Visibility of non-self-similar sets

2025/05/09 by Yi Cai, Yang Yang, Cai, Yi +1
Engineering · #Artificial Immune Systems Applications #FOS: Mathematics #Number Theory (math.NT)

paper · pdf · doi:10.48550/arxiv.2505.05808

openalex publication_date 2025/05/09 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

Abstract

The visible problem is related to the arithmetic on the fractals. The visibility of self-similar set has been studied in the past. In this work, we investigate the visibility of non-self-similar sets. We begin by analyzing the structure of F²_λ, where F²_λ:=\setx²:x∈ F_λ and F_λ is the middle 1-2λ Cantor set, we show that it lacks self-similarity. Due to the nonlinear phenomena exhibited by F²_λ, we develop a different approach to characterize the visible set. %combining methods from fractal theory, numerical computation, and dynamical systems theory. Our results also reveal that the visible set may contain a closed interval within a large range of λ.

Citations

• On dimensions of visible parts of self-similar sets with finite rotation groups
• Visible part of dominated self-affine sets in the plane
• On continuous images of self-similar sets
• Multiplication on self-similar sets with overlaps
• Cantor set arithmetic
• On the arithmetic difference of middle Cantor sets

Related

• A P.C.F. Self-Similar Set with no Self-Similar Energy
• Lipschitz Equivalence of Self-Similar Sets: Algebraic and Geometric Properties
• Projections of self-similar and related fractals: a survey of recent developments
• Statistically self-similar fractal sets
• A general theory of self-similarity II: recognition