{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,5,13]],"date-time":"2026-05-13T21:43:30Z","timestamp":1778708610299,"version":"3.51.4"},"reference-count":9,"publisher":"Cambridge University Press (CUP)","issue":"4","license":[{"start":{"date-parts":[[2015,9,30]],"date-time":"2015-09-30T00:00:00Z","timestamp":1443571200000},"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":[[2016,7]]},"abstract":"<jats:p>Let <jats:italic>p<\/jats:italic><jats:sub>1<\/jats:sub>, <jats:italic>p<\/jats:italic><jats:sub>2<\/jats:sub>, <jats:italic>p<\/jats:italic><jats:sub>3<\/jats:sub> be three noncollinear points in the plane, and let <jats:italic>P<\/jats:italic> be a set of <jats:italic>n<\/jats:italic> other points in the plane. We show that the number of distinct distances between <jats:italic>p<\/jats:italic><jats:sub>1<\/jats:sub>, <jats:italic>p<\/jats:italic><jats:sub>2<\/jats:sub>, <jats:italic>p<\/jats:italic><jats:sub>3<\/jats:sub> and the points of <jats:italic>P<\/jats:italic> is \u03a9(<jats:italic>n<\/jats:italic><jats:sup>6\/11<\/jats:sup>), improving the lower bound \u03a9(<jats:italic>n<\/jats:italic><jats:sup>0.502<\/jats:sup>) of Elekes and Szab\u00f3 [4] (and considerably simplifying the analysis).<\/jats:p>","DOI":"10.1017\/s0963548315000231","type":"journal-article","created":{"date-parts":[[2015,9,29]],"date-time":"2015-09-29T23:06:33Z","timestamp":1443567993000},"page":"623-632","source":"Crossref","is-referenced-by-count":8,"title":["Distinct Distances from Three Points"],"prefix":"10.1017","volume":"25","author":[{"given":"MICHA","family":"SHARIR","sequence":"first","affiliation":[],"role":[{"role":"author","vocabulary":"crossref"}]},{"given":"J\u00d3ZSEF","family":"SOLYMOSI","sequence":"additional","affiliation":[],"role":[{"role":"author","vocabulary":"crossref"}]}],"member":"56","published-online":{"date-parts":[[2015,9,30]]},"reference":[{"key":"S0963548315000231_ref9","doi-asserted-by":"publisher","DOI":"10.1017\/S0963548397002976"},{"key":"S0963548315000231_ref7","doi-asserted-by":"crossref","unstructured":"Pach J. and de Zeeuw F. (2014) Distinct distances on algebraic curves in the plane. In Proc. 30th Symposium on Computational Geometry, pp. 549\u2013557. Also in arXiv:1308.0177.","DOI":"10.1145\/2582112.2582135"},{"key":"S0963548315000231_ref1","unstructured":"Elekes G. (2002) Sums versus products in number theory, algebra and Erd\u0151s geometry: A survey. In Paul Erd\u0151s and his Mathematics II, Vol. 11 of Bolyai Mathematical Society Studies, Budapest, pp. 241\u2013290."},{"key":"S0963548315000231_ref8","doi-asserted-by":"publisher","DOI":"10.1016\/j.jcta.2013.06.009"},{"key":"S0963548315000231_ref2","doi-asserted-by":"publisher","DOI":"10.1006\/jcta.1999.2976"},{"key":"S0963548315000231_ref6","doi-asserted-by":"publisher","DOI":"10.1017\/S0963548397003192"},{"key":"S0963548315000231_ref5","doi-asserted-by":"publisher","DOI":"10.1007\/BF02187746"},{"key":"S0963548315000231_ref4","doi-asserted-by":"publisher","DOI":"10.1007\/s00493-012-2505-6"},{"key":"S0963548315000231_ref3","doi-asserted-by":"publisher","DOI":"10.1017\/S0963548309990265"}],"container-title":["Combinatorics, Probability and Computing"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/www.cambridge.org\/core\/services\/aop-cambridge-core\/content\/view\/S0963548315000231","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2019,4,18]],"date-time":"2019-04-18T22:15:15Z","timestamp":1555625715000},"score":1,"resource":{"primary":{"URL":"https:\/\/www.cambridge.org\/core\/product\/identifier\/S0963548315000231\/type\/journal_article"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2015,9,30]]},"references-count":9,"journal-issue":{"issue":"4","published-print":{"date-parts":[[2016,7]]}},"alternative-id":["S0963548315000231"],"URL":"https:\/\/doi.org\/10.1017\/s0963548315000231","relation":{},"ISSN":["0963-5483","1469-2163"],"issn-type":[{"value":"0963-5483","type":"print"},{"value":"1469-2163","type":"electronic"}],"subject":[],"published":{"date-parts":[[2015,9,30]]}}}