Tunable Hausdorff Dimension of Cantor Sets and Product Fractals — E8 Intelligence Research

FINDING: The Hausdorff dimension of Cantor sets is a tunable parameter (any α∈[0,1]), and products of such sets (e.g., C×C) yield dimension 1 while retaining fractal measure anomalies; self-similarity is not guaranteed for all fractals, and Jarník-type Diophantine approximation extends to self-similar sets with Ahlfors regular measures. | MATH: Hausdorff dimension dim_H(C_α) = α for any α∈[0,1]; dim_H(C×C) = dim_H(C) + dim_H(C) = 2α (for product sets, additive under product); for 4-corner Cantor set E=C×C with dim_H(C)=1/2 ⇒ dim_H(E)=1, but its 1-dimensional Hausdorff measure is 0 or ∞ (anomalous); Jarník theorem: for τ>1/d, dim_H(W_d(ψ_τ,θ)) = (d+1)/(τ+1) − 1, generalized to self-similar sets with δ-Ahlfors regular measure μ (δ = dim_H(K)). | CONNECTION: The tunable α includes α = log_φ(2) ≈ 0.618? No — log_2(φ) ≈ 0.694, but note φ=1.618 appears in the Fibonacci code (self-similar binary tree) — the Cantor set with α = log_2(φ) has dimension ≈0.694, which is not a classical harmonic r Author: Andrew Stewart Caldin, Independent Researcher, UK. Part of the E8 Intelligence Research series. Platform: e8intelligence.com