Arithmetic on self-similar sets

2019/08/01 by Bing Zhao, Zhao, Bing, Xiaomin Ren +5
Mathematics · #Advanced Topology and Set Theory #Dynamical Systems (math.DS) #FOS: Mathematics #Mathematical Dynamics and Fractals #Number Theory (math.NT)

paper · pdf · doi:10.48550/arxiv.1908.00224

openalex publication_date 2019/08/01 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

Abstract

Let K₁ and K₂ be two one-dimensional homogeneous self-similar sets. Let f be a continuous function defined on an open set U⊂ ℝ². Denote the continuous image of f by f_U(K₁,K₂)=\f(x,y):(x,y)∈ (K₁× K₂)∩ U\. In this paper we give an sufficient condition which guarantees that f_U(K₁,K₂) contains some interiors. Our result is different from Simon and Taylor's \cite[Proposition 2.9]ST as we do not need the condition that the multiplication of the thickness of K₁ and K₂ is strictly greater than 1. As a consequence, we give an application to the univoque sets in the setting of q-expansions.

Citations

• Resonance between Cantor sets

Related

• On continuous images of self-similar sets
• The interior of randomly perturbed self-similar sets on the line
• Self-similar sets and measures on the line
• Self-embeddings of homogeneous self-similar sets generated by three maps
• Thickness theorems with partial derivatives