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

TL;DR

The paper addresses the minimal-distance problem for point-line pairs in the unit square by constructing point-line configurations with a distance lower bound that improves upon trivial constructions. The authors introduce a two-stage approach: a probabilistic base bound that achieves $n(\delta) \ge c_1\delta^{-1}\log(1/\delta)$, followed by a self-affine fractal amplification using affine rescalings to boost the minimal distance and enlarge the configuration size. This yields constants $c,\gamma>0$ with $n(\delta) \ge c\,\delta^{-1-\gamma}$, establishing a fractal-type configuration that achieves $d(p_i,\ell_j) \ge c\,n^{\gamma-1}$ for all $i\neq j$ and all $n$. The work extends to higher dimensions and suggests a new constructive paradigm for related problems, though it does not imply improvements for Heilbronn-type problems, it indicates potential for stronger lower bounds via self-similar constructions.

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 $c n^{γ-1}$ for some universal constants $c, γ>0$. This is better than a trivial construction by a polynomial factor.

Theorems & Definitions (7)

• Theorem 1.1
• Theorem 2.1
• Lemma 2.1
• Lemma 2.2
• proof : Proof of Theorem \ref{['thm']}
• proof : Proof of Lemma \ref{['lem1']}
• proof : Proof of Lemma \ref{['lem2']}