2020/08/19 by Kan Jiang, Jiang, Kan
Mathematics · #Advanced Topology and Set Theory #Dynamical Systems (math.DS) #FOS: Mathematics #Functional Equations Stability Results #Mathematical Dynamics and Fractals #Number Theory (math.NT)
paper · pdf · doi:10.48550/arxiv.2008.08229
openalex publication_date 2020/08/19 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
Let C1 and C2 be two Cantor sets with convex hull [0,1]. Newhouse proved if τ(C1)⋅ τ(C2)≥ 1, then the arithmetic sum C1+C2 is an interval, where τ(Ci), 1≤ i≤ 2 denotes the thickness of Ci. In this paper, we generalize this thickness theorem as follows. Let Ki⊂ ℝ, i=1,⋯, d, be some Cantor sets (perfect and nowhere dense) with convex hull [0,1]. Suppose f(x1,⋯, xd-1,z)∈ C1 is a continuous function defined on ℝd. Denote the continuous image of f by f(K1,⋯, Kd)=\f(x1, ⋯ xd-1,z):xi∈ Ki,z∈ Kd, 1≤ i≤ d-1\. If for any (x1, ⋯, xd-1,z)∈ [0,1]d, we have (τ(Ki))-1≤ |\dfrac∂xi f∂z f|≤ τ(Kd),1≤ i≤ d-1 then f(K1,⋯, Kd) is a closed interval. We give two applications. Firstly, we partially answer some questions posed by Takahashi. Secondly, we obtain various nonlinear identities, associated with the continued fractions with restricted partial quotients, which can represent real numbers.