Lacunarity of self-similar and stochastically self-similar sets

Abstract:

Let $K$ be a self-similar set in $\mathbb R^d$, of Hausdorff dimension $D$, and denote by $|K(\epsilon )|$ the $d$-dimensional Lebesgue measure of its $\epsilon$-neighborhood. We study the limiting behavior of the quantity $\epsilon ^{-(d-D)}|K(\epsilon )|$ as $\epsilon \rightarrow 0$. It turns out that this quantity does not have a limit in many interesting cases, including the usual ternary Cantor set and the Sierpinski carpet. We also study the above asymptotics for stochastically self-similar sets. The latter results then apply to zero-sets of stable bridges, which are stochastically self-similar (in the sense of the present paper), and then, more generally, to level-sets of stable processes. Specifically, it follows that, if $K_t$ is the zero-set of a real-valued stable process of index $\alpha \in (1,2]$, run up to time $t$, then $\epsilon ^{-1/\alpha }| K_t(\epsilon )|$ converges to a constant multiple of the local time at $0$, simultaneously for all $t\geq 0$, on a set of probability one. The asymptotics for deterministic sets are obtained via the renewal theorem. The renewal theorem also yields asymptotics for the mean $\mathbb E [|K(\epsilon )| ]$ in the random case, while the almost sure asymptotics in this case are obtained via an analogue of the renewal theorem for branching random walks.