On the Hausdorff dimension of ultrametric subsets in R^n

TL;DR

This paper establishes a quantitative relationship between the Hausdorff dimension of subsets in R^n and their ultrametric distortion, showing that higher dimension implies greater distortion.

Contribution

It provides a new lower bound on ultrametric distortion for subsets of R^n based on their Hausdorff dimension, linking geometric measure theory with metric geometry.

Findings

Subsets with Hausdorff dimension > (1-e)n have ultrametric distortion > 1/(4e)

Quantitative bounds connecting dimension and ultrametric distortion

Insight into the geometric structure of high-dimensional subsets

Abstract

For every e>0, any subset of R^n with Hausdorff dimension larger than (1-e)n must have ultrametric distortion larger than 1/(4e).