{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2025,9,25]],"date-time":"2025-09-25T18:20:45Z","timestamp":1758824445613},"reference-count":51,"publisher":"Cambridge University Press (CUP)","issue":"3","license":[{"start":{"date-parts":[[2014,10,2]],"date-time":"2014-10-02T00:00:00Z","timestamp":1412208000000},"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":[[2015,5]]},"abstract":"<jats:p>We establish an improved upper bound for the number of incidences between<jats:italic>m<\/jats:italic>points and<jats:italic>n<\/jats:italic>circles in three dimensions. The previous best known bound, originally established for the planar case and later extended to any dimension \u2265 2, is<jats:italic>O<\/jats:italic>*(<jats:italic>m<\/jats:italic><jats:sup>2\/3<\/jats:sup><jats:italic>n<\/jats:italic><jats:sup>2\/3<\/jats:sup>+<jats:italic>m<\/jats:italic><jats:sup>6\/11<\/jats:sup><jats:italic>n<\/jats:italic><jats:sup>9\/11<\/jats:sup>+<jats:italic>m<\/jats:italic>+<jats:italic>n<\/jats:italic>), where the<jats:italic>O<\/jats:italic>*(\u22c5) notation hides polylogarithmic factors. Since all the points and circles may lie on a common plane (or sphere), it is impossible to improve the bound in \u211d<jats:sup>3<\/jats:sup>without first improving it in the plane.<\/jats:p><jats:p>Nevertheless, we show that if the set of circles is required to be \u2018truly three-dimensional\u2019 in the sense that no sphere or plane contains more than<jats:italic>q<\/jats:italic>of the circles, for some<jats:italic>q<\/jats:italic>\u226a<jats:italic>n<\/jats:italic>, then for any \u03f5 &gt; 0 the bound can be improved to<jats:disp-formula-group><jats:disp-formula><jats:alternatives><jats:graphic xmlns:xlink=\"http:\/\/www.w3.org\/1999\/xlink\" orientation=\"portrait\" mime-subtype=\"gif\" mimetype=\"image\" position=\"float\" xlink:type=\"simple\" xlink:href=\"S0963548314000534_eqnU1\" \/><jats:tex-math>\\[ O\\bigl(m^{3\/7+\\eps}n^{6\/7} + m^{2\/3+\\eps}n^{1\/2}q^{1\/6} + m^{6\/11+\\eps}n^{15\/22}q^{3\/22} + m + n\\bigr). \\]<\/jats:tex-math><\/jats:alternatives><\/jats:disp-formula><\/jats:disp-formula-group>For various ranges of parameters (<jats:italic>e.g.<\/jats:italic>, when<jats:italic>m<\/jats:italic>= \u0398(<jats:italic>n<\/jats:italic>) and<jats:italic>q<\/jats:italic>=<jats:italic>o<\/jats:italic>(<jats:italic>n<\/jats:italic><jats:sup>7\/9<\/jats:sup>)), this bound is smaller than the lower bound \u03a9*(<jats:italic>m<\/jats:italic><jats:sup>2\/3<\/jats:sup><jats:italic>n<\/jats:italic><jats:sup>2\/3<\/jats:sup>+<jats:italic>m<\/jats:italic>+<jats:italic>n<\/jats:italic>), which holds in two dimensions.<\/jats:p><jats:p>We present several extensions and applications of the new bound.<jats:list list-type=\"number\"><jats:list-item><jats:label>(i)<\/jats:label><jats:p>For the special case where all the circles have the same radius, we obtain the improved bound<jats:italic>O<\/jats:italic>(<jats:italic>m<\/jats:italic><jats:sup>5\/11+\u03f5<\/jats:sup><jats:italic>n<\/jats:italic><jats:sup>9\/11<\/jats:sup>+<jats:italic>m<\/jats:italic><jats:sup>2\/3+\u03f5<\/jats:sup><jats:italic>n<\/jats:italic><jats:sup>1\/2<\/jats:sup><jats:italic>q<\/jats:italic><jats:sup>1\/6<\/jats:sup>+<jats:italic>m<\/jats:italic>+<jats:italic>n<\/jats:italic>).<\/jats:p><\/jats:list-item><jats:list-item><jats:label>(ii)<\/jats:label><jats:p>We present an improved analysis that removes the subpolynomial factors from the bound when<jats:italic>m<\/jats:italic>=<jats:italic>O<\/jats:italic>(<jats:italic>n<\/jats:italic><jats:sup>3\/2\u2212\u03f5<\/jats:sup>) for any fixed \u03f5 &lt; 0.<\/jats:p><\/jats:list-item><jats:list-item><jats:label>(iii)<\/jats:label><jats:p>We use our results to obtain the improved bound<jats:italic>O<\/jats:italic>(<jats:italic>m<\/jats:italic><jats:sup>15\/7<\/jats:sup>) for the number of mutually similar triangles determined by any set of<jats:italic>m<\/jats:italic>points in \u211d<jats:sup>3<\/jats:sup>.<\/jats:p><\/jats:list-item><\/jats:list>Our result is obtained by applying the polynomial partitioning technique of Guth and Katz using a constant-degree partitioning polynomial (as was also recently used by Solymosi and Tao). We also rely on various additional tools from analytic, algebraic, and combinatorial geometry.<\/jats:p>","DOI":"10.1017\/s0963548314000534","type":"journal-article","created":{"date-parts":[[2014,10,2]],"date-time":"2014-10-02T10:30:32Z","timestamp":1412245832000},"page":"490-520","source":"Crossref","is-referenced-by-count":13,"title":["Improved Bounds for Incidences Between Points and Circles"],"prefix":"10.1017","volume":"24","author":[{"given":"MICHA","family":"SHARIR","sequence":"first","affiliation":[]},{"given":"ADAM","family":"SHEFFER","sequence":"additional","affiliation":[]},{"given":"JOSHUA","family":"ZAHL","sequence":"additional","affiliation":[]}],"member":"56","published-online":{"date-parts":[[2014,10,2]]},"reference":[{"key":"S0963548314000534_ref15","doi-asserted-by":"publisher","DOI":"10.1007\/978-0-387-35651-8"},{"key":"S0963548314000534_ref13","doi-asserted-by":"publisher","DOI":"10.1007\/BF01955043"},{"key":"S0963548314000534_ref20","doi-asserted-by":"publisher","DOI":"10.1090\/mbk\/046"},{"key":"S0963548314000534_ref22","unstructured":"Guth L. 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