  In the setting of a doubling metric measure space $(X,d,\mu)$, we study
regularity of sets whose characteristic functions belong to the Besov class
$B^s_{1,1}(X)$. Following a result of Visintin in $\mathbb{R}^n$, we provide a
sufficient condition for membership in $B^s_{1,1}(X)$ given in terms of the
upper Minkowski codimension of the regularized boundary of the set. We also
show that if the characteristic function of a set belongs to $B^s_{1,1}(X)$,
then its measure theoretic boundary has codimension $s$ Hausdorff measure zero.
To the best of our knowledge, this result is new even in the Euclidean setting.
By studying certain fat Cantor sets, we provide examples illustrating that the
converses of these results do not hold in general. In the doubling metric
measure space setting, we then consider minimizers of the nonlocal perimeter
functional $\mathcal{J}_\Omega^s$, extending the definition introduced by
Caffarelli, Roquejoffre, and Savin in $\mathbb{R}^n$, and prove existence,
uniform density, and porosity results for minimizers.
