A Continuum Erdős-Beck Theorem

Abstract

We prove a version of the Erdős--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 $$\dim \mathcal{L}(X)\ge \min \{2\dim X,2k\}.$$ If we instead have that $X$ is not a subset of any $k$-plane, and $$0<\underset{H\in \mathcal{A}(n,k)}{\inf }\dim (X\setminus H)=t<\dim X,$$ we instead obtain the bound $$\dim \mathcal{L}(X)\ge \dim X+t.$$ 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.

15 pages, 3 figures