{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,8,23]],"date-time":"2026-08-23T17:37:23Z","timestamp":1787506643872,"version":"build-2736575974"},"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>\n                  <jats:p>\n                    We show that, for a constant-degree algebraic curve\n                    <jats:italic>\u03b3<\/jats:italic>\n                    in \u211d\n                    <jats:sup>\n                      <jats:italic>D<\/jats:italic>\n                    <\/jats:sup>\n                    , every set of\n                    <jats:italic>n<\/jats:italic>\n                    points on\n                    <jats:italic>\u03b3<\/jats:italic>\n                    spans at least \u03a9(\n                    <jats:italic>n<\/jats:italic>\n                    <jats:sup>4\/3<\/jats:sup>\n                    ) distinct distances, unless\n                    <jats:italic>\u03b3<\/jats:italic>\n                    is an\n                    <jats:italic>algebraic helix<\/jats:italic>\n                    , in the sense of Charalambides [2]. This improves the earlier bound \u03a9(\n                    <jats:italic>n<\/jats:italic>\n                    <jats:sup>5\/4<\/jats:sup>\n                    ) of Charalambides [2].\n                  <\/jats:p>\n                  <jats:p>\n                    We also show that, for every set\n                    <jats:italic>P<\/jats:italic>\n                    of\n                    <jats:italic>n<\/jats:italic>\n                    points that lie on a\n                    <jats:italic>d<\/jats:italic>\n                    -dimensional constant-degree algebraic variety\n                    <jats:italic>V<\/jats:italic>\n                    in \u211d\n                    <jats:sup>\n                      <jats:italic>D<\/jats:italic>\n                    <\/jats:sup>\n                    , there exists a subset\n                    <jats:italic>S<\/jats:italic>\n                    \u2282\n                    <jats:italic>P<\/jats:italic>\n                    of size at least \u03a9(\n                    <jats:italic>n<\/jats:italic>\n                    <jats:sup>\n                      4\/(9+12(\n                      <jats:italic>d<\/jats:italic>\n                      \u22121))\n                    <\/jats:sup>\n                    ), such that\n                    <jats:italic>S<\/jats:italic>\n                    spans\n                    <jats:inline-formula>\n                      <jats:alternatives>\n                        <jats:inline-graphic xmlns:xlink=\"http:\/\/www.w3.org\/1999\/xlink\" mime-subtype=\"png\" mimetype=\"image\" xlink:href=\"S096354832000022X_inline1.png\"\/>\n                        <jats:tex-math>$\\left({\\begin{array}{&amp;#x002A;{20}{c}} {|S|} \\\\ 2 \\\\\\end{array}} \\right)$<\/jats:tex-math>\n                      <\/jats:alternatives>\n                    <\/jats:inline-formula>\n                    distinct distances. This improves the earlier bound of \u03a9(\n                    <jats:italic>n<\/jats:italic>\n                    <jats:sup>\n                      1\/(3\n                      <jats:italic>d<\/jats:italic>\n                      )\n                    <\/jats:sup>\n                    ) of Conlon, Fox, Gasarch, Harris, Ulrich and Zbarsky [4].\n                  <\/jats:p>\n                  <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":"2020-07-16T02:27:02Z","timestamp":1594866422000},"page":"650-663","update-policy":"https:\/\/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":[],"role":[{"vocabulary":"crossref","role":"author"}]}],"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":"2020-09-14T08:06:25Z","timestamp":1600070785000},"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"}}]}}