Incidences Between Curves and Points on the Grid

Abstract

We derive an improved upper bound for the number of incidences between the n vertices of a uniform grid and m convex or concave curves, each pair of which intersect in at most s points, for some integer parameter s ≥ 1. For a square grid, our bound is O(n^{2/3}m^{2/3} + m^{1-1/(3s)} n^{(s+1)/3s} + m + n) . This improves a general bound of O(m n^{1/3}) on the number of incidences with respect to vertices of a grid and convex or concave curves.
For a rectangular grid, which fits inside a 1×K rectangle, for some integer K > 1 (which generally may depend on n), the bound also depends on how large K is. The precise result is stated in Theorem 2, but, roughly, we get the same bound as above when K is not too large. 
Our analysis competes with a celebrated result of Bombieri and Pila [E. Bombieri and J. Pila, 1989], which gives (usually) a sharper bound if we assume that the input curves are algebraic of constant degree and the input points are vertices of the square grid. However, the analysis in [E. Bombieri and J. Pila, 1989] strongly relies on these assumptions, and cannot be extended to handle the more general setup considered here.
As a main application, of independent interest, we present a variant of our technique for semi-algebraic range reporting on sets of points of "bounded spread" in the plane.