A fractal-like configuration of point-line pairs for the minimal distance problem

TL;DR

This paper constructs a fractal-like configuration of point-line pairs within the unit square, ensuring minimal distances between points and lines grow polynomially with the number of pairs, surpassing trivial solutions.

Contribution

It introduces a novel fractal-like arrangement of points and lines that achieves better minimal distance bounds than previous trivial constructions.

Findings

Constructs point-line configurations with polynomially large minimal distances

Demonstrates the existence of configurations for all natural numbers n

Improves upon trivial polynomial bounds in the minimal distance problem

Abstract

We show that for every $n \in \mathbb N$ there is a collection of points $p_1, \ldots, p_n$ and lines $\ell_1, \ldots, \ell_n$ in the unit square such that for any $i$ we have $p_i \in \ell_i$ and the distance from $p_i$ to any other line $\ell_j$ is at least $c n^{\gamma-1}$ for some universal constants $c, \gamma>0$. This is better than a trivial construction by a polynomial factor.