Incidences with curves in ℝd

Abstract

We prove that the number of incidences between m points and n bounded-degree curves with k degrees of freedom in ℝd is I(P, C) = where the constant of proportionality depends on k, ε and d, for any ε > 0, provided that no j-dimensional surface of degree cj (k, d, ε), a constant parameter depending on k, d, j, and ε, contains more than qj input curves, and that the qj’s satisfy certain mild conditions. This bound generalizes a recent result of Sharir and Solomon [20] concerning point-line incidences in four dimensions (where d = 4 and k = 2), and partly generalizes a recent result of Guth [8] (as well as the earlier bound of Guth and Katz [10]) in three dimensions (Guth’s three-dimensional bound has a better dependency on q). It also improves a recent d-dimensional general incidence bound by Fox, Pach, Sheffer, Suk, and Zahl [7], in the special case of incidences with algebraic curves. Our results are also related to recent works by Dvir and Gopi [4] and by Hablicsek and Scherr [11] concerning rich lines in high-dimensional spaces.