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

We prove that for every finite partition $G=A_1\cup\dots\cup A_n$ of a group $G$ either $cov(A_iA_i^{-1})\le n$ for all cells $A_i$ or else $cov(A_iA_i^{-1}A_i)<n$ for some cell $A_i$ of the partition. Here $cov(A)=\min\{|F|:F\subset G,\;G=FA\}$ is the covering number of $A$ in $G$. A similar result is proved also of partitions of $G$-spaces. This gives two partial answers to a problem of Protasov posed in 1995.

This paper was motivated by the following problem posed by I.V.Protasov in Kourovka Notebook [7]. Problem 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 G-spaces. 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 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 {gA : g ∈ G, A ∈ I} ⊂ I. A Boolean ideal I ⊂ B(G) on a group G will be called invariant if {xAy : 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 (resp. [X] ≤ω ) is a Boolean ideal on X if X is infinite (resp. uncountable).
For a subset A ⊂ X of an ideal G-space (X, I) by be the J -covering number of A. 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) = AA −1 for every subset A ⊂ G. That is why Problem 1 is a partial case of the following general problem. Problem 2. Is it true that for any partition X = A 1 ∪ · · · ∪ A n of an ideal G-space X 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 [3, 4.3]. The paper [3] gives a survey of available partial solutions of Protasov's Problems 1 and 2. Here we mention the following result of Banakh, Ravsky and Slobodianiuk [1]. Theorem 1. 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 and 2.
Theorem 2. For any partition X = A 1 ∪ · · · ∪ A n of an ideal G-space (X, I) either • cov(∆ I (A i )) ≤ n for all cells A i or else • cov J (∆ I (A i )) < n for some cell A i and some G-invariant ideal J ∋ ∆ I (A i ) on G.
Corollary 1. For any partition X = A 1 ∪ · · · ∪ A n of an ideal G-space (X, I) either cov(∆ I (A i )) ≤ n for all cells A i or else cov(∆ I (A i ) · ∆ I (A i )) < n for some cell A i .
Proof. By Theorem 2, 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 P(G). In the first case we are done. In the second case we can find a subset F ⊂ G of cardinality |F | < n such that Then for every x ∈ G the shift x∆ I (A i ) does not belong to the ideal 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: Theorem 3. Let G be a group and I be an invariant Boolean ideal on G with [G] ≤ω ⊂ I. For any partition Corollary 2. For any partition G = A 1 ∪ · · · ∪ A n of a group G either cov(A i A i ) ≤ n for all cells A i or else cov(A i A −1 i A i ) < n for some cell A i of the partition. Proof. On the group G consider the trivial ideal I = {∅}. By Theorem 3, either cov( 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 3 is G-invariant but not necessarily invariant, we can ask the following question. Problem 3. 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?
1. Minimal measures on G-spaces Theorems 2 and 3 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. The 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 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 : B(X) → {0, 1} such that µ −1 U (1) = 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 A ∈ I the G-invariance of the ideal I implies µ(A) Lemma 2. If a subset A of a group G has Is 12 (A) = 1, then cov(G \ A) ≥ ω.
Proof. It suffices to show that G = F (G \ A) for any finite set F ⊂ G. Consider the uniformly distributed measure µ = 1

|F |
x∈F δ x −1 (Ay), which implies that µ(Ay) = 1 and supp(µ) = F −1 ⊂ Ay. Then Remark 1. By Theorem 3.8 of [2], for every subset A of a group G we get Is 12 for B ⊂ G and P ω (G) 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

