We obtain a new bound on the number of two-rich points spanned by an arrangement of low degree algebraic curves in \(\mathbb {R}^4\). Specifically, we show that an arrangement of n algebraic curves determines at most \(C_\varepsilon n^{4/3+3\varepsilon }\) two-rich points, provided at most \(n^{2/3-2\varepsilon }\) curves lie in any low degree hypersurface and at most \(n^{1/3-\varepsilon }\) curves lie in any low degree surface. This result follows from a structure theorem about arrangements of curves that determine many two-rich points.
['Larry Guth', 'Joshua Zahl']
Curves in $\mathbb{R}^4$ and two-rich points