A Szemerédi–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 $O(m^{{k}/({2k-1})}n^{({2k-2})/({2k-1})}+m+n)$ incidences, provided that the algebraic surfaces behave like pseudoflats with k degrees of freedom, and that $m\le n^{(2k+2)/3k}$. 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.