Almost constant sequences and well-distribution in compact groups (Q1061243)

We study the class \(C_ u(G)\) \((C_ w(G)\) resp.) of sequences \((c_ n)\) in a compact group G, such that \((x_ n)\) being well-distributed implies \((c_ nx_ n)\) being well-distributed. Generalizing widely a result of \textit{G. M. Petersen} and \textit{A. Zame} [ibid. 73, 147-158 (1969; Zbl 0176.329)] we proof that \((c_ n)\in C_ u(G)\) if and only if \[ \lim_{N\to \infty}\sup_{k\geq 0}(1/N)\sum^{N+k}_{n=k+1}d(c_ n,c_{n+1})=0 \] holds for all bounded left-invariant pseudometrics d that generate the topology of G. \(C_ u(G)\) may also be described as the closure (in a certain topology of \(G^ N)\) of all sequences which are constant on blocks whose lengths tend to infinity in an appropriate way. Analogous results apply to \(C_ w(G)\). Furthermore, extensions to non-compact groups are given.