A Continuum Erd{o}s-Beck Theorem

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