Deterministic functions on amenable semigroups and a generalization of the Kamae-Weiss theorem on normality preservation

Abstract

A classical Kamae-Weiss theorem states that an increasing sequence $(n_i)_{i\in\mathbb N}$ of positive lower density is \emph{normality preserving}, i.e. has the property that for any normal binary sequence $(b_n)_{n\in\mathbb N}$, the sequence $(b_{n_i})_{i\in\mathbb N}$ is normal, if and only if $(n_i)_{i\in\mathbb N}$ is a deterministic sequence. Given a countable cancellative amenable semigroup $G$, and a F\o lner sequence $\mathcal F=(F_n)_{n\in\mathbb N}$ in $G$, we introduce the notions of normality preservation, determinism and subexponential complexity for subsets of $G$ with respect to $\mathcal F$, and show that for sets of positive lower $\mathcal F$-density these three notions are equivalent. The proof utilizes the apparatus of the theory of tilings of amenable groups and the notion of tile-entropy. We also prove that under a natural assumption on $\mathcal F$, positive lower $\mathcal F$-density follows from normality preservation. Finally, we provide numerous examples of normality preserving sets in various semigroups