Let $(X,\mu)$ be a standard probability space and $G\curvearrowright (X,\mu)$ be a measure-preserving action of a group $G$ on $X$. The general problem that we consider is to understand the structure of measurable tilings $F\oplus A=X$ of $X$ by a measurable tile $A\subseteq X$ shifted by a finite set $F\subseteq G$, thus the shifts $f\cdot A$, $f\in F$ partition $X$ up to null sets. The motivation comes from the theory of (paradoxical) equidecompositions and tilings in $\mathbb{R}^n$. After a summary of recent results that concern the spheres and tori, I will focus on the intersection of these cases, that is, the case of the circle. Using the structure theorem of Greenfeld and Tao for tilings of $\mathbb{Z}^d$, we show that measurable tilings of the circle can be reduced to tilings of finite cyclic groups.
This is a joint work with Conley and Pikhurko, and Greenfeld, Rozhon and Tao.
