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Transactions of the American Mathematical Society

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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
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Abstract:

Let $K$ be a self-similar set in $\mathbb R^d$, of Hausdorff dimension $D$, and denote by $\vert K(\epsilon)\vert$the $d$-dimensional Lebesgue measure of its $\epsilon$-neighborhood. We study the limiting behavior of the quantity $\epsilon^{-(d-D)}\vert K(\epsilon)\vert$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 $\epsilon^{-1/\alpha}\vert K_t(\epsilon)\vert$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 $\mathbb E [\vert K(\epsilon)\vert ]$ 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

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