More pointsets with many rich lines
Gabriel Currier
TL;DR
Addresses sharpness in Szemerédi-Trotter-type incidence bounds for points and lines by constructing point sets from arbitrary number fields using incidence geometry rather than number theory. The main approach constructs $P = A_{n^{\alpha}}(\Lambda)\times A_{n^{1-\alpha}}(\Lambda)$ from a nice basis $\Lambda$, achieving $\Omega_\Lambda(n^2/r^3)$ $r$-rich lines for $r \le C'n^{\alpha}$. This unifies and extends prior Erdős– and Elekes-style constructions as well as Guth and Silier, and ME's GAP-based examples, while providing simpler proofs. The results advance the inverse Szemerédi-Trotter problem and yield sharper, scalable incidence constructions applicable to broader incidence-geometry questions.
Abstract
We present some new sharp constructions for the Szemerédi-Trotter theorem. These constructions generalize previous work of Erdős, Elekes, Sheffer and Silier, Guth and Silier, and the author. In the past, arguments showing the optimality of many of these constructions have required some elementary number theory and have been rather technical, thus limiting the scope of the results. We replace these number-theoretic arguments with purely incidence-geometric ones, allowing for simpler proofs and more general results.