Non-Autonomous Cantor Sets from Decimal Expansions

Abstract

We construct irregular Cantor sets C_ξ ⊂ [0,1] using the decimal digits of a real number ξ (typically transcendental) as level-dependent removal ratios. Under the mild assumption that only finitely many digits are zero (satisfied, for example, by ξ = π − 3 and ξ = e − 2 after discarding finitely many initial digits), the resulting sets are compact, perfect, totally disconnected, and nowhere dense. The natural probability measure μ_ξ splits mass equally at each construction step. We prove that the Hausdorff dimension is given by dim_HC_ξ = lim inf_k→∞ log 2 / (−(1/k) ∑_j=1^k log a_j) where a_k = ½(1 − d_k/10), and that the associated Cantor function F_ξ(x) = μ_ξ([0,x]) is Hölder continuous with optimal exponent equal to this dimension. The construction extends to Cartesian products K_n = C_ξ1 × ⋯ × C_ξn; the Hausdorff dimension adds, and the product Cantor function F_n(x₁,…,x_n) = ∏_iF_ξi(x_i) is Hölder continuous with exponent min_i dim_HC_ξi. Connections to normal numbers (which yield dimension ≈ 0.874) and Liouville numbers (which can realise any dimension in (0,1]) are discussed, and several open problems are formulated.