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

TL;DR

This paper demonstrates that sums and images of certain Cantor sets under smooth functions can have non-empty interior, leading to new insights on the structure and universality of Cantor sets and related conjectures.

Contribution

It proves that the sum of any Cantor set with another can have interior under smooth maps, and shows that all Cantor sets are not topologically universal, advancing the Erdős similarity conjecture.

Findings

Existence of Cantor sets with sums having interior

Construction of Cantor sets with interior distance sets

Robustness of interior property under perturbations

Abstract

We show that for all Cantor set $K_1$ on ${\mathbb R}^d$, it is always possible to find another Cantor set $K_2$ so that the sum $g(K_1)+ K_2$ (where $g$ is a $C^1$ 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 $H(\alpha, K_1,K_2)$, where $H$ is some $C^1$ function on ${\mathbb R}^N\times{\mathbb R}^d\times{\mathbb R}^d$ with non-vanishing Jacobian, have non-empty interior for $\alpha$ all in an open ball of ${\mathbb R}^N$. This result allows us to show that all Cantor sets are not topologically universal using $C^1$ local diffeomorphism, proving a stronger version of the topological Erd\H{o}s 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.