2024/06/14 by Paige Bright, Bright, Paige, Caleb Marshall +1
Computer Science · Mathematics · #28A75 #28A78 #Classical Analysis and ODEs (math.CA) #Combinatorics (math.CO) #FOS: Mathematics #Mathematics and Applications #Matrix Theory and Algorithms #Metric Geometry (math.MG) #Rings, Modules, and Algebras
paper · pdf · doi:10.48550/arxiv.2406.10058
openalex publication_date 2024/06/14 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
We prove a version of the Erdős--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 dim (X ∖ H) = dim X for every k-dimensional hyperplane H ∈ 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 L(X) ≥ min \2 dim X, 2k\. If we instead have that X is not a subset of any k-plane, and 0lt;infH ∈ A(n,k) dim (X ∖ H) = t lt; dim X, we instead obtain the bound dim L(X) ≥ 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) ≥ max\dim X + t, min\2dim 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.