Pablo Shmerkin (Buenos Aires): Patterns in random fractals

A popular line of research in recent years concerns finding conditions on a sparse set that ensure that the set contains geometric configurations such as arithmetic progressions or certain angles. I will present my joint work with Ville Suomala, in which we find the precise threshold that ensures the presence of many patterns of this kind in natural random fractals such as fractal percolation. I will also discuss the variant of this problem in which patterns are sought in sets of positive measure, for the natural measure on the random fractal.