On G–transitive version of perfectly meager sets

We study the G− invariant version of perfectly meager sets (a generalization of the notion of AFC′ sets). We find the necessary and sufficient conditions for the inclusion AFCG ⊆ I. In particular, we partially characterize for which groups G of automorphisms of the Cantor space every AFCG set is Lebesgue null. 1. Definitions and notation We consider the Cantor space 2ω as a topological group (where (x + y)(k) = x(k)+y(k) mod 2). By 2<ω let us denote the collection of all finite binary sequences: 2<ω = {f : n→ 2 where n ∈ ω} For any s ∈ 2<ω by [s] denote the base open set detemined by s: [s] = {x ∈ 2ω : s ⊆ x}. Let Perf stand for the family of all perfect subsets of the space 2ω. Recall that a proper collection of subsets of 2ω: I ⊆ P (2ω) is called a σ ideal iff it is closed under taking subsets and countable sums. Throughout the paper we assume that every σ ideal I contains all singletons: ∀x∈X{x} ∈ I. Let I ⊆ P (2ω) be a σ ideal. Define the following cardinal numbers: Definition 1. cov(I) = min{|A| : A ⊆ I ∧⋃A = 2ω} and cof (I) = min{|A| : A ⊆ I ∧ ∀Z∈I∃A∈AZ ⊆ A}. Notice that we always have cov(I) ≤ cof (I). We assume that the reader is familiar with basic concept of arithmetic of cardinal numbers. In particular, we need the notion of cofinality; recall that an uncountable cardinal number κ is called regular iff cf(κ) = κ. By Hom(X) we denote the group of all homeomorphisms of the topological space X. We always assume that G is a fixed subgroup of Hom(2ω). The following additional terminology will be useful in our proof. 2010 Mathematics Subject Classification. Primary: 03E15; Secondary 03E20, 28E15.


Definitions and notation
We consider the Cantor space 2 ω as a topological group (where (x + y)(k) = x(k) + y(k) mod 2). By 2 <ω let us denote the collection of all finite binary sequences: 2 <ω = {f : n → 2 where n ∈ ω} For any s ∈ 2 <ω by [s] denote the base open set detemined by s: [s] = {x ∈ 2 ω : s ⊆ x}. Let Perf stand for the family of all perfect subsets of the space 2 ω . Recall that a proper collection of subsets of 2 ω : I ⊆ P (2 ω ) is called a σ -ideal iff it is closed under taking subsets and countable sums. Throughout the paper we assume that every σ -ideal I contains all singletons: ∀ x∈X {x} ∈ I.
Let I ⊆ P (2 ω ) be a σ -ideal. Define the following cardinal numbers: Notice that we always have cov (I) ≤ cof (I).
We assume that the reader is familiar with basic concept of arithmetic of cardinal numbers. In particular, we need the notion of cofinality; recall that an uncountable cardinal number κ is called regular iff cf(κ) = κ.
By Hom(X) we denote the group of all homeomorphisms of the topological space X. We always assume that G is a fixed subgroup of Hom(2 ω ).
The following additional terminology will be useful in our proof.
For an arbitrary g ∈ G and Q ∈ Perf we often abbreviate the image g(Q) = {gx : x ∈ Q} and write simply gQ. Also for any t ∈ 2 ω and A ⊆ 2 ω we write We denote by M(P ) the collection of all first category sets on P , where P ∈ P erf (X).
We use a letter N to denote the sigma ideal of Lebesgue measure zero sets of 2 ω .
We denote by Trans(2 ω ) the subgroup of all translations of 2 ω .

Introduction
Let us start with the following, classical definition: and only if, it is uncountable and has countable intersection with any set of measure zero.
Notice that under the assumption of Continuum Hypothesis there exists a Sierpiński set (see [9]) and, on the other hand, it is consistent that there is no Sierpiński set.
A special variation of the notion of a Sierpiński set is a κ -Sierpiński set with respect to the σ-ideal I, namely: Definition 3. Suppose that κ is a cardinal number and I ⊆ P (2 ω ) a σideal. A set X ⊆ 2 ω is called a κ -Sierpiński set X with respect to I iff |X| = κ and ∀ A∈I |A ∩ X| < κ.
Notice that if T is a σ-ideal (which contains singletons) and κ = cof (I) = cov (I) then there exists a κ -Sierpiński set X with respect to I.
Recall the classical definition of perfectly meager sets (called also always of the first category sets): Definition 4. A set X of 2 ω is a perfectly meager (AFC) set iff for every P ∈ Perf, X ∩ P is a first category set in the topology of P .
The following notion of sets was first defined in [5] and then it has been studied most extensively in papers [6] and [7].

