Haar-smallest sets

We will be interested in the following notions of smallness: a subset A of an abelian Polish group X is called Haar-countable/Haar-finite/Haar-n if there are a Borel set B containing A and a copy C of the Cantor set in X such that (C+x)\cap B is countable/finite/of cardinality at most n, for all x from X. We will indicate connections of the considered families to some notions raised in the literature, study their basic properties, and give suitable examples distinguishing them.