Incidences between points and curves with almost two degrees of freedom

On rich lenses in planar arrangements of circles and related problems

“…Aside from the log factor, this bound generalizes the recent result of Sharir and Zlydenko [12] (see also Sharir, Solomon, and Zlydenko [10]) on incidences between so-called directed points and circles. A directed point is a pair (p, u) where p is a point in the plane and u is a direction, and (p, u) is incident to a circle c if p ∈ c and u is the direction of the tangent to c at p. The bound in [10,12] is O(m 3/5 n 3/5 + m + n) which is similar, albeit slightly sharper, than the bound in Theorem 10. The two setups are indeed related, as a directed point of degree at least two is a limiting case of a lens, and the resulting infinitesimal limit lenses are clearly pairwise non-overlapping.…”

“…The novelty in Theorem 10 is that lenses are 4-parameterizable, that is, each lens is specified by four real parameters (the coordinates of its vertices p, q), whereas directed points are 3-parameterizable. This makes the analysis in [10,12] inapplicable to the case of lenses, and yet the bound is more or less preserved.…”

Distinct distances in the complex plane

“…The crucial step is to establish the following incidence result for points and curves in R 3 . The following proof is closely modeled on the arguments in [29] by the second author, which are in turn based on arguments of Sharir and Zlydenko [19].…”

Sphere tangencies, line incidences, and Lie's line-sphere correspondence

“…where C is an absolute constant Proof. Our proof is closely modeled on the techniques of Sharir and Zlydenko from [21]. We will prove the result by induction on m; the base case for our induction will be when m ≤ m 0 , where m 0 is an absolute constant to be specified below.…”

“…Statements of this form appear frequently in the literature (see e.g. Theorem 1.9 from [21]), and follow directly from the machinery of Guth and the author developed in [12]. We will briefly outline the proof here.…”