Definition 5.
A set X ⊆ 2 ω is an AFC -set if for each perfect set P there exists an F σ -set F such that X ⊆ F and for each t ∈ 2 ω , (F + t) ∩ P is a first category set in the topology of P .
Notice that the notion AFC is a strengthening of the classical perfectly meager sets.
Notice that K. Prikry conjectured that the collection of strongly meager sets form a σ-ideal but it turned out that it is consistent that strongly meager sets are exactly the countable sets (see [3]) and that it is consistent that even the sum of two strongly meager sets need not be strongly meager set (see [2]).
It is known (see for example [5] and [7]), that AFC ⊆ AFC and every strongly meager set is an AFC set.
It is also known (see [8]) that every Sierpiński set is strongly meager. We can summarize all these inclusions in Fig. 1 Andrzej Nowik 3 the sum of two strongly meager sets need not be strongly meager set (see [2]). It is known (see for example [5] and [?]), that AFC ⊆ AFC and every strongly meager set is an AFC set.
It is also known (see [8]) that every Sierpiński set is strongly meager. We can summarize all these inclusions in Fig. 2 Sierpiński set SFC AFC AFC --- The AFC G -sets Suppose that G is a subgroup of Hom(2 ω ) and let X be an arbitrary subset of 2 ω .
This notion is a natural generalization of the notion of AFC sets.

Remarks:
It is obvious that AFC Trans(2 ω ) = AFC , AFC {id} = AFC and Fig. 2 (where arrows denote inclusions). Let us define: Definition 2.7. Suppose that I is a σ -ideal of subsets of the space 2 ω . We say that a group G ≤ Hom(2 ω ) has the (Em) I property iff there exists a perfect set Q ∈ Perf such that for each P ∈ Perf \ I there exists g ∈ G such that P ∩ gQ ∈ M(gQ).

Remarks:
One can prove that Trans(2 ω ) does not have the (Em) N property. Let us define the main notion of this article.
The AFC G -sets Suppose that G is a subgroup of Hom(2 ω ) and let X be an arbitrary subset of 2 ω .
This notion is a natural generalization of the notion of AFC sets.

Remarks:
It is obvious that It is also evident that if All inclusions are summarized in Fig. 2 (where arrows denote inclusions).  Let us define: Definition 8. Suppose that I is a σ -ideal of subsets of the space 2 ω . We say that a group G ≤ Hom(2 ω ) has the (Em) I property iff there exists a perfect set Q ∈ Perf such that for each P ∈ Perf \ I there exists g ∈ G such that P ∩ gQ ∈ M(gQ).

