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

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

2012-03-20

arxiv.org/abs/1203.4600v1

Mathematics

Combinatorics

45 pages, 1 figure

Scientific paper

We show that under suitable non-degeneracy conditions, $m$ points and $n$ 2--dimensional algebraic surfaces in $\mathbb{R}^4$ satisfying certain "pseudoflat" requirements can have at most $O(m^{2/3}n^{2/3} + m + n)$ incidences, provided that $m\leq n^{2-\epsilon}$ for any $\epsilon>0$ (where the implicit constant in the above bound depends on $\epsilon$), or $m\geq n^2$. As a special case, we obtain the Szemer\'edi-Trotter theorem for 2--planes in $\mathbb{R}^4,$ again provided $m\leq n^{2-\epsilon}$ or $m\geq n^2$. As a further special case we recover the Szemer\'edi-Trotter theorem for complex lines in $\mathbb{C}^2$ with no restrictions on $m$ and $n$ (this theorem was originally proved by T\'oth using a different method). As a second special case, we obtain the Szemer\'edi-Trotter theorem for complex unit circles in $\mathbb{C}^2$, which has applications to the complex unit distance problem. We obtain our results by combining the discrete polynomial ham sandwich theorem with the crossing number inequality.

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