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Improved Bounds for Incidences between Points and Circles
, 2013
"... We establish an improved upper bound for the number of incidences between m points and n arbitrary circles in three dimensions. The previous best known bound, originally established for the planar case and later extended to any dimension ≥ 2, is O ∗ m 2/3 n 2/3 +m 6/11 n 9/11 +m+n (where the O ∗ (·) ..."
Abstract

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We establish an improved upper bound for the number of incidences between m points and n arbitrary circles in three dimensions. The previous best known bound, originally established for the planar case and later extended to any dimension ≥ 2, is O ∗ m 2/3 n 2/3 +m 6/11 n 9/11 +m+n (where the O ∗ (·) notation hides subpolynomial factors). Since all the points and circles may lie on a common plane or sphere, it is impossible to improve the bound in R 3 without first improving it in the plane. Nevertheless, we show that if the set of circles is required to be “truly threedimensional”in the sense that no sphere or plane contains more than q of the circles, for some q ≪ n, then the bound can be improved to O ∗ ( m 3/7 n 6/7 +m 2/3 n 1/2 q 1/6 +m 6/11 n 15/22 q 3/22 +m+n). For various ranges of parameters (e.g., when m = Θ(n) and q = o(n 7/9)), this bound is smaller than the best known twodimensional worstcase lower bound Ω ∗ (m 2/3 n 2/3 +m+n). We present several extensions and applications of the new bound: (i) For the special case where all the circles have the same radius, we obtain the improved bound