Amenability, proximality and higher-order syndeticity

Abstract We show that the universal minimal proximal flow and strongly of a discrete group can be realized as Stone spaces translation-invariant Boolean algebras subsets satisfying higher-order notion syndeticity. establish algebraic, combinatorial topological dynamical characterizations these we use to obtain new necessary sufficient conditions for strong amenability amenability. also characterize dense orbit sets, answering question Glasner, Tsankov, Weiss Zucker.