BERNSTEIN SETS AND κ-COVERINGS

JAN KRASZEWSKI, ROBERT RAŁOWSKI, PRZEMYSŁAW SZCZEPANIAK AND SZYMON ŻEBERSKI

Abstract. In this paper we study a notion of a κ-covering in connection with Bernstein sets and other types of nonmeasurability. Our results correspond to those obtained by Muthuvel in [7] and Nowik in [8]. We consider also other types of coverings.

1. Definitions and notation

In 1993 Carlson in his paper [3] introduced a notion of κ-coverings and used it for investigating whether some ideals are or are not κ-translatable. Later on κ-coverings were studied by other authors, e.g. Muthuvel (cf. [7]) and Nowik (cf. [8], [9]). In this paper we present new results on κ-coverings in connection with Bernstein sets. We also introduce two natural generalizations of the notion of κ-coverings, namely κ-S-coverings and κ-I-coverings.

We use standard set-theoretical notation and terminology from [1]. Recall that the cardinality of the set of all real numbers ℝ is denoted by 𝔠. The cardinality of a set A is denoted by |A|. If κ is a cardinal number then

\begin{matrix}[A]^{κ} = \{B ⊆ A : |B| = κ\}; \\ [A]^{<κ} = \{B ⊆ A : |B| < κ\}. \end{matrix}

The cofinality of κ is denoted by cf(κ). The power set of a set A is denoted by P(A).

For a given uncountable Abelian Polish group (X, +), the family of all uncountable perfect subsets of X is denoted by Perf(X) and the family of all Borel subsets of X is denoted by Borel(X). We say that a set B ⊆ X is a Bernstein set if for every uncountable set Z ∈ Borel(X) both sets Z ∩ B and Z \ B are nonempty.

In this paper I stands for a σ-ideal of subsets of a given uncountable Abelian Polish group (X, +). We will always assume that I is proper and group invariant, contains singletons and has a Borel base (i.e. for every set A ∈ I we can find a Borel set B ∈ I such that A ⊆ B). We will use three cardinal characteristics of an ideal I: the additivity number add(I), the covering number cov(I) and the uniformity number non(I), defined as follows:

add(I) = min{|A| : A ⊆ I ∧ ⋃ A ∉ I};
cov(I) = min{|A| : A ⊆ I ∧ ⋃ A = X};
non(I) = min{|A| : A ⊆ X ∧ A ∉ I}.

Let us recall the notion investigated for instance in [4].

2000 Mathematics Subject Classification. Primary 03E35, 03E75; Secondary 28A99.

Key words and phrases. nonmeasurable set, real line, Bernstein set, κ-covering.

Definition 1. Let N ⊆ X. We say that the set N is completely I-nonmeasurable if

(∀A ∈ Borel(X) \ I)(A ∩ N ∉ I ∧ A ∩ (X \ N) ∉ I).

In particular, for the σ-ideal of Lebesgue null sets N ⊆ P(ℝ) we have that a set N ⊆ ℝ is completely N -nonmeasurable if and only if the inner measure of N and the inner measure of the complement of N are zero. One can observe that if I is a σ-ideal of our interest (i.e. having properties mentioned above) then every Bernstein set is completely I-nonmeasurable. