Iterated Point-Line Configurations Grow Doubly-Exponentially

TL;DR

This paper investigates the iterative process of adding intersection points from initial four points in the plane, demonstrating that the number of points grows doubly-exponentially and providing bounds using combinatorial geometry techniques.

Contribution

It establishes doubly exponential bounds on the growth of points in iterated point-line configurations, extending previous density results with new quantitative estimates.

Findings

Number of points grows doubly-exponentially at each iteration

Provides upper and lower bounds on point counts

Uses a variant of the Szemerédi-Trotter Theorem for analysis

Abstract

Begin with a set of four points in the real plane in general position. Add to this collection the intersection of all lines through pairs of these points. Iterate. Ismailescu and Radoi\v{c}i\'{c} (2003) showed that the limiting set is dense in the plane. We give doubly exponential upper and lower bounds on the number of points at each stage. The proof employs a variant of the Szemer\'edi-Trotter Theorem and an analysis of the ``minimum degree'' of the growing configuration.