New characterizations of Muckenhoupt Ap distance weights for p > 1

We characterize the collection of sets $E\subset R^n$ for which there exists $\theta \in R\setminus \left\{0\right\}$ such that the distance weight $w\left(x\right)=\mathrm{dist}{(x,E)}^{\theta }$ belongs to the Muckenhoupt class $A_p$, where $p>1$. These sets exhibit a certain balance between the small-scale and large-scale pores that constitute their complement—a property we show to be more general than the so-called weak porosity condition, which in turn, and according to recent results, characterizes the sets with associated distance weights in the $A_1$ case. Furthermore, we verify the agreement between this new characterization and the properties of known examples of distance weights, that are either $A_p$ weights or merely doubling weights, by means of a probabilistic approach that may be of interest by itself.