{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,3,27]],"date-time":"2026-03-27T15:16:13Z","timestamp":1774624573161,"version":"3.50.1"},"reference-count":8,"publisher":"Cambridge University Press (CUP)","issue":"2","license":[{"start":{"date-parts":[[2014,3,12]],"date-time":"2014-03-12T00:00:00Z","timestamp":1394582400000},"content-version":"unspecified","delay-in-days":4667,"URL":"https:\/\/www.cambridge.org\/core\/terms"}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["J. symb. log."],"published-print":{"date-parts":[[2001,6]]},"abstract":"<jats:title>Abstract<\/jats:title><jats:p>Answers are given to two questions concerning the existence of some sparse subsets of <jats:italic>\u210b<\/jats:italic> = {0, 1 \u2026 ., <jats:italic>H<\/jats:italic> \u2013 1} \u2286 *\u2115. where <jats:italic>H<\/jats:italic> is a hyperfinite integer. In \u00a71. we answer a question of Kanovei by showing that for a given cut <jats:italic>U<\/jats:italic> in <jats:italic>\u210b<\/jats:italic>, there exists a countably determined set <jats:italic>X<\/jats:italic> \u2286 <jats:italic>\u210b<\/jats:italic> which contains exactly one element in each <jats:italic>U<\/jats:italic>-monad, if and only if <jats:italic>U<\/jats:italic> = <jats:italic>a<\/jats:italic> \u00b7 \u2115 for some <jats:italic>a<\/jats:italic> \u0404 <jats:italic>\u210b<\/jats:italic> \u2216 {0}. In \u00a72, we deal with a question of Keisler and Leth in [6]. We show that there is a cut <jats:italic>V<\/jats:italic> \u2286 <jats:italic>\u210b<\/jats:italic> such that for any cut <jats:italic>U<\/jats:italic>, (i) there exists a <jats:italic>U<\/jats:italic>-discrete set <jats:italic>X<\/jats:italic> \u2286 <jats:italic>\u210b<\/jats:italic> with <jats:italic>X<\/jats:italic> + <jats:italic>X<\/jats:italic> = <jats:italic>\u210b<\/jats:italic> (<jats:italic>mod H<\/jats:italic>) provided <jats:inline-graphic xmlns:xlink=\"http:\/\/www.w3.org\/1999\/xlink\" mime-subtype=\"gif\" xlink:type=\"simple\" xlink:href=\"S0022481200011117_inline01\"\/>, (ii) there does not exist any <jats:italic>U<\/jats:italic>-discrete set <jats:italic>X<\/jats:italic> \u2286 <jats:italic>\u210b<\/jats:italic> with <jats:italic>X<\/jats:italic> + <jats:italic>X<\/jats:italic> = <jats:italic>\u210b<\/jats:italic> (<jats:italic>mod H<\/jats:italic>) provided <jats:inline-graphic xmlns:xlink=\"http:\/\/www.w3.org\/1999\/xlink\" mime-subtype=\"gif\" xlink:type=\"simple\" xlink:href=\"S0022481200011117_inline02\"\/>. We obtain some partial results for the case <jats:italic>U<\/jats:italic> = <jats:italic>V<\/jats:italic>.<\/jats:p>","DOI":"10.2307\/2695055","type":"journal-article","created":{"date-parts":[[2006,5,6]],"date-time":"2006-05-06T14:05:52Z","timestamp":1146924352000},"page":"959-973","source":"Crossref","is-referenced-by-count":2,"title":["Existence of some sparse sets of nonstandard natural numbers"],"prefix":"10.1017","volume":"66","author":[{"given":"Renling","family":"Jin","sequence":"first","affiliation":[]}],"member":"56","published-online":{"date-parts":[[2014,3,12]]},"reference":[{"key":"S0022481200011117_ref006","first-page":"71\u2013102","volume":"56","author":"Keisler","year":"1991","journal-title":"Meager sets on the hyperfinite timeline"},{"key":"S0022481200011117_ref005","first-page":"1167\u20131180","volume":"54","author":"Keisler","year":"1989","journal-title":"Descriptive set theory over hyperfinite sets"},{"key":"S0022481200011117_ref004","doi-asserted-by":"publisher","DOI":"10.1090\/S0002-9939-1993-1126195-9"},{"key":"S0022481200011117_ref002","doi-asserted-by":"publisher","DOI":"10.4153\/CJM-1979-066-0"},{"key":"S0022481200011117_ref008","first-page":"906\u2013914","volume":"56","author":"\u017divaljevi\u0107","year":"1991","journal-title":"U-meager sets when the cofinality and the coinitiality of U are uncountable"},{"key":"S0022481200011117_ref007","first-page":"1","volume-title":"Nonstandard analysis and its applications","author":"Lindstr\u00f8m","year":"1988"},{"key":"S0022481200011117_ref001","volume-title":"Model Theory","author":"Chang","year":"1990"},{"key":"S0022481200011117_ref003","first-page":"1","volume-title":"Nonstandard analysis: Theory and applications","author":"Henson","year":"1997"}],"container-title":["Journal of Symbolic Logic"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/www.cambridge.org\/core\/services\/aop-cambridge-core\/content\/view\/S0022481200011117","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2019,5,7]],"date-time":"2019-05-07T20:21:03Z","timestamp":1557260463000},"score":1,"resource":{"primary":{"URL":"https:\/\/www.cambridge.org\/core\/product\/identifier\/S0022481200011117\/type\/journal_article"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2001,6]]},"references-count":8,"journal-issue":{"issue":"2","published-print":{"date-parts":[[2001,6]]}},"alternative-id":["S0022481200011117"],"URL":"https:\/\/doi.org\/10.2307\/2695055","relation":{},"ISSN":["0022-4812","1943-5886"],"issn-type":[{"value":"0022-4812","type":"print"},{"value":"1943-5886","type":"electronic"}],"subject":[],"published":{"date-parts":[[2001,6]]}}}