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We investigate Hartman functions on a topological group G. Recall that (ι,C) is a group compactification of G if C is a compact group, ι: G → C is a continuous group homomorphism and ι(G) ⊆ C is dense. A bounded function f: G → ℂ is a Hartman function if there exists a group compactification (ι,C) and F: C → ℂ such that f = F∘ι and F is Riemann integrable, i.e. the set of discontinuities of F is a null set with respect to the Haar measure. In particular, we determine how large a compactification...

Let $T:x\mapsto x+g$ be an ergodic translation on the compact group $C$ and $M\subseteq C$ a continuity set, i.e. a subset with topological boundary of Haar measure 0. An infinite binary sequence $\mathbf{a}:\mathbb{Z}\mapsto \{0,1\}$ defined by $\mathbf{a}\left(k\right)=1$ if $T^k\left(0_C\right)\in M$ and $\mathbf{a}\left(k\right)=0$ otherwise, is called a Hartman sequence. This paper studies the growth rate of $P_{\mathbf{a}}\left(n\right)$, where $P_{\mathbf{a}}\left(n\right)$ denotes the number of binary words of length $n\in \mathbb{N}$ occurring in $\mathbf{a}$. The growth rate is always subexponential and this result is optimal. If $T$ is an ergodic translation $x\mapsto x+\alpha$ $(\alpha =({\alpha }_1,...,{\alpha }_s))$ on ${𝕋}^s$ and $M$ is a box with side lengths...

Let X be a metrizable one-dimensional continuum. We describe the fundamental group of X as a subgroup of its Čech homotopy group. In particular, the elements of the Čech homotopy group are represented by sequences of words. Among these sequences the elements of the fundamental group are characterized by a simple stabilization condition. This description of the fundamental group is used to give a new algebro-combinatorial proof of a result due to Eda on continuity properties of homomorphisms from...