Pablo Shmerkin visited the University of St Andrews from 26 February to 1 March to work with the fractal and dynamical systems group.  He also gave an analysis seminar on “Multifractal structure of Bernoulli convolutions” and a pure mathematics colloquium on “On sets containing circles/squares centered at every point“.  The abstract for the colloqium follows:
Let A be a subset of the plane with the property that, for each point of the plane, A contains a circle centered at that point. A deep classical result due independently to Marstrand and to Bourgain says that A must have positive Lebesgue measure.
I will review the history of the circle problem and tell about some recent developments on the similar problem where circles are replaced by (boundaries of) squares. Unlike the circle case, it was known that a set containing a square centered at every point may have zero Lebesgue measure. The question we study is: how small can A be? As we will see, the answer surprisingly depends on the notion of  “smallness” used. The talk is based on joint work with T. Keleti and D. Nagy.