The covering number of the difference sets in partitions of G-spaces and groups

We prove that for every finite partition G=A1∪⋯∪An\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$G=A_1\cup \cdots \cup A_n$$\end{document} of a group G there are a cell Ai\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$A_i$$\end{document} of the partition and a subset F⊂G\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$F\subset G$$\end{document} of cardinality |F|⩽n\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$|F|\leqslant n$$\end{document} such that G=FAiAi-1Ai\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$G=FA_iA_i^{-1}A_i$$\end{document}. A similar result is proved also for partitions of G-spaces. This gives two partial answers to a problem of Protasov posed in 1995.


Introduction
This paper was motivated by the following problem posed by Protasov in Kourovka Notebook [7]. Problem 1.1 (Protasov, 1995) Is it true that for any partition G = A 1 ∪ · · · ∪ A n of a group G some cell A i of the partition has cov(A i A −1 i ) n? Here for a non-empty subset A ⊂ G by cov(A) = min{|F| : F ⊂ G, G = F A} we denote the covering number of A.
In fact, Protasov's problem can be posed in a more general context of ideal Gspaces. Let us recall that a G-space is a set X endowed with an action G × X → X , (g, x) → gx, of a group G. An ideal G-space is a pair (X, I) consisting of a G-space X and a G-invariant Boolean ideal I ⊂ B(X ) in the Boolean algebra B(X ) of all subsets of X . A Boolean ideal on X is a proper non-empty subfamily I B(X ) such that for any A, B ∈ I any subset C ⊂ A ∪ B belongs to I. A Boolean ideal I is G-invariant if {g A : g ∈ G, A ∈ I} ⊂ I. A Boolean ideal I ⊂ B(G) on a group G will be called invariant if {x Ay : x, y ∈ G, A ∈ I} ⊂ I. By [X ] <ω and [X ] ω we denote the families of all finite and countable subsets of a set X , respectively. The family [X ] <ω (respectively [X ] ω ) is a Boolean ideal on X if X is infinite (respectively uncountable).
For a subset A ⊂ X of an ideal G-space (X, I) by For the empty subset we put cov J (∅) = ∞ and assume that ∞ is larger than any cardinal number.
Observe that for the left action of the group G on itself we get (A) = A A −1 for every subset A ⊂ G. That is why Problem 1.1 is a partial case of the following general problem. Problem 1.2 Is it true that for any partition X = A 1 ∪ · · · ∪ A n of an ideal G-space (X, I) some cell A i of the partition has cov( I (A i )) n?
This problem has an affirmative answer for G-spaces with amenable acting group G, see [2,Theorem 4.3]. The paper [2] gives a survey of available partial solutions of Protasov's Problems 1.1 and 1.2. Here we mention the following result of Banakh, Ravsky and Slobodianiuk [3]. Theorem 1.3 For any partition X = A 1 ∪ · · · ∪ A n of an ideal G-space (X, I) some cell A i of the partition has In this paper we shall give another two partial solutions to Protasov's Problems 1.1 and 1.2.

Corollary 1.5 For any partition X
Proof By Theorem 1.4, either cov( I (A i )) n for all cells A i or else there is a cell A i of the partition such that cov J ( I (A i )) < n for some G-invariant ideal J I (A i ) on X . In the first case we are done. In the second case we can find a F ⊂ G of cardinality |F| < n such that F · I (A i )= J G. It follows that for every x ∈ G the shift x I (A i ) does not belong to J and hence intersects the set For groups G (considered as G-spaces endowed with the left action of G on itself), we can prove a bit more.
In the first case we are done. In the second case, choose a finite subset F ⊂ G of cardinality |F| < n such that the set Taking into account that the ideal J appearing in Theorem 1.6 is G-invariant but not necessarily invariant, we can ask the following question. Problem 1.8 Is it true that for any partition G = A 1 ∪ · · · ∪ A n of a group G some cell A i of the partition has cov J (A i A −1 i ) n for some invariant Boolean ideal J on G?
2 Minimal measures on G-spaces Theorems 1.4 and 1.6 will be proved with help of minimal probability measures on X and right quasi-invariant idempotent measures on G.
For a G-space X by P(X ) we denote the (compact Hausdorff) space of all finitely additive probability measures on X . The action of the group G on X extends to an action of the convolution semigroup P(G) on P(X ): for two measures μ ∈ P(G) and ν ∈ P(X ) their convolution is defined as the measure μ * ν ∈ P(X ) assigning to each bounded function ϕ : X → R the real number The convolution map * : P(G)× P(X ) → P(X ) is right-continuous in the sense that for any fixed measure ν ∈ P(X ) the right shift P(G) → P(X ), μ → μ * ν, is continuous. This implies that the P(G)-orbit P(G) * ν = {μ * ν : μ ∈ P(G)} of ν coincides with the closure conv(G ·ν) of the convex hull of the G-orbit G ·ν of ν in P(X ).
A measure μ ∈ P(X ) will be called minimal if for any measure ν ∈ P(G) * μ we get P(G) * ν = P(G) * μ. Zorn's Lemma combined with the compactness of the orbits implies that the orbit P(G) * μ of each measure μ ∈ P(X ) contains a minimal measure.
It follows from Day's Fixed Point Theorem [8, 1.14] that for a G-space X with amenable acting group G each minimal measure μ on X is G-invariant, which implies that the set conv(G ·μ) coincides with the singleton {μ}.

