Title: Assouad dimension and regularity of microsets
Abstract: A microset of a compact set K is a type of tangent where the magnification need not be fixed at a single point. Whereas tangents can in general be very badly behaved, a key observation (due to Furstenberg) is that microsets can always be chosen to be a relatively 'nice' set. The goal of this talk is addressing the following question: how regular can microsets be? I will present some of the intuition and proofs behind known results concerning regularity and the relationship with the Assouad dimension, as well as some new related work concerning the lower dimension. We will then see some (counter)examples showing, for instance, that one cannot necessarily find any microsets with non-trivial Ahlfors-regular subsets. The new results are based on work in progress with Richárd Balka (Rényi) and Vilma Orgoványi (BME).
