Lacunarity of self-similar and stochastically self-similar sets

Abstract

Let K be a self-similar set in Rd, of Hausdorff dimension D, and denote by |K(ε)| the d-dimensional Lebesgue measure of its ε -neighborhood. We study the limiting behavior of the quantity ε -(d-D)|K(ε)| as ε → 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 Kt is the zero-set of a real-valued stable process of index α ∈ (1,2], run up to time t, then ε -(1/α)|Kt(ε)| converges to a constant multiple of the local time at 0, simultaneously for all t≥ 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 E[|K(ε)|] 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.