DENSITIES, SUBMEASURES AND PARTITIONS OF GROUPS TARAS BANAKH, IGOR PROTASOV, SERGIY SLOBODIANIUK Abstract. In 1995 in Kourovka notebook the second author asked the following problem: is it true that for each partition G = A 1 ∪···∪ An of a group G there is a cell A i of the partition such that G = FA i A -1 i for some set F ⊂ G of cardinality |F |≤ n? In this paper we survey several partial solutions of this problem, in particular those involving certain canonical invariant densities and submeasures on groups. In particular, we show that for any partition G = A 1 ∪···∪ An of a group G there are cells A i , A j of the partition such that • G = FA j A -1 j for some finite set F ⊂ G of cardinality |F |≤ max 0<k≤n ∑ n-k p=0 k p ≤ n!; • G = F ·  x∈E xA i A -1 i x -1 for some finite sets F, E ⊂ G with |F |≤ n; • G = FA i A -1 i A i for some finite set F ⊂ G of cardinality |F |≤ n; • the set (A i A -1 i ) 4 n-1 is a subgroup of index ≤ n in G. The last three statements are derived from the corresponding density results. 1. Introduction In this paper we survey partial solutions to the following open problem posed by I.V.Protasov in 1995 in the Kourovka notebook [14, Problem 13.44]. Problem 1.1. Is it true that for any finite partition G = A 1 ∪···∪ A n of a group G there is a cell A i of the partition and a subset F ⊂ G of cardinality |F |≤ n such that G = FA i A −1 i ? In [14] it was observed that this problem has simple affirmative solution for amenable groups (see Theorem 4.3 below). Problem 1.1 is a partial case of its “idealized” G-space version. Let us recall that a G-space is a set X endowed with a left action α : G × X → X, α :(g,x) → gx, of a group G. Each group G will be considered as a G-space endowed with the left action α : G × G → G, α :(g,x) → gx. A non-empty family I of subsets of a set X is called a Boolean ideal if for any A, B ∈I and C ⊂ X we get A ∪ B ∈I and A ∩ C ∈I . A Boolean ideal I on a set X will be called trivial if it coincides with the Boolean ideal B(X) of all subsets of X. By [X] <ω we shall denote the Boolean ideal consisting of all finite subsets of X. A Boolean ideal I on a G-space X is called G-invariant if for any A ∈I and g ∈ G the shift gA of A belongs to the ideal I . By an ideal G-space we shall understand a pair (X, I ) consisting of a G-space X and a non-trivial G-invariant Boolean ideal I⊂B(X). For an ideal G-space (X, I ) and a subset A ⊂ X the set Δ I (A)= {x ∈ G : A ∩ xA / ∈ I} ⊂ G will be called the I -difference set of A. It is not empty if and only if A/ ∈I . For a non-empty subset A ⊂ G of a group G its covering number is defined as cov(A) = min{|F | : F ⊂ G, G = FA}. More generally, for a Boolean ideal J⊂B(G) on a group G and a non-empty subset A ⊂ G let cov J (A) = min{|F | : F ⊂ G, G \ FA ∈J} be the J -covering number of A. Observe that for the smallest Boolean ideal I = {∅} on a group G and a subset A ⊂ G the I -difference set Δ I (A) is equal to AA −1 . That is why Problem 1.1 is a partial case of the following more general Problem 1.2. Is it true that for any finite partition X = A 1 ∪···∪ A n of an ideal G-space (X, I ) some cell A i of the partition has • cov(Δ I (A i )) ≤ n? • cov J (Δ I (A i )) ≤ n for some non-trivial G-invariant Boolean ideal J on the acting group G? 1991 Mathematics Subject Classification. 05E15; 05D10; 28C10. Key words and phrases. partition of a group; density; submeasure; amenable group. 1