{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,3,11]],"date-time":"2026-03-11T22:26:10Z","timestamp":1773267970968,"version":"3.50.1"},"reference-count":24,"publisher":"Cambridge University Press (CUP)","issue":"4","license":[{"start":{"date-parts":[[2011,4,4]],"date-time":"2011-04-04T00:00:00Z","timestamp":1301875200000},"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":[[2011,7]]},"abstract":"<jats:p>We first describe a reduction from the problem of lower-bounding the number of distinct distances determined by a set <jats:italic>S<\/jats:italic> of <jats:italic>s<\/jats:italic> points in the plane to an incidence problem between points and a certain class of helices (or parabolas) in three dimensions. We offer conjectures involving the new set-up, but are still unable to fully resolve them.<\/jats:p><jats:p>Instead, we adapt the recent new algebraic analysis technique of Guth and Katz [9], as further developed by Elekes, Kaplan and Sharir [6], to obtain sharp bounds on the number of incidences between these helices or parabolas and points in \u211d<jats:sup>3<\/jats:sup>. Applying these bounds, we obtain, among several other results, the upper bound <jats:italic>O<\/jats:italic>(<jats:italic>s<\/jats:italic><jats:sup>3<\/jats:sup>) on the number of rotations (rigid motions) which map (at least) three points of <jats:italic>S<\/jats:italic> to three other points of <jats:italic>S<\/jats:italic>. In fact, we show that the number of such rotations which map at least <jats:italic>k<\/jats:italic> \u2265 3 points of <jats:italic>S<\/jats:italic> to <jats:italic>k<\/jats:italic> other points of <jats:italic>S<\/jats:italic> is close to <jats:italic>O<\/jats:italic>(<jats:italic>s<\/jats:italic><jats:sup>3<\/jats:sup>\/<jats:italic>k<\/jats:italic><jats:sup>12\/7<\/jats:sup>).<\/jats:p><jats:p>One of our unresolved conjectures is that this number is <jats:italic>O<\/jats:italic>(<jats:italic>s<\/jats:italic><jats:sup>3<\/jats:sup>\/<jats:italic>k<\/jats:italic><jats:sup>2<\/jats:sup>), for <jats:italic>k<\/jats:italic> \u2265 2. If true, it would imply the lower bound \u03a9(<jats:italic>s<\/jats:italic>\/log<jats:italic>s<\/jats:italic>) on the number of distinct distances in the plane.<\/jats:p>","DOI":"10.1017\/s0963548311000137","type":"journal-article","created":{"date-parts":[[2011,4,4]],"date-time":"2011-04-04T07:57:32Z","timestamp":1301903852000},"page":"571-608","source":"Crossref","is-referenced-by-count":33,"title":["Incidences in Three Dimensions and Distinct Distances in the Plane"],"prefix":"10.1017","volume":"20","author":[{"given":"GY\u00d6RGY","family":"ELEKES","sequence":"first","affiliation":[],"role":[{"role":"author","vocabulary":"crossref"}]},{"given":"MICHA","family":"SHARIR","sequence":"additional","affiliation":[],"role":[{"role":"author","vocabulary":"crossref"}]}],"member":"56","published-online":{"date-parts":[[2011,4,4]]},"reference":[{"key":"S0963548311000137_ref8","volume-title":"Modern Differential Geometry of Curves and Surfaces with Mathematica","author":"Gray","year":"1997"},{"key":"S0963548311000137_ref3","volume-title":"Research Problems in Discrete Geometry","author":"Brass","year":"2005"},{"key":"S0963548311000137_ref2","doi-asserted-by":"publisher","DOI":"10.1007\/s00454-004-1111-9"},{"key":"S0963548311000137_ref4","doi-asserted-by":"publisher","DOI":"10.1016\/0097-3165(84)90041-4"},{"key":"S0963548311000137_ref15","doi-asserted-by":"publisher","DOI":"10.1007\/978-1-4471-3696-5"},{"key":"S0963548311000137_ref22","doi-asserted-by":"publisher","DOI":"10.1017\/S0963548397002976"},{"key":"S0963548311000137_ref5","doi-asserted-by":"publisher","DOI":"10.1007\/BF02187820"},{"key":"S0963548311000137_ref9","doi-asserted-by":"publisher","DOI":"10.1016\/j.aim.2010.05.015"},{"key":"S0963548311000137_ref23","doi-asserted-by":"publisher","DOI":"10.1007\/BF02579194"},{"key":"S0963548311000137_ref10","unstructured":"[10] Guth L. and Katz N. H. On the Erd\u0151s distinct distances problem in the plane. arXiv:1011.4105."},{"key":"S0963548311000137_ref1","volume-title":"The Probabilistic Method","author":"Alon","year":"1992"},{"key":"S0963548311000137_ref16","doi-asserted-by":"publisher","DOI":"10.1137\/090763160"},{"key":"S0963548311000137_ref18","first-page":"75","article-title":"On distinct distances and incidences: Elekes's transformation and the new algebraic developments.","volume":"52","author":"Sharir","year":"2009","journal-title":"Annales Univ. Sci. 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