Packing dimension and Ahlfors regularity of porous sets in metric spaces
Let $X$ be a metric measure space with an $s$-regular measure $μ$. We prove that if $A\subset X$ is $\varrho$-porous, then $\dim_{\mathrm{p}}(A)\le s-c\varrho^s$ where $\dim_{\mathrm{p}}$ 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 $\dim_{\mathrm{p}}(A)\le\dim_{\mathrm{p}}(X)-c(\log\tfrac 1\varrho)^{-1}\varrho^t$ 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 $A\subset F$.