@ Appl. Gen. Topol. 16, no. 2(2015), 217-224doi:10.4995/agt.2015.3584 c© AGT, UPV, 2015 The dynamical look at the subsets of a group Igor Protasov a and Serhii Slobodianiuk b a Department of Cybernetics, Kyiv University, Volodymyrska 64, Kyiv 01033, Ukraine (i.v.protasov@gmail.com) b Department of Mathematics and Mechanics, Kyiv University, Volodymyrska 64, Kyiv 01033, Ukraine (slobodianiuks@gmail.com) Abstract We consider the action of a group G on the family P(G) of all subsets of G by the right shifts A 7→ Ag and give the dynamical characterizations of thin, n-thin, sparse and scattered subsets. For n ∈ N, a subset A of a group G is called n-thin if g0A ∩ · · · ∩ gnA is finite for all distinct g0, . . . , gn ∈ G. Each n-thin subset of a group of cardinality ℵ0 can be partitioned into n 1-thin subsets but there is a 2-thin subset in some Abelian group of cardinality ℵ2 which cannot be partitioned into two 1-thin subsets. We eliminate the gap between ℵ0 and ℵ2 proving that each n-thin subset of an Abelian group of cardinality ℵ1 can be partitioned into n 1-thin subsets. 2010 MSC: 54H20; 05C15. Keywords: Thin; sparse and scatterad subsets of a group; recurrent point; chromatic number of a graph. 1. Introduction Let G be a group with the identity e, P(G) denotes the family of all subsets of G, [G]<ω = {F ⊆ G : F is finite}, [G]n = {F ⊆ G : |F | = n}, n ∈ N. We say that a subset A of G is • thin if A ∩ gA is finite for every g ∈ G \ {e}; • n-thin if g0A ∩ · · · ∩ gnA is finite for any distinct g0, . . . , gn ∈ G; • sparse if, for every infinite subset X ⊆ G, there exists a finite subset F ⊂ S such that ⋂ g∈F gA is finite; Received 18 February 2015 – Accepted 21 June 2015 http://dx.doi.org/10.4995/agt.2015.3584 I. Protasov and S. Slobodianiuk • scattered if, for any subset B ⊆ A, there exists F ∈ [G]<ω such that, for each H ∈ [G]<ω, F ∩ H = ∅, we can find b ∈ B such that Hb ∩ B = ∅; • thick if, for any F ∈ [G]<ω, there exists g ∈ G such that Fg ⊆ A. In [3] C. Chou used thin subsets to prove that there are 22 |G| distinct left invariant Banach measures on each infinite amenable groups. Clearly, thin subsets are precisely 1-thin subsets, n-thin subsets appeared in [10] in attempt to characterize the ideal in the Boolean algebra P(G) generated by thin subsets. Sparse subsets appeared in [4] for characterization of strongly prime ultra- filters in the semigroup G∗ of free ultrafilters on G and studied in [9]. Scattered subsets were introduced in [1] as asymptotic counterparts of scat- tered topological spaces. Unexplicitely, thick subsets were used in [11] to partition of an infinite totally bounded group G into |G| dense subsets. As to our knowledge, the name ”thick subset” appeared in [2]. For every infinite group G, we have thin ⇒ 2-thin ⇒ · · · ⇒ n-thin ⇒ · · · ⇒ sparse ⇒ scattered and none of these arrows could be reversed. For ”scattered ✟✟⇒ sparse” see Remark 3.6. More on these subsets and their applications one can find in the surveys [12], [16]. In this paper, we identify P(G) with {0, 1}G, endow P(G) with the product topology and consider the action of G on P(G) by the right shifts A 7→ Ag. After short preliminary section 2, we give the dynamical characterizations to all above defined subsets in section 3. It should be mentioned that the dynamical approach is especially effective for finite partitions of groups [5]. By [10], every n-thin subset of a countable group G can be partitioned into n thin subsets (some cells of the partitions could be empty). Answering the question from [10], G. Bergman (see [15]) constructed an Abelian group G of cardinality ℵ2 and a 2-thin subset of G which cannot be partitioned into two thin subsets. On the other hand [15], every n-thin subset of an Abelian group of cardinality ℵm can be partitioned into n m+1 thin subsets but there is a 2-thin subset in some group of cardinality ℵω which cannot be finitely partitioned. In section 4 we eliminate the gap between ℵ0 and ℵ2 proving that every n-thin subsets of an Abelian group of cardinality ℵ1 is a union of n thin subsets. 