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{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2022,3,29]],"date-time":"2022-03-29T11:41:49Z","timestamp":1648554109295},"reference-count":15,"publisher":"Cambridge University Press (CUP)","issue":"4","license":[{"start":{"date-parts":[[2020,10,5]],"date-time":"2020-10-05T00:00:00Z","timestamp":1601856000000},"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":["J. symb. log."],"published-print":{"date-parts":[[2021,12]]},"abstract":"Abstract<\/jats:title>We consider the structures

\n$(\\mathbb {Z}; \\mathrm {SF}^{\\mathbb {Z}})$\n<\/jats:tex-math><\/jats:alternatives><\/jats:inline-formula>,

\n$(\\mathbb {Z}; <, \\mathrm {SF}^{\\mathbb {Z}})$\n<\/jats:tex-math><\/jats:alternatives><\/jats:inline-formula>,

\n$(\\mathbb {Q}; \\mathrm {SF}^{\\mathbb {Q}})$\n<\/jats:tex-math><\/jats:alternatives><\/jats:inline-formula>, and

\n$(\\mathbb {Q}; <, \\mathrm {SF}^{\\mathbb {Q}})$\n<\/jats:tex-math><\/jats:alternatives><\/jats:inline-formula> where

\n$\\mathbb {Z}$\n<\/jats:tex-math><\/jats:alternatives><\/jats:inline-formula> is the additive group of integers,

\n$\\mathrm {SF}^{\\mathbb {Z}}$\n<\/jats:tex-math><\/jats:alternatives><\/jats:inline-formula> is the set of

\n$a \\in \\mathbb {Z}$\n<\/jats:tex-math><\/jats:alternatives><\/jats:inline-formula> such that

\n$v_{p}(a) < 2$\n<\/jats:tex-math><\/jats:alternatives><\/jats:inline-formula> for \u00a0every prime p<\/jats:italic> and corresponding p<\/jats:italic>-adic valuation

\n$v_{p}$\n<\/jats:tex-math><\/jats:alternatives><\/jats:inline-formula>,

\n$\\mathbb {Q}$\n<\/jats:tex-math><\/jats:alternatives><\/jats:inline-formula> and

\n$\\mathrm {SF}^{\\mathbb {Q}}$\n<\/jats:tex-math><\/jats:alternatives><\/jats:inline-formula> are defined likewise for rational numbers, and

\n$<$\n<\/jats:tex-math><\/jats:alternatives><\/jats:inline-formula> denotes the natural ordering on each of these domains. We prove that the second structure is model-theoretically wild while the other three structures are model-theoretically tame. Moreover, all these results can be seen as examples where number-theoretic randomness yields model-theoretic consequences.<\/jats:p>","DOI":"10.1017\/jsl.2020.30","type":"journal-article","created":{"date-parts":[[2020,10,5]],"date-time":"2020-10-05T07:58:44Z","timestamp":1601884724000},"page":"1324-1349","update-policy":"http:\/\/dx.doi.org\/10.1017\/policypage","source":"Crossref","is-referenced-by-count":0,"title":["THE ADDITIVE GROUPS OF AND WITH PREDICATES FOR BEING SQUARE-FREE"],"prefix":"10.1017","volume":"86","author":[{"given":"NEER","family":"BHARDWAJ","sequence":"first","affiliation":[]},{"given":"CHIEU-MINH","family":"TRAN","sequence":"additional","affiliation":[]}],"member":"56","published-online":{"date-parts":[[2020,10,5]]},"reference":[{"key":"S0022481220000304_r7","doi-asserted-by":"publisher","DOI":"10.4064\/fm256-5-2016"},{"key":"S0022481220000304_r5","doi-asserted-by":"publisher","DOI":"10.1215\/00294527-2019-0002"},{"key":"S0022481220000304_r9","first-page":"1041","article-title":"Decidability and classification of the theory of integers with primes","volume":"82","author":"Kaplan","year":"2017","journal-title":"this Journal"},{"key":"S0022481220000304_r2","first-page":"672","article-title":"Decidability and undecidability of theories with a predicate for the primes","volume":"58","author":"Bateman","year":"1993","journal-title":"this Journal"},{"key":"S0022481220000304_r15","doi-asserted-by":"publisher","DOI":"10.1017\/CBO9780511566080"},{"key":"S0022481220000304_r8","doi-asserted-by":"publisher","DOI":"10.1090\/S0002-9947-1984-0742419-0"},{"key":"S0022481220000304_r14","unstructured":"[14] Tran, M. C. , Tame structures via multiplicative character sums on varieties over finite fields, arXiv preprint, 2017, arXiv:1704.03853."},{"key":"S0022481220000304_r1","doi-asserted-by":"publisher","DOI":"10.1126\/stke.3902007tw202"},{"key":"S0022481220000304_r4","doi-asserted-by":"publisher","DOI":"10.1016\/j.apal.2013.10.002"},{"key":"S0022481220000304_r6","first-page":"187","article-title":"There are no intermediate structures between the group of integers and Presburger arithmetic","volume":"83","author":"Conant","year":"2018","journal-title":"this Journal"},{"key":"S0022481220000304_r10","volume-title":"Simplicity Theory","volume":"53","author":"Kim","year":"2014"},{"key":"S0022481220000304_r3","first-page":"1115","article-title":"Quasi-o-minimal structures","volume":"65","author":"Belegradek","year":"2000","journal-title":"this Journal"},{"key":"S0022481220000304_r13","first-page":"515","article-title":"The Schnirelmann density of the squarefree integers","volume":"15","author":"Rogers","year":"1964","journal-title":"Proceedings of the American Mathematical Society"},{"key":"S0022481220000304_r12","doi-asserted-by":"publisher","DOI":"10.1093\/qmath\/os-18.1.178"},{"key":"S0022481220000304_r11","doi-asserted-by":"publisher","DOI":"10.1007\/s00153-013-0363-6"}],"container-title":["The Journal of Symbolic Logic"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/www.cambridge.org\/core\/services\/aop-cambridge-core\/content\/view\/S0022481220000304","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2022,1,12]],"date-time":"2022-01-12T03:33:49Z","timestamp":1641958429000},"score":1,"resource":{"primary":{"URL":"https:\/\/www.cambridge.org\/core\/product\/identifier\/S0022481220000304\/type\/journal_article"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2020,10,5]]},"references-count":15,"journal-issue":{"issue":"4","published-print":{"date-parts":[[2021,12]]}},"alternative-id":["S0022481220000304"],"URL":"https:\/\/doi.org\/10.1017\/jsl.2020.30","relation":{},"ISSN":["0022-4812","1943-5886"],"issn-type":[{"value":"0022-4812","type":"print"},{"value":"1943-5886","type":"electronic"}],"subject":[],"published":{"date-parts":[[2020,10,5]]},"assertion":[{"value":"\u00a9 Association for Symbolic Logic 2020","name":"copyright","label":"Copyright","group":{"name":"copyright_and_licensing","label":"Copyright and Licensing"}}]}}