Point configurations in random fractal sets

Fractal Geometry and Dynamics

19 September 15:00 - 15:50

Ville Suomala - University of Oulu

We present various results on the existence of patterns in random fractal sets. We focus on a canonical model, the fractal percolation. We characterize in terms of the dimension of the limit set $A$ the existence of geometric configurations in $A$ such as homothetic copies of all finite sets with a given number of elements, all angles and simples of all small volumes. In the spirit of relative Szemer\'{e}di theorems for random discrete sets, we also consider the corresponding problem for sets of positive measure (with respect to the natural measure on $A$). This talk is based on a joint work with Pablo Shmerkin.
Kenneth J. Falconer
University of St Andrews
Maarit Järvenpää
University of Oulu
Antti Kupiainen
University of Helsinki
Francois Ledrappier
University of Notre Dame
Pertti Mattila
University of Helsinki


Maarit Järvenpää


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