{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2022,4,2]],"date-time":"2022-04-02T21:05:02Z","timestamp":1648933502665},"reference-count":5,"publisher":"Cambridge University Press (CUP)","issue":"6","license":[{"start":{"date-parts":[[2021,4,29]],"date-time":"2021-04-29T00:00:00Z","timestamp":1619654400000},"content-version":"unspecified","delay-in-days":0,"URL":"https:\/\/www.cambridge.org\/core\/terms"}],"content-domain":{"domain":["cambridge.org"],"crossmark-restriction":true},"short-container-title":["Combinator. Probab. Comp."],"published-print":{"date-parts":[[2021,11]]},"abstract":"<jats:title>Abstract<\/jats:title><jats:p>Extending a result by Alon, Linial, and Meshulam to abelian groups, we prove that if <jats:italic>G<\/jats:italic> is a finite abelian group of exponent <jats:italic>m<\/jats:italic> and <jats:italic>S<\/jats:italic> is a sequence of elements of <jats:italic>G<\/jats:italic> such that any subsequence of <jats:italic>S<\/jats:italic> consisting of at least <jats:inline-formula><jats:alternatives><jats:inline-graphic xmlns:xlink=\"http:\/\/www.w3.org\/1999\/xlink\" mime-subtype=\"png\" xlink:href=\"S0963548321000109_inline1.png\" \/><jats:tex-math>\n$$|S| - m\\ln |G|$$\n<\/jats:tex-math><\/jats:alternatives><\/jats:inline-formula> elements generates <jats:italic>G<\/jats:italic>, then <jats:italic>S<\/jats:italic> is an additive basis of <jats:italic>G<\/jats:italic> . We also prove that the additive span of any <jats:italic>l<\/jats:italic> generating sets of <jats:italic>G<\/jats:italic> contains a coset of a subgroup of size at least <jats:inline-formula><jats:alternatives><jats:inline-graphic xmlns:xlink=\"http:\/\/www.w3.org\/1999\/xlink\" mime-subtype=\"png\" xlink:href=\"S0963548321000109_inline2.png\" \/><jats:tex-math>\n$$|G{|^{1 - c{ \\in ^l}}}$$\n<\/jats:tex-math><\/jats:alternatives><\/jats:inline-formula> for certain <jats:italic>c<\/jats:italic>=<jats:italic>c<\/jats:italic>(<jats:italic>m<\/jats:italic>) and <jats:inline-formula><jats:alternatives><jats:inline-graphic xmlns:xlink=\"http:\/\/www.w3.org\/1999\/xlink\" mime-subtype=\"png\" xlink:href=\"S0963548321000109_inline3.png\" \/><jats:tex-math>\n$$ \\in  =  \\in (m) &lt; 1$$\n<\/jats:tex-math><\/jats:alternatives><\/jats:inline-formula>; we use the probabilistic method to give sharper values of <jats:italic>c<\/jats:italic>(<jats:italic>m<\/jats:italic>) and <jats:inline-formula><jats:alternatives><jats:inline-graphic xmlns:xlink=\"http:\/\/www.w3.org\/1999\/xlink\" mime-subtype=\"png\" xlink:href=\"S0963548321000109_inline4.png\" \/><jats:tex-math>\n$$ \\in (m)$$\n<\/jats:tex-math><\/jats:alternatives><\/jats:inline-formula> in the case when <jats:italic>G<\/jats:italic> is a vector space; and we give new proofs of related known results.<\/jats:p>","DOI":"10.1017\/s0963548321000109","type":"journal-article","created":{"date-parts":[[2021,4,29]],"date-time":"2021-04-29T05:09:55Z","timestamp":1619672995000},"page":"930-941","update-policy":"http:\/\/dx.doi.org\/10.1017\/policypage","source":"Crossref","is-referenced-by-count":0,"title":["Additive bases via Fourier analysis"],"prefix":"10.1017","volume":"30","author":[{"given":"Bodan","family":"Arsovski","sequence":"first","affiliation":[],"role":[{"role":"author","vocabulary":"crossref"}]}],"member":"56","published-online":{"date-parts":[[2021,4,29]]},"reference":[{"key":"S0963548321000109_ref4","doi-asserted-by":"publisher","DOI":"10.1142\/S1793042110003216"},{"key":"S0963548321000109_ref5","doi-asserted-by":"publisher","DOI":"10.4310\/MRL.2005.v12.n1.a11"},{"key":"S0963548321000109_ref3","doi-asserted-by":"publisher","DOI":"10.1016\/0095-8956(92)90016-Q"},{"key":"S0963548321000109_ref2","first-page":"P3","article-title":"On the additive bases problem in finite fields","volume":"23","author":"Hatami","year":"2016","journal-title":"Electron. J. Comb."},{"key":"S0963548321000109_ref1","doi-asserted-by":"publisher","DOI":"10.1016\/0097-3165(91)90045-I"}],"container-title":["Combinatorics, Probability and Computing"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/www.cambridge.org\/core\/services\/aop-cambridge-core\/content\/view\/S0963548321000109","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2021,10,15]],"date-time":"2021-10-15T15:51:46Z","timestamp":1634313106000},"score":1,"resource":{"primary":{"URL":"https:\/\/www.cambridge.org\/core\/product\/identifier\/S0963548321000109\/type\/journal_article"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2021,4,29]]},"references-count":5,"journal-issue":{"issue":"6","published-print":{"date-parts":[[2021,11]]}},"alternative-id":["S0963548321000109"],"URL":"https:\/\/doi.org\/10.1017\/s0963548321000109","relation":{},"ISSN":["0963-5483","1469-2163"],"issn-type":[{"value":"0963-5483","type":"print"},{"value":"1469-2163","type":"electronic"}],"subject":[],"published":{"date-parts":[[2021,4,29]]},"assertion":[{"value":"\u00a9 The Author(s), 2021. Published by Cambridge University Press","name":"copyright","label":"Copyright","group":{"name":"copyright_and_licensing","label":"Copyright and Licensing"}}]}}