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Fractal Percolation, Porosity, and Dimension

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Abstract

We study the porosity properties of fractal percolation sets \(E\subset \mathbb {R}^d\). Among other things, for all \(0<\varepsilon <\tfrac{1}{2}\), we obtain dimension bounds for the set of exceptional points where the upper porosity of E is less than \(\tfrac{1}{2}-\varepsilon \), or the lower porosity is larger than \(\varepsilon \). Our method works also for inhomogeneous fractal percolation and more general random sets whose offspring distribution gives rise to a Galton–Watson process.

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Notes

  1. The amount of selected sub-cubes is iid, but the way these sub-cubes are distributed inside the parent cube is basically free. Their distributions can be different for different parent cubes, and all kinds of dependencies are allowed.

  2. Recall that \(\mu \) is the natural measure defined in (1.2).

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Acknowledgments

CC and ER acknowledge the support of the Vilho, Yrjö, and Kalle Väisälä foundation. CC and VS acknowledge support from the Centre of Excellence in Analysis and Dynamics Research funded by the Academy of Finland.

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Authors and Affiliations

  1. Department of Mathematical Sciences, University of Oulu, P.O. Box 3000, 90014, Oulu, Finland

    Changhao Chen & Ville Suomala

  2. Department of Mathematics and Statistics, University of Jyvaskyla, P.O. Box 35 (MaD), 40014, University of Jyvaskyla, Finland

    Tuomo Ojala & Eino Rossi

Authors
  1. Changhao Chen
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  2. Tuomo Ojala
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  3. Eino Rossi
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  4. Ville Suomala
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Correspondence to Tuomo Ojala.

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Chen, C., Ojala, T., Rossi, E. et al. Fractal Percolation, Porosity, and Dimension. J Theor Probab 30, 1471–1498 (2017). https://doi.org/10.1007/s10959-016-0680-x

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  • DOI: https://doi.org/10.1007/s10959-016-0680-x

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