On (alpha)_n-regular sets

Abstract

Abstract. We define (α n )-regular sets in uniformly perfect metric spaces. This definition is quasisymmetrically invariant and the construction resembles generalized dyadic cubes in metric spaces. For these sets we then determine the necessary and sufficient conditions to be fat (or thin). In addition we discuss restrictions of doubling measures to these sets, and, in particular, give a sufficient condition to retain at least some of the restricted measures doubling on the set. Our main result generalizes and extends analogous results that were previously known to hold on the real line.

On thin carpets for doubling measures

“…Lemma 1.1 implies that every subset E of R d with Hausdorff dimension zero is thin for doubling measures (the same argument as mass distribution principle, see [5,Chapter 4]). Various examples of thin sets for doubling measures relate to the concept of porosity, see [9,21,23,24,29]. Doubling measures give zero weight to any smooth hyper-surface, see [25, p.40].…”