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.