Incidences of not-too-degenerate hyperplanes

Abstract

We present a multi-dimensional generalization of the Szemerédi-Trotter Theorem, and give a sharp bound on the number of incidences of points and not-too-degenerate hyperplanes in three-or higher-dimensional Euclidean spaces. We call a hyperplane not-too-degenerate if at most a constant portion of its incident points lie in a lower dimensional affine subspace.

Geometric incidence theorems via Fourier analysis

“…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.…”

Incidence Theorems for Pseudoflats

“…Another line of work concerns point-surface, especially pointhyperplane, incidences in higher dimensions; see [15] for an excellent survey. Recently, Elekes and Tóth [10] obtained a sharp Szemerédi-Trotter type bound for point-hyperplane incidences in R n , assuming a certain nondegeneracy condition; this bound was refined further by Solymosi and Tóth [16] under the additional assumption that the point set in question is homogeneous (see below).…”

“…We then use it to prove the main result of our paper, namely an analogous incidence theorem for a class of 2-dimensional surfaces in R 3 (pseudoplanes). For 2-dimensional planes in R 3 , our bound differs from that of [10], [16] only by the additional logarithmic factors; on the other hand, our non-degeneracy assumption is weaker than that of [10], [16]. Definition 1.1.…”

“…It is well known (cf. [10], Section 2) that bounds on the number of k-rich curves or surfaces are equivalent to the more standard formulation in terms of a bound on the total number of incidences between the point set and the objects in question. Essentially ) on the number of incidences between M type r pseudoplanes and N well-distributed points in R 3 .…”

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.…”