Interior of certain sums and continuous images of very thin Cantor sets

2024/10/02 by Jung, Yeonwook, Lai, Chun-Kit · 2 citations
#Classical Analysis and ODEs (math.CA) #Dynamical Systems (math.DS) #FOS: Mathematics #Metric Geometry (math.MG)

paper · doi:10.48550/arxiv.2410.01267

Abstract

We show that for all Cantor set K₁ on \mathbb R^d, it is always possible to find another Cantor set K₂ so that the sum g(K₁)+ K₂ (where g is a C¹ local diffeomorphism) has non-empty interior, and the existence of the interior is robust under small perturbation of the mapping. More generally, we can also show that the image set H(α, K₁,K₂), where H is some C¹ function on \mathbb R^N×\mathbb R^d×\mathbb R^d with non-vanishing Jacobian, have non-empty interior for α all in an open ball of \mathbb R^N. This result allows us to show that all Cantor sets are not topologically universal using C¹ local diffeomorphism, proving a stronger version of the topological Erdős similarity conjecture. Moreover, we are also able to construct a Cantor set of dimension d on \mathbb R^2d, whose distance set has an interior.

Cited by

• Interior of distance trees over thin Cantor sets
• Triangles in the Plane and arithmetic progressions in thick compact subsets of ℝ^d

Related

• Interiors of continuous images of the middle-third Cantor set
• Sums of two homogeneous Cantor sets
• Interior of distance trees over thin Cantor sets
• On a topological Erdős similarity problem
• Full measure universality for Cantor Sets