A Continuum Erd\H{o}s-Beck Theorem

TL;DR

This paper extends the Erd ext{"o}s--Beck Theorem to fractal sets in all dimensions, providing new lower bounds on the dimension of line sets containing points of the fractal, and introduces the concept of the trapping number.

Contribution

It generalizes the Erd ext{"o}s--Beck Theorem to fractal sets, introduces the trapping number, and proposes a conjecture for the exact dimension of line sets.

Findings

Established lower bounds for the dimension of line sets in fractal sets.

Introduced the trapping number to refine bounds on line set dimensions.

Proposed a conjecture for the exact dimension of line sets in fractals.

Abstract

We prove a version of the Erd\H{o}s--Beck Theorem from discrete geometry for fractal sets in all dimensions. More precisely, let $X\subset \mathbb{R}^n$ Borel and $k \in [0, n-1]$ be an integer. Let $\dim (X \setminus H) = \dim X$ for every $k$-dimensional hyperplane $H \in \mathcal{A}(n,k)$, and let $\mathcal L(X)$ be the set of lines that contain at least two distinct points of $X$. Then, a recent result of Ren shows $$ \dim \mathcal{L}(X) \geq \min \{2 \dim X, 2k\}. $$ If we instead have that $X$ is not a subset of any $k$-plane, and $$ 0<\inf_{H \in \mathcal{A}(n,k)} \dim (X \setminus H) = t < \dim X, $$ we instead obtain the bound $$ \dim \mathcal{L}(X) \geq \dim X + t. $$ We then strengthen this lower bound by introducing the notion of the "trapping number" of a set, $T(X)$, and obtain \[ \dim \mathcal L(X) \geq \max\{\dim X + t, \min\{2\dim X, 2(T(X)-1)\}\}, \] as consequence of our main result and of Ren's result in $\mathbb{R}^n$. Finally, we introduce a conjectured equality for the dimension of the line set $\mathcal{L}(X)$, which would in particular imply our results if proven to be true.