Ville Suomala: Point configurations in random fractal sets

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.