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

Abstract: We show that $m$ points and $n$ two-dimensional algebraic surfaces in $\mathbb{R}<sup>4$ can have at most $$O(m<sup>{\frac{k}{2k-1}}n<sup>{\frac{2k-2}{2k-1}}+m+n)$$ incidences, provided that the algebraic surfaces behave like pseudoflats with $k$ degrees of freedom, and that $m\leq n<sup>{\frac{2k+2}{3k}}$. As a special case, we obtain a Szemer\'edi-Trotter type theorem for 2--planes in $\mathbb{R}<sup>4$, provided $m\leq n$ and the planes intersect transversely. As a further special case, we obtain a Szemer\'edi-Trotter type theorem for complex lines in $\mathbb{C}<sup>2$ with no restrictions on $m$ and $n$ (this theorem was originally proved by T\'oth using a different method). As a third special case, we obtain a Szemer\'edi-Trotter type theorem for complex unit circles in $\mathbb{C}<sup>2$. We obtain our results by combining several tools, including a two-level analogue of the discrete polynomial partitioning theorem and the crossing lemma.