A Szemeredi-Trotter type theorem in ${\mathbb{R}}^4$

Abstract

We show that $m$ points and $n$ two-dimensional algebraic surfaces in ${\mathbb{R}}^4$ can have at most $O(m^{\frac{k}{2k-1}}{n}^{\frac{2k-2}{2k-1}}+m+n)$ incidences, provided that the algebraic surfaces behave like pseudoflats with $k$ degrees of freedom, and that $m\le n^{\frac{2k+2}{3k}}$. As a special case, we obtain a Szemerédi-Trotter type theorem for 2--planes in ${\mathbb{R}}^4$, provided $m\le n$ and the planes intersect transversely. As a further special case, we obtain a Szemerédi-Trotter type theorem for complex lines in ${\mathbb{C}}^2$ with no restrictions on $m$ and $n$ (this theorem was originally proved by Tóth using a different method). As a third special case, we obtain a Szemerédi-Trotter type theorem for complex unit circles in ${\mathbb{C}}^2$. We obtain our results by combining several tools, including a two-level analogue of the discrete polynomial partitioning theorem and the crossing lemma.

50 pages. V3: final version. To appear in Discrete and Computational Geometry

References in corpus (1)

• Improved bounds for incidences between points and circles

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