Remarks:
One can prove that Trans(2 ω ) does not have the (Em) N property.
Without loss of generality we may assume that in Definition 8 P is only closed set such that P ∈ I.
We will start with the following theorem.
Then we have: AFC G ⊆ I.
Proof. Let X ⊆ 2 ω be a set such that X ∈ I. By the definition of the notion (Em) I there is a perfect set Q such that for each closed E ∈ I we have where cl(F n ) = F n , so there exists n 0 < ω such that F n 0 ∈ I. Now there exists g ∈ G such that F n 0 ∩ gQ is not meager in gQ. So we conclude, that X is not an AFC G set.
The implication given in Theorem 1 is reversible under some additional set theoretical assumptions. Indeed, we have the following theorem.
Theorem 2. Let us assume like in Theorem 1 that I is an arbitrary σ -ideal of subsets of 2 ω such that ∀ x∈2 ω {x} ∈ I and G ≤ Hom(2 ω ) is a subgroup of Hom(2 ω ). Moreover, assume that Then the following conditions are equivalent: (1) AFC G ⊆ I (2) G fulfills (Em) I .
Proof. Theorem 1 gives us immediately the implication (2) ⇒ (1). Now suppose that G fulfills ¬(Em) I . Since κ = cof (I) = cov (I) and I contains singletons we conclude that there exists a κ-Sierpiński set X with respect to I (see Def. 3). Let Q ∈ Perf be arbitrary. From the assumption ¬(Em) I there exists a perfect set P such that P ∈ I and ∀ g∈G gQ ∩ P ∈ M(gQ). Pick a countable set C ⊆ 2 ω such that 2 ω \ (C + P ) ∈ I.
Since κ ≤ non(AFC G ) we obtain [2 ω \(P +C)]∩X ∈ AFC G , so there exists E ∈ F σ , E ⊇ X \ (P + C) such that ∀ g∈G gQ ∩ E ∈ M(gQ). Finally, define E * = E ∪(P +C). It is easy to see that X ⊆ E * and ∀ g∈G gQ∩E * ∈ M(gQ). Hence X ∈ AFC G and the proof is completed, since X does not belong to I.
Unfortunately, we don't know whether this theorem is true under weaker assumptions. Thus we think that the following question may be of some interest. Assume that G has the property that for each x ∈ 2 ω the orbit Gx is dense in 2 ω . Then the following conditions are equivalent: (1) G fulfills (Em) I ; (2) |Em(Perf \ I, G)| ≤ ℵ 0 . We will need the following technical lemma (folklore for the group G = Trans(2 ω )): is a group such that for each x ∈ 2 ω , Gx is dense in 2 ω , then for every sequence Q n of perfect subsets of 2 ω there exists a perfect P ∈ Perf such that ∀ n∈ω ∃ g∈G gQ n ∩ P ∈ M(P ).

Proof.
Let v k = [(0, . . . , 0, 1)] (0 k times). For each k choose x k ∈ Q k and g k ∈ G such that g k x k ∈ V k . Define P = k∈ω g k Q k ∩ V k , then P is a perfect set and if k ∈ ω then g k Q k ∩ P ⊇ g k Q k ∩ V k ∈ M(P ).
Next we give an useful characterization of the property (Em) N .
By way of contradiction suppose that there exists g ∈ G such that gQ ∩ U = ∅. Then gQ ⊆ C + P , hence there exists c 0 ∈ C and an open set I such that  For any m < ω we choose, using the assumption (2), an open set U m such that This can be done, since I m ∩ R is a perfect or an empty set. Now put We see that Define F = 2 ω \ U , then we have µ(F ) > 0 so choose a perfect P ⊆ F of positive measure.
Let g ∈ G and I m 0 be given such that R ∩ I m 0 = ∅. Now U m 0 ∩ g(R ∩ I m 0 ) = ∅. Moreover, since U m 0 ∩ P = ∅ we obtain that g(R ∩ I m 0 ) ⊆ P . This means that (1) holds.
Notice that in the proof of implication (2)⇒(1) we did not use the assumption that Trans(2 ω ) ≤ G.
In the next part we will prove theorems about relations between AFC G and different classes of peculiar small sets of the real line.
Theorem 5. Assume that G is a subgroup of Hom(2 ω ) which contains Trans(2 ω ). If G fulfills ¬(Em) N , then every strongly meager set is an AFC G set.

Remark:
This implication is reversible under CH. Namely: Theorem 6. Suppose that G ≤ Hom(2 ω ) and assume that G has the (Em) N property. Moreover, assume CH. Then there exists a strongly meager set X ⊆ 2 ω such that X ∈ AFC G .
Proof. Let X ⊆ 2 ω be arbitrary Sierpiński set. Then X is strongly meager ( [8]). From the (Em) N property we obtain that there exists Q ∈ Perf such that ∀ P ∈Perf \N ∃ g∈G P ∩ g(Q) ∈ M(gQ). Suppose that E is an F σ -set such that X ⊆ E. Since X ∈ N it follows that E ∈ N . Hence there exists P ∈ Perf \ N such that P ⊆ E Therefore ∃ g∈G P ∩ g(Q) ∈ M(gQ), hence E ∩ g(Q) ∈ M(g(Q)). This yields X ∈ AFC G , which finishes the proof.
Corollary 1. Assume that cov (N ) = cof (N ) and cov (N ) is a regular cardinal. Let G ≤ Hom(2 ω ) and suppose that Trans(2 ω ) ≤ G. Then the following conditions are equivalent: (1) G has the (Em) N property.