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

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 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.