The number of translates of a closed nowhere dense set required to cover   a Polish group
  For a Polish group G let cov_G be the minimal number of translates of a fixed closed nowhere dense subset of G required to cover G. For many locally compact G this cardinal is known to be consistently larger than cov(meager) which is the smallest cardinality of a covering of the real line by meagre sets. It is shown that for several non-locally compact groups cov_G=cov(meager). For example the equality holds for the group of permutations of the integers, the additive group of a separable Banach space with an unconditional basis and the group of homeomorphisms of various compact spaces.   Most recent version at: www.math.wisc.edu/~miller 
Arnold W. Miller and Juris Steprans