A Proto Inverse Szemerédi-Trotter Theorem

A point-line incidence is a point-line pair such that the point is on the line. The Szemer\'edi-Trotter theorem says the number of point-line incidences for $n$ (distinct) points and lines in R^2 is tightly upperbounded by O(n^{4/3}). We advance the inverse problem: we geometrically characterize `sharp' examples which saturate the bound by proving the existence of a nice cell decomposition we call the \textit{two bush cell decomposition}. The proof crucially relies on the crossing number inequality from graph theory and has a traditional analysis flavor. 

Our two bush cell decomposition also holds in the analogous point-unit circle incidence problem. This constitutes an important step towards obtaining an $\epsilon$ improvement in the unit-distance problem. (Ongoing work with Nets Katz)

No background required, all welcome!