Abstract

abstract:

We characterize the existence of certain geometric configurations in the fractal percolation limit set $A$ in terms of the almost sure dimension of $A$. Some examples of the configurations we study are: homothetic copies of finite sets, angles, distances, and volumes of simplices. In the spirit of relative Szemer\'{e}di theorems for random discrete sets, we also consider the corresponding problem for sets of positive $\nu$-measure, where $\nu$ is the natural measure on $A$. In both cases we identify the dimension threshold for each class of configurations. These results are obtained by investigating the intersections of the products of $m$ independent realizations of $A$ with transversal planes and, more generally, algebraic varieties, and extend some well-known features of independent percolation on trees to a setting with long-range dependencies.

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Additional Information

ISSN
1080-6377
Print ISSN
0002-9327
Pages
pp. 683-749
Launched on MUSE
2020-05-14
Open Access
No
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