Answer to Raczkowski's quest on convergent sequences of integers

We answer a question of Raczkowski on totally bounded Hausdorff group topologies on the integers with a convergent sequence (un). More specifically, we show that for fast growing and for slowly growing sequences (un) the asymptotic behaviour of the ratio un+1/un leads to rather specific properties of the topologies in question. (a)If un+1/un→∞, then there exists a totally bounded Hausdorff group topology of weight c on Z that makes (un) converging to 0. (b)If un+1/un is bounded, then every group topology as in (a) must be metrizable (i.e., has weight א0). We also show (under the assumption of Martin's Axiom) that there exists a precompact group topology τ on Z without non-trivial convergent sequences generated by a measure-zero subgroup H of T