by Toth [@Toth] and then proved using different methods by Zahl [@Zahl]. For a finite set of points $V$, we denote by $\cL_r(V)$ the set of $r$-rich lines in $V$. The following is equivalent to Theorem-\[thm-Szemeredi-Trotter\] (but stated over $\C$).

\[thm-r-richlines-2dimensions\] Given a set $V$ of $n$ points in $\C^2$, for $r\ge 2$, $$|\cL_r(V)|\lesssim \frac{n^2}{r^3}+\frac{n}{r}.$$

Theorem \[thm-r-richlines-2dimensions\] is tight since a two dimensional square grid of $n$ points contains $\gtrsim n^2/r^3$ lines that are $r$-rich. We might then ask whether a $d$-dimensional grid $G_d = \{1,2,\ldots,h\}^d$, with $h \approx n^{1/d}$, has asymptotically the maximal number of $r$-rich lines among all $n$-point configurations that do not have a large low dimensional subset. It can be shown that for $r\ll_d n^{1/d}$,$$|\cL_r(G_d)| \approx_d \frac{n^2}{r^{d+1}},$$ where the subscript $d$ denotes that the constants in the inequalities may depend on the dimension $d$ [@SoVu04]. Clearly, we can obtain a larger number of rich lines in $\C^d$ if $V$ is a union of several low dimensional grids. For example, for some $\alpha\gg_d 1$ (We use $A \gg B$ to mean $A \geq C\cdot B$ for some sufficiently large constant $C$) and $d>\ell>1$, we can take a disjoint union of $r^{d-\ell}/\alpha$ $\ell$-dimensional grids $G_{\ell}$ of size $\alpha 