Interior of distance trees over thin Cantor sets

Abstract

It is known that if a compact set $E$ in ${\mathbb{R}}^d$ has Hausdorff dimension greater than $(d+1)/2$, then its $n$-chain distance set $$\hat{I}{”}^n(E)=\left\{\left(\left|x^1-x^2\right|,\cdots ,\left|x^n-x^{n+1}\right|\right)\in {\mathbb{R}}^n:x^i\in E,x^i\ne x^j\text{ for }i\ne j\right\}$$ has nonempty interior for any $n\in \mathbb{N}$. In this paper, we prove that for every Cantor set $K\subset {\mathbb{R}}^d$ and for every $n\in \mathbb{N}$, there exists $\tilde{K}\subset {\mathbb{R}}^d$ such that the pinned $n$-chain distance set of $K\times \tilde{K}\subset {\mathbb{R}}^{2d}$ has nonempty interior, and hence, that $\hat{I}{”}^n(K\times \tilde{K})$ has nonempty interior. Our results do not depend on the Newhouse gap lemma but rather on the containment lemma recently introduced by Jung and Lai. Our results generalize three-fold: to arbitrary finite trees, to higher dimensions, and to maps that have non-vanishing partials. As an application, we provide a class of examples of Cantor sets $E\subset {\mathbb{R}}^{2d}$ so that for any $s\ge d$, ${\dim }_{\mathrm{H}}(E)=s$ and $\hat{I}{”}_x^n(E{)}^{\circ }\ne \varnothing$ for some $x\in E$.

16 pages, 5 figures