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

TL;DR

This paper extends previous geometric results by demonstrating that sets in Euclidean space with sufficient Hausdorff dimension contain complex tree structures based on dot products, with implications for measure and regularity.

Contribution

It introduces the first results on trees of dot products in thin sets, establishing their prevalence and measure-theoretic properties under dimensional and regularity conditions.

Findings

Sets with Hausdorff dimension > (d+1)/2 contain trees of dot products.

Gaps in embedded trees are prevalent in positive measure sets.

Number of trees with specified gaps matches regular value theorem predictions.

Abstract

A. Iosevich and K. Taylor showed that compact subsets of $\mathbb R^d$ with Hausdorff dimension greater than $(d+1)/2$ 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.