Patterns in random fractals - Shmerkin, Pablo Sebastian (2020)
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´edi theorems for random discrete sets, we also consider the corresponding problem for sets of positive ν-measure, where ν 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.