publisher","DOI":"10.1017\/S0963548310000520"},{"key":"S0963548311000447_ref4","article-title":"A sumset problem","volume":"2","author":"Cilleruelo","year":"2010","journal-title":"J. Combin. Number Theory"},{"key":"S0963548311000447_ref2","first-page":"99","article-title":"Recherches sur les nombres.","volume":"9","author":"Cauchy","year":"1813","journal-title":"J. \u00c9cole Polytechnique"},{"key":"S0963548311000447_ref9","article-title":"Distribution of points on arcs","volume":"5","author":"Lev","year":"2005","journal-title":"Integers: Electron. J. Combin. Number Theory"},{"key":"S0963548311000447_ref11","unstructured":"[11] Ljujic Z. A lower bound for the size of a sum of dilates. arXiv.1101.5425"},{"key":"S0963548311000447_ref12","first-page":"151","article-title":"Inverse problems for linear forms over finite sets of integers.","volume":"23","author":"Nathanson","year":"2008","journal-title":"J. Ramanujan Math. Soc."},{"key":"S0963548311000447_ref5","unstructured":"[5] Du S.-S. , Cao H.-Q. and Sun Z.-W. On a sumset problem for the integers. arXiv:1011.5438"},{"key":"S0963548311000447_ref1","doi-asserted-by":"publisher","DOI":"10.1017\/S096354830800919X"},{"key":"S0963548311000447_ref3","doi-asserted-by":"publisher","DOI":"10.1017\/S0963548309990307"},{"key":"S0963548311000447_ref7","doi-asserted-by":"publisher","DOI":"10.4064\/aa103-2-4"}],"container-title":["Combinatorics, Probability and Computing"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/www.cambridge.org\/core\/services\/aop-cambridge-core\/content\/view\/S0963548311000447","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2019,4,27]],"date-time":"2019-04-27T03:41:00Z","timestamp":1556336460000},"score":1,"resource":{"primary":{"URL":"https:\/\/www.cambridge.org\/core\/product\/identifier\/S0963548311000447\/type\/journal_article"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2011,10,6]]},"references-count":12,"journal-issue":{"issue":"6","published-print":{"date-parts":[[2011,11]]}},"alternative-id":["S0963548311000447"],"URL":"https:\/\/doi.org\/10.1017\/s0963548311000447","relation":{},"ISSN":["0963-5483","1469-2163"],"issn-type":[{"value":"0963-5483","type":"print"},{"value":"1469-2163","type":"electronic"}],"subject":[],"published":{"date-parts":[[2011,10,6]]}}}