Interior of distance trees over thin Cantor sets

TL;DR

This paper demonstrates that for certain Cantor sets in Euclidean space, the $n$-chain distance set has a nonempty interior, extending previous results to more general sets and higher dimensions without relying on the Newhouse gap lemma.

Contribution

It proves the existence of subsets within product Cantor sets whose pinned $n$-chain distance sets have nonempty interior, generalizing to arbitrary finite trees and higher dimensions.

Findings

Pinned $n$-chain distance sets of certain Cantor sets have nonempty interior.

Results apply to higher dimensions and more general maps.

Provides examples of Cantor sets with full Hausdorff dimension and nonempty interior distance 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 $$\Delta^n(E) = \left\{\left(\left|x^1-x^2\right|,\cdots, \left|x^{n}- x^{n+1}\right|\right)\in \mathbb{R}^n: x^i \in E, x^i\neq x^j \text{ for } i\neq j \right\}$$ 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 $\widetilde{K}\subset \mathbb{R}^d$ such that the pinned $n$-chain distance set of $K\times \widetilde{K}\subset \mathbb{R}^{2d}$ has nonempty interior, and hence, that $\Delta^n(K\times \widetilde{K})$ 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 $s\geq d$, $\dim_{\rm H}(E)= s$ and $\Delta_x^n(E)^\circ{}\neq \varnothing$ for some $x\in E$.