Incidences with Curves in ℝ^d

Abstract

We prove that the number of incidences between m points and n bounded-degree curves with k degrees of freedom in R^d is I(P, C) = O (mdk-d+1/^k +ε_n^(dk)−d−+1/dk-d + ∑^(d-1)/_(j=2) m/jk−j+1+/k ε_n d(j−1)(k−1)/(d−1)(jk−j+1) q_j /(d−j)(k−1) (d−1)(jk−j+1) +m + n), where the constant of proportionality depends on k, ε and d, for any ε > 0, provided that no j-dimensional surface of degree c_j (k, d, ε), a constant parameter depending on k, d, j, and ε, contains more than q_j input curves, and that the q_j '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.

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