On the Hausdorff dimension of ultrametric subsets in R ⁿ

For every ε < 0, any subset of R ⁿ with Hausdor dimension larger than (1 -ε)n must have ultrametric distortion larger than 1/(4ε). We prove the following theorem. Theorem 1. For every D < 1, every n ∈ N, and every norm || . || on R ⁿ, any subset S ⊂ R ⁿ having ultrametric distortion at most D must have Hausdor dimension at most {equation presented} An ultrametric space (X,ρ) is a metric space satisfying ρ(x;y) ≤ max{ρ(x,z),ρ(y,z)} for all x,y,z∈ X. The ultrametric distortion of a metric space (X,d), written cUM(X,d), is the infimum over D such that there exists an ultrametric ρ on X satisfying d(x,y) ≤ (x,y) ≤ D.d(x,y) for all x,y ∈ X. The Euclidean distortion c ₂(X,d) of (X,d) is defined similarly with respect to Hilbertian metrics over X. The diameter of a metric space (X,d) is given by {equation presented} The αHausdor content of a metric space (X; d) is defined as {equation presented} and the Hausdor dimension of X is dim _H(X) = inf{α<0: C ^α(X)=0}.