Let $G$ be an infinitely countable amenable group and let $K$ be a
finite subset of $G$ containing the unit and at least one more element.
A subset $C$ of $G$ is *$K$-separated* if the sets $Kc$ are disjoint as $c$ ranges
over $C$. A $K$-separated set $C$ is *maximal* if no proper superset of $C$ is
$K$-separated. The collection of the indicator functions of all maximal
$K$-separated sets is closed and shift-invariant. We call it the *$K$-shift*.
Last year, Benjy Weiss proposed to show that $K$-shifts are *universal*
in the sense that any free ergodic measure-preserving action whose
entropy is less than that of the $K$-shift is isomorphic to some invariant
measure on the $K$-shift. Together with Benjy, Mateusz Więcek and
Guohua Zhang we are trying to prove this. However an unexpected
obstacle has been revealed (exactly two days ago) which makes the
subject even more interesting.

Meeting ID: 852 4277 3200
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