{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,2,2]],"date-time":"2026-02-02T06:18:20Z","timestamp":1770013100860,"version":"3.49.0"},"reference-count":15,"publisher":"Cambridge University Press (CUP)","issue":"3","license":[{"start":{"date-parts":[[2013,3,19]],"date-time":"2013-03-19T00:00:00Z","timestamp":1363651200000},"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":[[2013,5]]},"abstract":"<jats:p>We prove results about the <jats:italic>L<jats:sup>p<\/jats:sup><\/jats:italic>-almost-periodicity of convolutions. One of these follows from a simple but rather general lemma about approximating a sum of functions in <jats:italic>L<jats:sup>p<\/jats:sup><\/jats:italic>, and gives a very short proof of a theorem of Green that if <jats:italic>A<\/jats:italic> and <jats:italic>B<\/jats:italic> are subsets of {1,.\u00a0.\u00a0.,<jats:italic>N<\/jats:italic>} of sizes \u03b1<jats:italic>N<\/jats:italic> and \u03b2<jats:italic>N<\/jats:italic> then <jats:italic>A+B<\/jats:italic> contains an arithmetic progression of length at least\n<jats:disp-formula-group><jats:disp-formula><jats:alternatives><jats:graphic xmlns:xlink=\"http:\/\/www.w3.org\/1999\/xlink\" orientation=\"portrait\" mime-subtype=\"gif\" mimetype=\"image\" position=\"float\" xlink:type=\"simple\" xlink:href=\"S0963548313000060_eqnU1\"\/><jats:tex-math>\n\\begin{equation}\n\\exp ( c (\\alpha \\beta \\log N)^{1\/2} - \\log\\log N).\n\\end{equation}\n<\/jats:tex-math><\/jats:alternatives><\/jats:disp-formula><\/jats:disp-formula-group>\nAnother almost-periodicity result improves this bound for densities decreasing with <jats:italic>N<\/jats:italic>: we show that under the above hypotheses the sumset <jats:italic>A+B<\/jats:italic> contains an arithmetic progression of length at least\n<jats:disp-formula-group><jats:disp-formula><jats:alternatives><jats:graphic xmlns:xlink=\"http:\/\/www.w3.org\/1999\/xlink\" orientation=\"portrait\" mime-subtype=\"gif\" mimetype=\"image\" position=\"float\" xlink:type=\"simple\" xlink:href=\"S0963548313000060_eqnU2\"\/><jats:tex-math>\n\\begin{equation}\n\\exp\\biggl( c \\biggl(\\frac{\\alpha \\log N}{\\log^3 2\\beta^{-1}} \\biggr)^{1\/2} - \\log( \\beta^{-1} \\log N) \\biggr).\n\\end{equation}\n<\/jats:tex-math><\/jats:alternatives><\/jats:disp-formula><\/jats:disp-formula-group><\/jats:p>","DOI":"10.1017\/s0963548313000060","type":"journal-article","created":{"date-parts":[[2013,3,19]],"date-time":"2013-03-19T13:49:31Z","timestamp":1363700971000},"page":"351-365","source":"Crossref","is-referenced-by-count":17,"title":["Arithmetic Progressions in Sumsets and <i>L<sup>p<\/sup><\/i>-Almost-Periodicity"],"prefix":"10.1017","volume":"22","author":[{"given":"ERNIE","family":"CROOT","sequence":"first","affiliation":[],"role":[{"role":"author","vocabulary":"crossref"}]},{"given":"IZABELLA","family":"\u0141ABA","sequence":"additional","affiliation":[],"role":[{"role":"author","vocabulary":"crossref"}]},{"given":"OLOF","family":"SISASK","sequence":"additional","affiliation":[],"role":[{"role":"author","vocabulary":"crossref"}]}],"member":"56","published-online":{"date-parts":[[2013,3,19]]},"reference":[{"key":"S0963548313000060_ref15","doi-asserted-by":"publisher","DOI":"10.1017\/CBO9780511755149"},{"key":"S0963548313000060_ref14","doi-asserted-by":"publisher","DOI":"10.2140\/apde.2012.5.627"},{"key":"S0963548313000060_ref13","doi-asserted-by":"publisher","DOI":"10.4064\/aa146-1-6"},{"key":"S0963548313000060_ref10","volume-title":"Seminar on Functional Analysis, 1980\u20131981","author":"Pisier","year":"1981"},{"key":"S0963548313000060_ref8","doi-asserted-by":"publisher","DOI":"10.1007\/BF01192399"},{"key":"S0963548313000060_ref7","volume-title":"Probability: A Graduate Course","author":"Gut","year":"2005"},{"key":"S0963548313000060_ref3","doi-asserted-by":"publisher","DOI":"10.1007\/s00039-010-0101-8"},{"key":"S0963548313000060_ref1","doi-asserted-by":"publisher","DOI":"10.1017\/CBO9780511983917.008"},{"key":"S0963548313000060_ref11","first-page":"87","volume-title":"Combinatorial Number Theory and Additive Group Theory","author":"Ruzsa","year":"2009"},{"key":"S0963548313000060_ref5","unstructured":"Green B. (2002) Restriction and Kakeya Phenomena, lecture notes. http:\/\/www.dpmms.cam.ac.uk\/~bjg23\/rkp.html"},{"key":"S0963548313000060_ref2","doi-asserted-by":"publisher","DOI":"10.1215\/S0012-7094-02-11331-3"},{"key":"S0963548313000060_ref9","article-title":"New proofs of Pl\u00fcnnecke-type estimates for product sets in groups","author":"Petridis","year":"2013","journal-title":"Combinatorica"},{"key":"S0963548313000060_ref12","doi-asserted-by":"publisher","DOI":"10.1017\/S030500410700093X"},{"key":"S0963548313000060_ref6","first-page":"1","volume-title":"Surveys in Combinatorics 2005","author":"Green","year":"2005"},{"key":"S0963548313000060_ref4","doi-asserted-by":"publisher","DOI":"10.1007\/s00039-002-8258-4"}],"container-title":["Combinatorics, Probability and Computing"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/www.cambridge.org\/core\/services\/aop-cambridge-core\/content\/view\/S0963548313000060","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2019,4,23]],"date-time":"2019-04-23T22:00:50Z","timestamp":1556056850000},"score":1,"resource":{"primary":{"URL":"https:\/\/www.cambridge.org\/core\/product\/identifier\/S0963548313000060\/type\/journal_article"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2013,3,19]]},"references-count":15,"journal-issue":{"issue":"3","published-print":{"date-parts":[[2013,5]]}},"alternative-id":["S0963548313000060"],"URL":"https:\/\/doi.org\/10.1017\/s0963548313000060","relation":{},"ISSN":["0963-5483","1469-2163"],"issn-type":[{"value":"0963-5483","type":"print"},{"value":"1469-2163","type":"electronic"}],"subject":[],"published":{"date-parts":[[2013,3,19]]}}}