Trees of Dot Products in Thin Subsets of ${\mathbb{R}}^d$

A. Iosevich and K. Taylor showed that compact subsets of ${\mathbb{R}}^d$ with Hausdorff dimension greater than $(d+1)\mathrm{/}2$ contain trees with gaps in an open interval. Under the same dimensional threshold, we prove the analogous result where distance is replaced by the dot product. We additionally show that the gaps of embedded trees of dot products are prevalent in a set of positive Lebesgue measure, and for Ahlfors-David regular sets, the number of trees with given gaps agrees with the regular value theorem.