Sublinear measures, Menger curvature, and Hausdorff dimension

Abstract

We study metric properties of sets which are characterized by their approximation properties by smooth curves at scales of positive density and their relationships with known problems in complex dynamics and analysis. Three direct applications towards removability for bounded analytic functions, optimal bounds for Traveling Salesman problem, and Hausdorff dimension of unfolding attractors are discussed in the paper.The main technical ingredient is a construction, for every continuum K, of a finite Radon measure of bounded Menger curvature such that on every ball B(x,r), x∈K, the measure is bounded by a universal constant multiple of rexp⁡(−g(x,r)), where g(x,r)≥0 is an explicit function. Theorem A is the main analytical result of the paper and provides a precise estimate on the growth of the function g(x,r) in terms of Jones' β numbers. Its dimension theory counterpart is stated as Theorems 1 and 2 and yields sharp estimates of the Hausdorff dimension of compacta in terms of asymptotic properties of log⁡g(x,r)/log⁡r for all x∈K except for a set of finite linear measure.