Interior of pinned distance trees over thin Cantor sets

Abstract

We show that all Cantor sets in ${\mathbb{R}}^d$ can be accompanied by another Cantor set in ${\mathbb{R}}^d$ so that their product has a pinned tree distance set with nonempty interior. As a corollary, we construct Cantor sets of Hausdorff dimension $d/2$ in ${\mathbb{R}}^d$ for even $d$ that have a pinned tree distance set with nonempty interior. Our results generalize to the setting in which the Euclidean distance, $|x-y|$, is replaced by a general function, $\varphi (x,y)$, satisfying a mild derivative condition.