Interior of pinned distance trees over thin Cantor sets

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, \phi(x,y) , satisfying a mild derivative condition.