Weakly P-Small not P-Small Subsets in Groups

Answering a question of D.Dikranjan and I.Protasov we prove that each infinite group G contains a weakly P-small subset A ⊂ G which is not P-small. This means that for every n there are n pairwise disjoint translation copies of A in G but there is no infinitely many such disjoint copies. In this paper we answer a question of D.Dikranjan and I.Protasov concerning the relation between P-small and weakly P-small sets in groups. We recall that a subset A of a group G is called • P-small (or else small in the sense of Prodanov) if there is an infinite set B ⊂ G such that xA ∩ yA = ∅ for all distinct elements x, y ∈ B; • weakly P-small if for every n ∈ N the re is a subset Bn ⊂ G of size |Bn| ≥ n such that xA ∩ yA = ∅ for all distinct elements x, y ∈ Bn. It is clear that each P-small set is weakly P-small. The converse is not true: each infinite Abelian group G contains a weakly P-small subset A that is not P-small, see [1]. In this paper we prove a non-commutative version of this result. Main Theorem. Each infinite group G contains a weakly P-small subset A ⊂ G that is not P-small. The proof of this theorem relies on the existence of a subset B = B−1 of G with the following properties: (1) for every n ∈ N there is a subset Bn ⊂ G of size |Bn| ≥ n such that B−1 n Bn ⊂ B; (2) c1B∩c2B∩c3B is finite for any pairwise distinct points c1, c2, c3 ∈ G; (3) FBF 6= G for any subset F ⊂ G of size |F | < |G|. Here by |A| we denote the cardinality of a set A. Assuming for a moment that such a set B exists we shall construct a weakly P-small subset A ⊂ G which is not P-small.