Lemma 2.1
For any ideal G-space (X, I) the set P I (X ) contains some minimal probability measure.
Proof Let U be any ultrafilter on X , which contains the filter F = {F ⊂ X : X \ F ∈ I}. This ultrafilter U can be identified with the 2-valued measure μ U : It follows that μ U (A) = 0 for any subset A ∈ I. In the P(G)orbit P(G) * μ U choose any minimal measure μ = ν * μ U and observe that for every For a subset A of a group G put

Lemma 2.2 If a subset A of a group G has
Proof If cov(A −1 ) ω, then for every non-empty finite subset T ⊂ G we could find a point Then for the uniformly distributed measure μ T = 1/|T |· t∈T δ x −1 T t on the set x −1 T T we get μ T (AT ) = 0. By the compactness of the space P(G), the net (μ T ) T ∈[G] <ω has a limit point μ ∞ ∈ P(X ), which means that for every set B ⊂ G, finite subset denotes the set of finitely supported probability measures on G.
For a probability measure μ ∈ P(X ) on a G-space X and a subset A ⊂ X put μ(A) = sup x∈G μ(x A).

A density version of Theorem 1.4
In this section we shall prove the following density theorem, which will be used in the proof of Theorem 1.4 presented in the next section. Proof By the compactness of P(G) * μ = conv(G ·μ), there is a measure μ ∈ P(G) * μ ⊂ P I (X ) such that μ (A) = sup{ν(A) : ν ∈ P(G) * μ} = μ(A). We can replace the measure μ by μ and assume that μ(A) = μ(A). Choose a positive ε such that where r = max{n ∈ Z : n r } denotes the integer part of a real number r . Consider the set L = {x ∈ G : μ(x A) > μ(A) − ε} and choose a maximal subset F ⊂ L such that μ(x A ∩ y A) = 0 for any distinct points x, y ∈ F. The additivity of the measure μ implies that 1 x∈F μ(x A) > |F|(μ(A) − ε) and hence |F| 1/(μ(A) − ε) = 1/(μ(A)) 1/μ(A). By the maximality of F, for every x ∈ L there is y ∈ F such that μ(x A ∩ y A) > 0. Then x A ∩ y A / ∈ I and y −1 x ∈ I (A). It follows that x ∈ y · I (A) ⊂ F · I (A) and L ⊂ F · I (A).

Proof of Theorem 1.4
Let X = A 1 ∪ · · · ∪ A n be a partition of an ideal G-space (X, I). By Lemma 2.1, there exists a minimal probability measure μ ∈ P(X ) such that I ⊂ {A ∈ B(G) : μ(A) = 0}.
For every i ∈ {1, . . . , n} consider the number μ(A i ) = sup x∈G μ(x A i ) and observe that n i=1 μ(A i ) 1. There are two cases. Case 1. For every i ∈ {1, . . . , n}, μ(A i ) 1/n. In this case for every x ∈ G we get and hence μ(x A i ) = 1/n for every i ∈ {1, . . . , n}. For every i ∈ {1, . . . , n} fix a maximal subset F i ⊂ G such that μ(x A i ∩ y A i ) = 0 for any distinct points x, y ∈ F i . The additivity of the measure μ implies that 1 x∈F i μ(x A i ) |F i |/n and hence |F i | n. By the maximality of F i , for every x ∈ G there is a point y ∈ F i such that μ(x A i ∩ y A i ) > 0 and hence x A i ∩ y A i / ∈ I. The G-invariance of the ideal I implies that y −1 x ∈ I (A i ) and so x ∈ y · I (A i ) ⊂ F i · I (A i ). Finally, we get G = F i · I (A i ) and cov( I (A i )) |F i | n.