A density version of Theorem 2
In this section we shall prove the following density theorem, which will be used in the proof of Theorem 2 presented in the next section.
Theorem 4. Let (X, I) be an ideal G-space and µ ∈ P I (X) be a minimal measure on X. If some subset . We can replace the measure µ by µ ′ and assume that µ(A) =μ(A). Choose a positive ε such that Consider the set L = {x ∈ G : µ(xA) >μ(A)−ε} and choose a maximal subset F ⊂ L such that µ(xA∩yA) = 0 for any distinct points x, y ∈ L. The additivity of the measure µ implies that 1 . By the maximality of F , for every x ∈ L there is y ∈ L such that µ(xA ∩ yA) > 0. Then xA ∩ yA / ∈ 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 2
Let X = A 1 ∪ · · · ∪ A n be a partition of an ideal G-space (X, I). By Lemma 1, there exists a minimal probability measure µ ∈ P (X) such that I ⊂ {A ∈ B(G) : µ(A) = 0}.
1) For every i ∈ {1, . . . , n}μ(A i ) ≤ 1 n . In this case for every x ∈ G we get and hence µ(xA i ) = 1 n for every i ∈ {1, . . . , n}. For every i ∈ {1, . . . , n} fix a maximal subset F i ⊂ G such that µ(xA i ∩ yA i ) = 0 for any distinct points x, y ∈ F i . The additivity of the measure µ implies that 1 ≥ x∈Fi µ(xA i ) ≥ |F i | 1 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(xA i ∩ yA i ) > 0 and hence xA i ∩ yA 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.
2) For some i we getμ(A i ) > 1 n . In this case Theorem 4 guarantees that cov J (∆ I (A i )) ≤ 1/μ(A i ) < n for some G-invariant ideal J ∋ ∆ I (A i ) on G.

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 3 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) : ∀g ∈ G λ * δ g * µ ∈ P I (X)} and observe that it is closed and convex in the compact Hausdorff space P (G).
Lemma 3. Let (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).
Proof. Choose any strictly positive function c : G → (0, 1] such that g∈G c(g) = 1 and consider the σ-additive probability measure λ = g∈G c(g)δ g −1 ∈ P (G). On the compact Hausdorff space P (G) consider the right shift Φ : P (G) → P (G), Φ : ν → ν * λ. We claim that Φ(P I (G; µ)) ⊂ P I (G; µ). Given any measure ν ∈ P I (G; µ) we need to check that Φ(ν) = ν * λ ∈ P I (G; µ), which means that ν * λ * δ x * µ ∈ P I (X) for all x ∈ G. It follows from ν ∈ P I (G; µ) that ν * δ g −1 x * µ ∈ P I (X). Since the set P I (X) is closed and convex in P (X), we get So, Φ(P I (G; µ)) ⊂ P I (G; µ) and by Schauder Fixed Point Theorem, the continuous map Φ 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 Ellis Theorem [6, 2.6]. Since ν * λ = ν, for every A ⊂ G and x ∈ G we get which means that ν is right quasi-invariant. Remark 2. Lemma 3 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 quasi-invariant measure. Assuming conversely that some measure µ ∈ P (F α ) is right quasi-invariant, 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.
1. Fix a point g ∈ L ε and observe that 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(z k L −1 δ g) = 1 for every g ∈ G by the right quasi-invariance of the measure λ. Choose k so large that λ(z k L −1 Then the set g∈E z k L −1 δ g −1 has measure > 1 − γ and hence it intersects the set z k M −1 a which has measure λ(z k M a ) ≥ γ. Consequently, the set M −1 a intersects g∈E L −1 δ g −1 , and the set M a intersects g∈E gL = G \ E(G \ L δ ) , which contradicts the choice of the set E.
3. To show that cov J δ (∆ I (A)) ≤ 1/δ, fix a maximal subset F ⊂ L δ such that ν(xA ∩ yA) = 0 for any distinct points x, y ∈ L δ . The additivity of the measure ν guarantees that 1 ≥ x∈F ν(xA) > |F | · δ and hence |F | < 1/δ. On the other hand, the maximality of F guarantees that for every x ∈ F there is y ∈ L δ such that ν(xA ∩ yA) > 0 and hence xA ∩ yA / ∈ I and y −1 x ∈ ∆ I (A). Then x ∈ y · ∆ I (A) ⊂ F · ∆ I (A) and hence Corollary 3. Let (X, I) be an ideal G-space with countable acting group G and µ ∈ P (X) be 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: (1) cov(∆ I (A i )) ≤ n for all cells A i or else (2) cov J (∆ I (A i )) < n for some cell A i and some G-invariant Boolean ideal J ⊂ P(G) such that {x ∈ G : Proof. By Lemma 3, the set P I (G; µ) contains an idempotent right quasi-invariant measure λ. Then for the measure ν = λ * µ ∈ P I (X) two cases are possible: 1) Every cell A i of the partition hasν(A i ) = sup x∈G ν(xA i ) ≤ 1 n . In this case we can proceed as in the proof of Theorem 2 and prove that cov(∆ I (A i )) ≤ n for all cells A i of the partition.
2) Some cell A i of the partition hasν(A i ) > 1 n . In this case Lemma 4 guarantees that cov J (∆ I (A i )) < n for the G-invariant Boolean ideal J ⊂ P(G) generated by the set {x ∈ G : ν(xA i ) ≤ 1 n }, and the set M = {x ∈ G : µ(xA i ) > 1 n } does not belong to the ideal J . Next, we extend Corollary 3 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 [5, 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. Let (X, I) be an ideal G-space and µ ∈ P (X) be 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: (1) cov(∆ I (A i )) ≤ n for all cells A i or else (2) cov J (∆ I (A i )) < n for some cell A i and some G-invariant Boolean ideal J ⊂ P(G) such that {x ∈ G : µ(xA i ) > 1 n } / ∈ J .  [4], [5, 4.4] of Fodor's Lemma, the stationary set H ∀ contains another stationary subset S ⊂ H ∀ such that for every i ∈ {1, . . . , n} the restriction f i |S is a constant function and hence f i (S) = {F i } for some finite set F i ⊂ G of cardinality |F i | ≤ n. We claim that G = F i · ∆ I (A i ). Indeed, given any element g ∈ G, by the stationarity of S there is a subgroup H ⊂ S such that g ∈ H. Then g ∈ H ⊂ f i (H) · ∆ I (A i ) = F i · ∆ I (A i ) and hence cov(∆ I (A i )) ≤ |F i | ≤ n for all i. Now assume that the family H ∃ is stationary in H. In this case for some i ∈ {1, . . . , n} the set H i = {H ∈ H ∃ : i H = i} is stationary in H ∃ . Since the function f : H ∃ → [G] <ω is regressive, by Jech's generalization [4], [5, 4.4] of Fodor's Lemma, the stationary set H i contains another stationary subset S ⊂ H i such that the restriction f |S is a constant function and hence f (S) = {F } for some finite set F ⊂ G of cardinality |F | < n. We claim that the set J = G \ (F · ∆ I (A i )) generates a G-invariant ideal J , which does not contain the set M = {x ∈ G : µ(xA i ) > 1 n }. Assume conversely that M ∈ J and hence M ⊂ EJ for some finite subset E ⊂ G. By the stationarity of the set S, there is a subgroup H ∈ S such that E ⊂ H.

