{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2025,7,20]],"date-time":"2025-07-20T03:26:30Z","timestamp":1752981990756},"reference-count":16,"publisher":"Cambridge University Press (CUP)","issue":"5","license":[{"start":{"date-parts":[[2020,7,16]],"date-time":"2020-07-16T00:00:00Z","timestamp":1594857600000},"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":[[2020,9]]},"abstract":"<jats:title>Abstract<\/jats:title><jats:p>We show that, for a constant-degree algebraic curve <jats:italic>\u03b3<\/jats:italic> in \u211d<jats:sup><jats:italic>D<\/jats:italic><\/jats:sup>, every set of <jats:italic>n<\/jats:italic> points on <jats:italic>\u03b3<\/jats:italic> spans at least \u03a9(<jats:italic>n<\/jats:italic><jats:sup>4\/3<\/jats:sup>) distinct distances, unless <jats:italic>\u03b3<\/jats:italic> is an <jats:italic>algebraic helix<\/jats:italic>, in the sense of Charalambides [2]. This improves the earlier bound \u03a9(<jats:italic>n<\/jats:italic><jats:sup>5\/4<\/jats:sup>) of Charalambides [2].<\/jats:p><jats:p>We also show that, for every set <jats:italic>P<\/jats:italic> of <jats:italic>n<\/jats:italic> points that lie on a <jats:italic>d<\/jats:italic>-dimensional constant-degree algebraic variety <jats:italic>V<\/jats:italic> in \u211d<jats:sup><jats:italic>D<\/jats:italic><\/jats:sup>, there exists a subset <jats:italic>S<\/jats:italic> \u2282 <jats:italic>P<\/jats:italic> of size at least \u03a9(<jats:italic>n<\/jats:italic><jats:sup>4\/(9+12(<jats:italic>d<\/jats:italic>\u22121))<\/jats:sup>), such that <jats:italic>S<\/jats:italic> spans <jats:inline-formula><jats:alternatives><jats:inline-graphic xmlns:xlink=\"http:\/\/www.w3.org\/1999\/xlink\" mime-subtype=\"png\" mimetype=\"image\" xlink:href=\"S096354832000022X_inline1.png\" \/><jats:tex-math>\n$\\left({\\begin{array}{&amp;#x002A;{20}{c}} {|S|} \\\\ 2 \\\\\\end{array}} \\right)$\n<\/jats:tex-math><\/jats:alternatives><\/jats:inline-formula> distinct distances. This improves the earlier bound of \u03a9(<jats:italic>n<\/jats:italic><jats:sup>1\/(3<jats:italic>d<\/jats:italic>)<\/jats:sup>) of Conlon, Fox, Gasarch, Harris, Ulrich and Zbarsky [4].<\/jats:p><jats:p>Both results are consequences of a common technical tool.<\/jats:p>","DOI":"10.1017\/s096354832000022x","type":"journal-article","created":{"date-parts":[[2020,7,16]],"date-time":"2020-07-16T06:27:02Z","timestamp":1594880822000},"page":"650-663","update-policy":"http:\/\/dx.doi.org\/10.1017\/policypage","source":"Crossref","is-referenced-by-count":4,"title":["A note on distinct distances"],"prefix":"10.1017","volume":"29","author":[{"given":"Orit E.","family":"Raz","sequence":"first","affiliation":[]}],"member":"56","published-online":{"date-parts":[[2020,7,16]]},"reference":[{"key":"S096354832000022X_ref16","doi-asserted-by":"publisher","DOI":"10.4153\/CJM-1961-059-8"},{"key":"S096354832000022X_ref15","unstructured":"[15] Thiele, T. (1995) Geometric selection problems and hypergraphs. PhD thesis, Institut f\u00fcr Mathematik, Freie Universit\u00e4t Berlin."},{"key":"S096354832000022X_ref13","doi-asserted-by":"publisher","DOI":"10.1215\/00127094-3674103"},{"key":"S096354832000022X_ref12","doi-asserted-by":"publisher","DOI":"10.1017\/S0963548316000225"},{"key":"S096354832000022X_ref11","doi-asserted-by":"publisher","DOI":"10.1007\/BF01895954"},{"key":"S096354832000022X_ref9","doi-asserted-by":"publisher","DOI":"10.1112\/jlms\/s1-16.4.212"},{"key":"S096354832000022X_ref7","doi-asserted-by":"publisher","DOI":"10.1007\/s00493-012-2505-6"},{"key":"S096354832000022X_ref6","doi-asserted-by":"publisher","DOI":"10.1006\/jcta.1999.2976"},{"key":"S096354832000022X_ref10","doi-asserted-by":"publisher","DOI":"10.4007\/annals.2015.181.1.2"},{"key":"S096354832000022X_ref5","doi-asserted-by":"publisher","DOI":"10.1080\/17476930902763801"},{"key":"S096354832000022X_ref2","doi-asserted-by":"publisher","DOI":"10.1007\/s00454-014-9586-5"},{"key":"S096354832000022X_ref1","volume-title":"Research Problems in Discrete Geometry","author":"Brass","year":"2005"},{"key":"S096354832000022X_ref4","doi-asserted-by":"publisher","DOI":"10.1137\/140954519"},{"key":"S096354832000022X_ref14","doi-asserted-by":"publisher","DOI":"10.1016\/j.jcta.2013.06.009"},{"key":"S096354832000022X_ref3","doi-asserted-by":"publisher","DOI":"10.1007\/s00022-013-0176-0"},{"key":"S096354832000022X_ref8","doi-asserted-by":"publisher","DOI":"10.1080\/00029890.1946.11991674"}],"container-title":["Combinatorics, Probability and Computing"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/www.cambridge.org\/core\/services\/aop-cambridge-core\/content\/view\/S096354832000022X","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2020,9,14]],"date-time":"2020-09-14T12:06:25Z","timestamp":1600085185000},"score":1,"resource":{"primary":{"URL":"https:\/\/www.cambridge.org\/core\/product\/identifier\/S096354832000022X\/type\/journal_article"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2020,7,16]]},"references-count":16,"journal-issue":{"issue":"5","published-print":{"date-parts":[[2020,9]]}},"alternative-id":["S096354832000022X"],"URL":"https:\/\/doi.org\/10.1017\/s096354832000022x","relation":{},"ISSN":["0963-5483","1469-2163"],"issn-type":[{"value":"0963-5483","type":"print"},{"value":"1469-2163","type":"electronic"}],"subject":[],"published":{"date-parts":[[2020,7,16]]},"assertion":[{"value":"\u00a9 The Author(s), 2020. Published by Cambridge University Press","name":"copyright","label":"Copyright","group":{"name":"copyright_and_licensing","label":"Copyright and Licensing"}}]}}