{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2025,10,17]],"date-time":"2025-10-17T19:34:36Z","timestamp":1760729676662},"reference-count":9,"publisher":"Cambridge University Press (CUP)","issue":"3","license":[{"start":{"date-parts":[[2014,3,12]],"date-time":"2014-03-12T00:00:00Z","timestamp":1394582400000},"content-version":"unspecified","delay-in-days":3479,"URL":"https:\/\/www.cambridge.org\/core\/terms"}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["J. symb. log."],"published-print":{"date-parts":[[2004,9]]},"abstract":"<jats:p>Let <jats:inline-graphic xmlns:xlink=\"http:\/\/www.w3.org\/1999\/xlink\" mime-subtype=\"gif\" xlink:type=\"simple\" xlink:href=\"S0022481200007672_inline1\" \/> be an o-minimal expansion of the ordered field of real numbers <jats:inline-graphic xmlns:xlink=\"http:\/\/www.w3.org\/1999\/xlink\" mime-subtype=\"gif\" xlink:type=\"simple\" xlink:href=\"S0022481200007672_inline2\" \/>, and let <jats:italic>S<\/jats:italic> be an <jats:inline-graphic xmlns:xlink=\"http:\/\/www.w3.org\/1999\/xlink\" mime-subtype=\"gif\" xlink:type=\"simple\" xlink:href=\"S0022481200007672_inline1\" \/>-definable subset (parameters allowed unless otherwise stated) of \u211d<jats:sup><jats:italic>n<\/jats:italic><\/jats:sup>. In this note I investigate questions concerning the distribution of points on <jats:italic>S<\/jats:italic> with <jats:italic>integer<\/jats:italic> coordinates. My main theorem gives an estimate which, though probably far from best possible, at least shows that the o-minimal assumption does have diophantine consequences. This is, perhaps, surprising in view of the flexibility that we now seem to have in constructing o-minimal expansions of <jats:inline-graphic xmlns:xlink=\"http:\/\/www.w3.org\/1999\/xlink\" mime-subtype=\"gif\" xlink:type=\"simple\" xlink:href=\"S0022481200007672_inline2\" \/> (see, e. g. [7], [8], [9]).<\/jats:p>","DOI":"10.2178\/jsl\/1096901771","type":"journal-article","created":{"date-parts":[[2005,3,2]],"date-time":"2005-03-02T16:44:39Z","timestamp":1109781879000},"page":"851-861","source":"Crossref","is-referenced-by-count":6,"title":["Diophantine properties of sets definable in o-minimal structures"],"prefix":"10.1017","volume":"69","author":[{"given":"A. J.","family":"Wilkie","sequence":"first","affiliation":[],"role":[{"role":"author","vocabulary":"crossref"}]}],"member":"56","published-online":{"date-parts":[[2014,3,12]]},"reference":[{"key":"S0022481200007672_ref009","doi-asserted-by":"publisher","DOI":"10.1007\/s000290050052"},{"key":"S0022481200007672_ref008","doi-asserted-by":"publisher","DOI":"10.1515\/crll.1999.508.189"},{"key":"S0022481200007672_ref007","doi-asserted-by":"publisher","DOI":"10.1090\/S0894-0347-03-00427-2"},{"key":"S0022481200007672_ref006","doi-asserted-by":"publisher","DOI":"10.1007\/b76882"},{"key":"S0022481200007672_ref005","first-page":"276","volume-title":"The Mathematical Gazette","author":"Halberstam","year":"1974"},{"key":"S0022481200007672_ref004","doi-asserted-by":"publisher","DOI":"10.1017\/CBO9780511525919"},{"key":"S0022481200007672_ref003","doi-asserted-by":"publisher","DOI":"10.1090\/S0273-0979-1986-15468-6"},{"key":"S0022481200007672_ref002","doi-asserted-by":"publisher","DOI":"10.2307\/1971463"},{"key":"S0022481200007672_ref001","doi-asserted-by":"publisher","DOI":"10.1017\/CBO9780511565977"}],"container-title":["Journal of Symbolic Logic"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/www.cambridge.org\/core\/services\/aop-cambridge-core\/content\/view\/S0022481200007672","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2019,5,6]],"date-time":"2019-05-06T16:26:59Z","timestamp":1557160019000},"score":1,"resource":{"primary":{"URL":"https:\/\/www.cambridge.org\/core\/product\/identifier\/S0022481200007672\/type\/journal_article"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2004,9]]},"references-count":9,"journal-issue":{"issue":"3","published-print":{"date-parts":[[2004,9]]}},"alternative-id":["S0022481200007672"],"URL":"https:\/\/doi.org\/10.2178\/jsl\/1096901771","relation":{},"ISSN":["0022-4812","1943-5886"],"issn-type":[{"value":"0022-4812","type":"print"},{"value":"1943-5886","type":"electronic"}],"subject":[],"published":{"date-parts":[[2004,9]]}}}