Thin subsets of topological groups

A subset A of a group G with the identity e is called thin if g A ∩ A and A g ∩ A are finite for each g ∈ G ∖ { e } . We prove that each countable totally bounded topological group has a dense thin subset and a thin subset X such that e is the unique limit point of X. On the other hand, for each thin subset T of a countable Abelian group G, we construct a non-discrete group topology on G in which T is closed and discrete.