Proof of Theorem 3
Theorem 3 is a simple corollary of Theorem 5. Indeed, assume that G = A 1 ∪ · · · ∪ A n is a partition of a group and I ⊂ P(G) is an invariant ideal on G which does not contain some countable subset and hence does not contain some countable subgroup H 0 ⊂ G. Let H be the family of all countable subgroups of G and µ = δ 1 be the Dirac measure supported by the unit 1 G of the group G. We claim that that for every subgroup H ∈ H that contains H 0 the set P I (H; µ) is not empty. It follows from H 0 / ∈ I that the family I H = {H ∩ A : A ∈ I} is an invariant Boolean ideal on the group H. Then the family {H\A : A ∈ I} is a filter on H, which can be enlarged to an ultrafilter U H . The ultrafilter U H determines a 2-valued measure µ H : P(H) → {0, 1} such that µ −1 H (1) = U H . By the right invariance of the ideal I, for every A ∈ I and x ∈ H we get µ H * δ x * µ(A) = µ H (Ax) = 0, which means that µ H ∈ P I (H; µ). So, the set H I = {H ∈ H : P I (H; µ) = ∅} ⊃ {H ∈ H : H ⊃ H 0 } is stationary in H.
Then by Theorem 5 either (1) cov(∆ I (A i )) ≤ n for all cells A i or else (2) cov J (∆ I (A i )) < n for some cell A i and some G-invariant Boolean ideal J ⊂ P(G) such that A −1 i = {x ∈ G : δ 1 (xA i ) > 1 n } / ∈ J .