Applying idempotent quasi-invariant measures
In this section we develop a technique involving idempotent right quasi-invariant measures, which will be used in the proof of Theorem 1.6 presented in the next section.
A measure μ ∈ P(G) on a group G will be called right quasi-invariant if for any y ∈ G there is a positive constant c > 0 such that c ·μ(Ay) μ(A) for any subset A ⊂ G.
For an ideal G-space (X, I) and a measure μ ∈ P(X ) consider the set P I (G; μ) = λ ∈ P(G) : λ * δ g * μ ∈ P I (X ) for all g ∈ G and observe that it is closed and convex in the compact Hausdorff space P(G). (X, I) be an ideal G-space with countable acting group G. If for some measure μ ∈ P(X ) the set P I (G; μ) is not empty, then it contains a right quasi-invariant idempotent measure ν ∈ P I (G; μ).
So, (P I (G; μ)) ⊂ P I (G; μ) and, by the Schauder Fixed Point Theorem, the continuous map on the non-empty compact convex set P I (G; μ) ⊂ P(G) has a fixed point, which implies that the closed set S = {ν ∈ P I (G; μ) : ν * λ = ν} is not empty. It is easy to check that S is a subsemigroup of the convolution semigroup (P(G), * ). Being a compact right-topological semigroup, S contains an idempotent ν ∈ S ⊂ P I (G; μ) according to the Ellis Theorem (see [4,Corollary 2.6] or [9, Theorem 4.1]). Since ν * λ = ν, for every A ⊂ G and x ∈ G we get which means that ν is right quasi-invariant.
Remark 5.2 Lemma 5.1 does not hold for uncountable groups, in particular for the free group F α with uncountable set α of generators. This group admits no right quasiinvariant measure. Assuming conversely that some measure μ ∈ P(F α ) is right quasiinvariant, fix a generator a ∈ α and consider the set A of all reduced words w ∈ F α that end with a n for some n ∈ Z\{0}. Observe that F α = Aa ∪ A and hence μ(A) > 0 or μ(Aa) > 0. Since μ is right quasi-invariant both cases imply that μ(A) > 0 and then μ(Ab) > 0 for any generator b ∈ α \{a}. But this is impossible since the family (Ab) b∈α \ {a} is disjoint and uncountable.
In the following lemma for a measure μ ∈ P(X ) we put μ(A) = sup x∈G μ(x A). (X, I) be an ideal G-space and μ ∈ P(X ) a measure on X such that the set P I (G; μ) contains an idempotent right quasi-invariant measure λ. For a subset A ⊂ X and numbers δ ε < sup x∈G λ * μ(x A) consider the sets M δ = {x ∈ G : μ(x A) > δ} and L ε = {x ∈ G : λ * μ(x A) > ε}. Then:

Lemma 5.3 Let
Proof Consider the measure ν = λ * μ and put ν(A) = sup x∈G ν(x A) for a subset A ⊂ X . (i) Fix a point g ∈ L ε and observe that (ii) To derive a contradiction, assume that the set M δ belongs to the G-invariant ideal generated by G \ L δ and hence M δ ⊂ E(G \ L δ ) for some finite subset E ⊂ G. Then Choose an increasing number sequence (ε k ) ∞ k=0 such that δ ε < ε 0 and lim k→∞ ε k = ν(A). For every k ∈ ω fix a point g k ∈ L ε k . The preceding item applied to the measure ν and set L δ (instead of μ and M δ ) yields the lower bound for every k ∈ ω. Then lim k→∞ λ(g k L −1 δ ) = 1 and hence lim k→∞ λ(g k L −1 δ g) = 1 for every g ∈ G by the right quasi-invariance and additivity of the measure λ. Choose k so large that λ(g k L −1 δ g −1 ) > 1 − γ /|E| for all g ∈ E. Then the set g∈E g k L −1 δ g −1 has measure > 1 − γ and hence it intersects the set g k M −1 δ which has measure λ(g k M δ ) γ . Consequently, the set M −1 δ intersects g∈E L −1 δ g −1 , and the set M δ intersects g∈E gL δ = G \ E(G \ L δ ), which contradicts the choice of the set E.
(iii) To show that cov J δ ( I (A)) < 1/δ, fix a maximal subset F ⊂ L δ such that ν(x A ∩ y A) = 0 for any distinct points x, y ∈ F. The additivity of the measure ν guarantees that 1 x∈F ν(x A) > |F|·δ and hence |F| < 1/δ. On the other hand, the maximality of F guarantees that for any x ∈ L δ \ F there is y ∈ F such that ν(x A ∩ y A) > 0 and hence x A ∩ y A / ∈ I and y −1 x ∈ I (A). Then x ∈ y · I (A) ⊂ F · I (A) and hence L δ ⊂ F · I (A). The inclusion G \(F · I (A)) ⊂ G \ L δ ∈ J δ implies cov J δ (F · I (A)) |F| < 1/δ.

Corollary 5.4
Let (X, I) be an ideal G-space with countable acting group G and μ ∈ P(X ) a measure on X such that the set P I (G; μ) is not empty. For any partition X = A 1 ∪ · · · ∪ A n of X either: Next, we extend Corollary 5.4 to G-spaces with arbitrary (not necessarily countable) acting group G. Given a G-space X , denote by H the family of all countable subgroups of the acting group G. A subfamily F ⊂ H will be called • closed if for each increasing sequence of countable subgroups {H n } n∈ω ⊂ F the union n∈ω H n belongs to F; • dominating if each countable subgroup H ∈ H is contained in some subgroup H ∈ F; • stationary if F ∩ C = ∅ for every closed dominating subset C ⊂ H.
It is known (see [6, 4.3]) that the intersection n∈ω C n of any countable family of closed dominating sets C n ⊂ H, n ∈ ω, is closed and dominating in H.
Theorem 5.5 Let (X, I) be an ideal G-space and μ ∈ P(X ) a measure on X such that the set H I = {H ∈ H : P I (H ; μ) = ∅} is stationary in H. For any partition X = A 1 ∪ · · · ∪ A n of X either: