Lacunarity of self-similar and stochastically self-similar sets

Author:
Dimitris Gatzouras

Journal:
Trans. Amer. Math. Soc. **352** (2000), 1953-1983

MSC (2000):
Primary 28A80, 28A75, 60D05; Secondary 60K05, 60G52

DOI:
https://doi.org/10.1090/S0002-9947-99-02539-8

Published electronically:
December 10, 1999

MathSciNet review:
1694290

Full-text PDF

Abstract | References | Similar Articles | Additional Information

Let be a self-similar set in , of Hausdorff dimension , and denote by the -dimensional Lebesgue measure of its -neighborhood. We study the limiting behavior of the quantity as . 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 is the zero-set of a real-valued stable process of index , run up to time , then converges to a constant multiple of the local time at , simultaneously for all , 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 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.

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Additional Information

**Dimitris Gatzouras**

Affiliation:
Statistical Laboratory, University of Cambridge, 16 Mill Lane, Cambridge CB2 1SB, England, U.K.

Address at time of publication:
Department of Mathematics, University of Crete, 714 09 Iraklion, Crete, Greece

Email:
gatzoura@math.uch.gr

DOI:
https://doi.org/10.1090/S0002-9947-99-02539-8

Keywords:
Cantor set,
$\epsilon$-neighborhood,
Minkowski content,
branching random walk,
renewal theorem,
stable process

Received by editor(s):
September 8, 1998

Received by editor(s) in revised form:
March 4, 1999

Published electronically:
December 10, 1999

Article copyright:
© Copyright 2000
American Mathematical Society