Rich line bound ($L_{\ge k}=O(n^2/k^3+n/k)$)

For $n$ distinct points in the real plane, $L_{\ge k}$ counts lines containing at least $k\ge 2$ points. The Szemerédi–Trotter theorem applied to these lines gives the bound by absorbing its additive line-count term; bounded $k$ follows by counting point pairs. Consequently, summing occupancies of lines with at least $k$ points by dyadic ranges gives $O(n^2/k^2+n\log n)$.