{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2022,4,4]],"date-time":"2022-04-04T17:33:58Z","timestamp":1649093638271},"reference-count":5,"publisher":"Cambridge University Press (CUP)","issue":"5","license":[{"start":{"date-parts":[[2011,8,18]],"date-time":"2011-08-18T00:00:00Z","timestamp":1313625600000},"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":"We give a short proof of the following result on the distribution of three-term arithmetic progressions in sparse subsets of F<\/jats:bold>p<\/jats:sub>n<\/jats:sup><\/jats:italic>. For every \u03b1 > 0 there exists a constant C<\/jats:italic> = C<\/jats:italic>(\u03b1) such that the following holds for all r<\/jats:italic> \u2265 Cp<\/jats:italic>n<\/jats:italic>\/2<\/jats:sup> and for almost all sets R<\/jats:italic> of size r<\/jats:italic> of F<\/jats:bold>p<\/jats:sub>n<\/jats:sup><\/jats:italic>. Let A<\/jats:italic> be any subset of R<\/jats:italic> of size at least \u03b1r<\/jats:italic>; then A<\/jats:italic> contains a non-trivial three-term arithmetic progression. This is an analogue of a hard theorem by Kohayakawa, \u0141uczak and R\u00f6dl. The proof uses a version of Green's regularity lemma for subsets of a typical random set, which is of interest in its own right.<\/jats:p>","DOI":"10.1017\/s0963548311000290","type":"journal-article","created":{"date-parts":[[2011,8,18]],"date-time":"2011-08-18T10:19:13Z","timestamp":1313662753000},"page":"777-791","source":"Crossref","is-referenced-by-count":0,"title":["On the Distribution of Three-Term Arithmetic Progressions in Sparse Subsets of Fp<\/sub>n<\/sup><\/i>"],"prefix":"10.1017","volume":"20","author":[{"given":"HOI H.","family":"NGUYEN","sequence":"first","affiliation":[]}],"member":"56","published-online":{"date-parts":[[2011,8,18]]},"reference":[{"key":"S0963548311000290_ref5","doi-asserted-by":"publisher","DOI":"10.1017\/CBO9780511755149"},{"key":"S0963548311000290_ref4","doi-asserted-by":"crossref","first-page":"133","DOI":"10.4064\/aa-75-2-133-163","article-title":"Arithmetic progressions of length three in subsets of a random set.","volume":"75","author":"Kohayakawa","year":"1996","journal-title":"Acta Arith."},{"key":"S0963548311000290_ref3","doi-asserted-by":"publisher","DOI":"10.1007\/s00039-005-0509-8"},{"key":"S0963548311000290_ref2","doi-asserted-by":"publisher","DOI":"10.1016\/0097-3165(87)90053-7"},{"key":"S0963548311000290_ref1","doi-asserted-by":"publisher","DOI":"10.1016\/0097-3165(84)90006-2"}],"container-title":["Combinatorics, Probability and Computing"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/www.cambridge.org\/core\/services\/aop-cambridge-core\/content\/view\/S0963548311000290","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2019,4,27]],"date-time":"2019-04-27T06:33:25Z","timestamp":1556346805000},"score":1,"resource":{"primary":{"URL":"https:\/\/www.cambridge.org\/core\/product\/identifier\/S0963548311000290\/type\/journal_article"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2011,8,18]]},"references-count":5,"journal-issue":{"issue":"5","published-print":{"date-parts":[[2011,9]]}},"alternative-id":["S0963548311000290"],"URL":"https:\/\/doi.org\/10.1017\/s0963548311000290","relation":{},"ISSN":["0963-5483","1469-2163"],"issn-type":[{"value":"0963-5483","type":"print"},{"value":"1469-2163","type":"electronic"}],"subject":[],"published":{"date-parts":[[2011,8,18]]}}}