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 ${A\subset X}$ is ${\varrho}$ -porous, then ${{\rm {dim}_p}(A)\le s-c\varrho^s}$ where dim_p 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 ${N\subset X}$ with μ(N) = 0 such that ${{\rm {dim}_p}(A)\le{\rm {dim}_p}(X)-c(\log \tfrac1\varrho)^{-1}\varrho^t}$ for all ${\varrho}$ -porous sets ${A \subset X{\setminus} N}$ . 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 < s and a t-regular set F such that ${A\subset F}$ .