Invariant random compacts

TL;DR

This paper studies invariant random compact measures under group actions on compact metric spaces, introduces IC-rigidity concepts, and explores their implications for set largeness on the circle.

Contribution

It provides sufficient conditions for IC-rigidity, introduces weak IC-rigidity, and applies these concepts to problems in multiplicative number theory.

Findings

The Chacon system is weakly IC-rigid but not IC-rigid.

Conditions for IC-rigidity are established and examples are given.

Results on multiplicative largeness of dilated sets on the circle are proved.

Abstract

For a compact metric space $X$ with a group $G$ acting on it continuously, an invariant random compact is a Borel probability measure on the space of nonempty compact subsets of $X$ that is invariant under the action of $G$. The action is IC-rigid if, with respect to every invariant random compact, every compact set is almost surely either finite or $X$. We give sufficient conditions for an action to be IC-rigid, and show there are natural examples of such actions. We further consider a notion of weak IC-rigidity, and prove that the Chacon system is weakly IC-rigid but not IC-rigid. As an application, we prove results concerning multiplicative largeness of dilations of sets on the circle.