Incidence Theorems for Pseudoflats

Abstract

We prove Pach-Sharir type incidence theorems for a class of curves in R n and surfaces in R 3 , which we call pseudoflats. In particular, our results apply to a wide class of generic irreducible real algebraic sets of bounded degree.

Geometric incidence theorems via Fourier analysis

“…We do not know of any combinatorial methods that would work in the latter case; simple extensions of the known methods are not sufficient, basically because it is difficult to control intersections of convex bodies in dimensions 3 and higher. There is some overlap between the results presented here and those of Laba and Solymosi [15], where combinatorial assumptions on the surfaces are made instead of analytic assumptions.…”

“…Such results have indeed been obtained by combinatorial methods. This was done, for instance, in [5], [25], [24] for spheres, in [6], [24] for k-dimensional affine subpaces, and in [15] for more algebraic affine surfaces.…”

“…The homogeneity assumption on the point set, formulated rigorously in Definition 2.1, means that the point set is uniformly distributed throughout a fixed cube and eliminates the lower-dimensional obstructions described earlier. This condition originated in analysis literature and was used in the context of incidence bounds, e.g., in [25], [24], [15].…”

An Incidence Theorem in Higher Dimensions

“…For instance, in [40] and [41] sharp incidence bounds between a homogeneous set of points and kdimensional subspaces were given. In [26] sharp point-pseudoplane incidence bounds were proved in R 3 . Similar bounds on point-surface incidences were proved for the non-homogenous case by Zahl [55].…”

From harmonic analysis to arithmetic combinatorics

“…The "right" bound on the number of plane-point incidences in R 3 appears to be O(m 3/4 n 3/4 ) (see e.g. Elekes-Tóth [49], Laba-Solymosi [120], and Solymosi-Tóth [163]), but the exact statement of the result depends on what additional assumptions are made. We omit the fine print.…”