Hausdorff dimension and doubling measures on metric spaces

Abstract

Vol′berg and Konyagin have proved that a compact metric space carries a nontrivial doubling measure if and only if it has finite uniform metric dimension. Their construction of doubling measures requires infinitely many adjustments. We give a simpler and more direct construction, and also prove that for any α > 0, the doubling measure may be chosen to have full measure on a set of Hausdorff dimension at most α. Let (X, ρ) be a compact metric space. Vol’berg and Konyagin proved in [VK] that (X, ρ) carries a nontrivial doubling measure μ (there exists Λ ≥ 1 so that μ(B(x, 2r)) ≤ Λμ(B(x, r)) for all x ∈ X and r > 0) if and only if (X, ρ) has finite uniform metric dimension (in each ball B(x, 2r), there exist at most N points with mutual distances at least r). Here B(x, r) = {y : ρ(x, y) 0, there exists a doubling measure on X that has full measure on a set of Hausdorff dimension at most α. Also we observe that a doubling measure may be concentrated on a countable set even when X is a set on the real line of positive length. Some ideas have been adapted from [FKP], [VK] and [T]. 1. Theorems and examples Assume, from now on, that (X, ρ) is a compact metric space of finite uniform metric dimension and that diam X < 1. For each k ≥ 0, let Sk = {xk,j : 1 ≤ j ≤ J(k)} be a maximal 10−k-net on X (points in Sk having mutual distances at least 10 −k, and points outside Sk having distances less than 10−k to Sk), satisfying S0 ⊆ S1 ⊆ · · · ⊆ Sk ⊆ Sk+1 ⊆ · · · . Note that S0 has only one point x0,1. For each k ≥ 0, let {Tk,j : 1 ≤ j ≤ J(k)} be a partition of Sk+1 satisfying Sk+1 ∩B(xk,j , 10−k/2) ⊆ Tk,j ⊆ Sk+1 ∩B(xk,j , 10−k). (1.1) Received by the editors October 24, 1996. 1991 Mathematics Subject Classification. Primary 28C15; Secondary 54E35, 54E45.