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 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.