2. Some dynamics Let G be a group. A topological space X is called a G-space if there is the action X × G → X : (x, g) 7→ xg such that, for each g ∈ G, the mapping X → X : x 7→ xg is continuous. Given any x ∈ X and U ⊆ X, we set [U]x = {g ∈ G : xg ∈ U} c© AGT, UPV, 2015 Appl. Gen. Topol. 16, no. 2 218 The dynamical look at the subsets of a group and denote O(x) = {xg : g ∈ G}, T (x) = clO(x), W(x) = {y ∈ T (X) : [U]x is infinite for each neighbourhood U of y}. We recall also that x ∈ X is a recurrent point if x ∈ W(x). Now we consider a group G, identify P(G) with the space {0, 1}G and endow P(G) with the product topology. Thus, the subsets U(F, H) = {A ⊆ G : F ⊆ A, H ∩ A = ∅}, where F ∈ [G]<ω, H ∈ [G]<ω, form the base for the open sets on P(G). In what follows, we consider P(G) as a G-space with the action defined by A 7→ Ag, Ag = {ag : a ∈ A}. We say that a subset A of G is recurrent if A is a recurrent point in (P(G), G). 3. Characterizations All groups in this sections are supposed to be infinite. Theorem 3.1. For a subset A of a group G, the following statements hold (i) A is finite if and only if W(A) = ∅; (ii) A is thick if and only if G ∈ W(A). Proof. (i) It suffices to note that A is finite if and only if, for every x ∈ G, the set {g ∈ G : x ∈ Ag} is finite. (ii) Suppose that G ∈ W(A) and take an arbitrary finite subset F of G. Since U(F, ∅) is a neighborhood of G in P(G), there exists g ∈ G such that Ag ∈ U(F, ∅), so Fg−1 ⊆ A and A is thick. Assume that A is thick and take an arbitrary finite subset F of G. Then we choose an injective sequence (gn)n∈ω in G such that Fgi ∩ Fgj = ∅ for all distinct i, j ∈ ω. For each in n ∈ ω, we take hn ∈ G such that (Fg0 ∪ · · · ∪ Fgn)hn ⊆ A, so F ⊆ Ah −1 n g −1 i , i ∈ {0, . . . , n}. It follows that U(F, ∅) contains infinitely many points of the orbit O(A), so G ∈ W(A). � Theorem 3.2. For a subset A of a group G, the following statements hold (i) A is n-thin if and only if |Y | ≤ n for every Y ∈ W(A); (ii) A is sparse if and only if each subset Y ∈ W(A) is finite; (iii) A is scattered if and only if, for every subset B ⊆ A there exists Y ∈ [G]<ω in the closure of {Bb−1 : b ∈ B}. Proof. Suppose that A is n-thin but |Y | > n for some Y ∈ W(A). Let {y0, . . . , yn} be distinct elements from Y . Since Y ∈ W(A) and the set U({y0, . . . , yn}, ∅) is a neighborhood of Y , the set W = {g ∈ G : {y0, . . . , yn} ⊆ Ag} is infinite. We note that W = {g ∈ G : {y0g −1, . . . , yng −1} ⊆ A} = {g ∈ G : g−1 ∈ y−10 A ∩ · · · ∩ y −1 n A}. Hence, A is not n-thin. Suppose that A is not n-thin. We take g0, . . . , gn ∈ G such that the subset B = g0A ∩ · · · ∩ gnA is infinite. If b ∈ B then {g −1 0 b, . . . , g −1 n b} ⊆ A so g−10 , . . . , g −1 n ⊆ Ab −1. We take an arbitrary limit point L of the set {Ab−1 : b ∈ B}. Then L ∈ W(A) but {g−10 , . . . , g −1 n } ⊆ L, so |L| > n. c© AGT, UPV, 2015 Appl. Gen. Topol. 16, no. 2 219 I. Protasov and S. Slobodianiuk (ii) Suppose that A is sparse but some subset Y ∈ W(A) is infinite. We take a countable subset {yn : n ∈ ω} of Y and put Un = U({y0, . . . , yn}, ∅). Then we choose an injective sequence (gn)n∈ω in G such that gn ∈ [Un]A for each n ∈ ω. We note that {y0, . . . , yn} ⊆ Agn so g −1 n ∈ y0A ∩ · · · ∩ ynA. We put X = {y−1n : n ∈ ω}. Then ⋂ g∈F gA is infinite for each finite subset F of X. Hence, A is not sparse. Assume that each subset Y ∈ W(A) is finite but A is not sparse. Then there exists an injective sequence (gn)n∈ω in G such that g0A ∩ · · · ∩ gnA is infinite for each n ∈ ω. We choose an injective sequence (yn)n∈ω in G such that yn ∈ g0A ∩ · · · ∩ gnA, so {g −1 0 , . . . , g −1 n } ⊆ Ay −1 n for each n ∈ ω. Let L be an arbitrary limit point of {Ay−1n : n ∈ ω}. Then {g −1 n : n ∈ ω} ⊆ L and L ∈ W(A). (iii) Suppose that A is scattered and B is a subset of G. We choose corre- sponding F ∈ [G]<ωand take an arbitrary H ∈ [G] k and xnym ∈ Y . Let K be a finite subset of G such that e /∈ K. We take n ∈ ω such that K ∩ H ⊆ Hn. and xnym ∈ Y for some m ∈ ω. By (2), Kxnym ∩ A = ∅, in particular, Kxnym ∩ Y = ∅. Case 2. There exists k ∈ ω such that Y ⊆ {xnym : n ≤ k, m ≥ n}. We take an arbitrary K ∈ [G]<ω such that K ∩ Hk = ∅. Then we choose n, m ∈ ω, such that K ∩ H ⊆ Hn, n > k and xnym ∈ Y . By (1), Kxnym ∩ {xnym : n ≤ k, m ≥ n} = ∅, in particular, Kxnym ∩ Y = ∅. 4. Partitions into thin subsets We fix a subset T of a group G and, for every g ∈ G \ {e}, consider the orbital graph Γg with the set of vertices T and the set of edges Eg = {{s, t} ∈ [T ] 2 : st−1 ∈ {g, g−1}}. c© AGT, UPV, 2015 Appl. Gen. Topol. 16, no. 2 221 I. Protasov and S. Slobodianiuk We say that a graph (T, E) is an orbital companion of T if, for each g ∈ G\{e}, the set E contains all but finitely many edges from Eg so E \ Eg if finite. For n ∈ N, a graph Γ is called n-discrete if each connected component of Γ has no more than n vertices. Given a graph (V, E), we take any subsets E′, E′′, from E such that E = E′ ∪ E′′ and say that (V, E) is a union of the graphs (V, E′) and (V, E′′). In this section we use the following equivalent definition of an n-thin subset T of a group (see Theorem 3.3): for every F ∈ [G]<ω, there exists H ∈ [G]<ω such that, for every g ∈ G \ H, we have |Fg ∩ T | ≤ n. Lemma 4.1. Every n-thin subset T of a countable group G has an n discrete orbital companion Γ. Proof. We write G as the union of an increasing chain {Fi : i < ω} of finite subsets such that e ∈ F0, Fi ⊂ Fi+1 and Fi = F −1 i . Then we choose a chain {Vi : i < ω} of finite subsets of G such that, for any i < ω and g ∈ G \ Vi, we have Vi ⊂ Vi+1 and |F m i g ∩ T | ≤ n. We define the set E of edges of Γ by the rule: {s, t} ∈ E ⇔ ∃k : s /∈ VK and st −1 ∈ Fk. Given any g ∈ G \ {e}, we pick k < ω such that g ∈ Fk. If t ∈ T and gt ∈ T then either {t, gt} ⊂ Vk or (t, gt) ∈ E. Since Vk is finite, we conclude that Γ is an orbital companion of T . Suppose that Γ is not n-discrete and choose a subset S ∈ [T ]n+1 such that the induced graph ΓS is connected. We find the minimal number k such that, for any {s1, s2} ∈ [S] 2 ∩ E, s1s −1 2 ∈ Fk and so there are {s, s ′} ∈ [S]2 ∩ E such that s′s−1 ∈ Fk \ Fk−1. It follows that s ∈ G \ Vk and S ⊂ F n k s but |S| > n and we get a contradiction with the choice of Vk. � Lemma 4.2. Let G be an Abelian group of cardinality ℵ1 and let T be an n-thin subset of G. Then some orbital companion of T is a union of two n-discrete graphs. Proof. Applying Lemma 2 from [15], we represent G as the union of increasing chain {Gi : i < ω1} of countable subgroups such that (∗) for any i < ω1 and g ∈ Gi+1 \ Gi, |Gig ∩ T | ≤ n. For every i < ω1, we consider the n-thin subset Ti = T ∩ (Gi+1 \ Gi) of Gi+1 and choose n-discrete companion (Ti, Ei) of Ti given by Lemma 4.1. We denote by E′i the union of Ei and the set of all {t, t ′} ∈ [Ti] 2 such that t−1t′ ∈ Gi. Since G is Abelian, we have Gig = gGi and by (∗), (Ti, E ′ i) is a union of two n-discrete graphs (Ti, Ei) and (Ti, E ′ i \ Ei). We put E = ⋃ i<ω1 E′i and note that (T, E) is a union of n-discrete graphs because the subsets {Ti : i < ω1} are pairwise disjoint. Now it remains to prove that (T, E) is an orbital companion of T . Given g ∈ G \ {e}, we choose i < ω1 such that g ∈ Gi+1 \ Gi. Since G is Abelian, T ∩T g−1 = T ∩g−1T and then the set {t ∈ T ∩g−1T : (t, gt) /∈ E} is the union c© AGT, UPV, 2015 Appl. Gen. Topol. 16, no. 2 222 The dynamical look at the subsets of a group of the following sets ⋃ i