Packing dimension and Ahlfors regularity of porous sets in metric spaces

Abstract

Let X be a metric measure space with an s-regular measure μ. We prove that if μ is ρ{variant}-porous, then dimp (A) ≤ s - cρ{variant}s where dimp is the packing dimension and c is a positive constant which depends on s and the structure constants of μ. This is an analogue of a well known asymptotically sharp result in Euclidean spaces. We illustrate by an example that the corresponding result is not valid if μ is a doubling measure. However, in the doubling case we find a fixed N ⊂ X with μ(N) = 0 such that dimp(X) - c(log 1/ρ{variant})-1 ρ{variant}t for all ρ{variant}-porous sets A ⊂ X\N. Here c and t are constants which depend on the structure constant of μ. Finally, we characterize uniformly porous sets in complete s-regular metric spaces in terms of regular sets by verifying that A is uniformly porous if and only if there is t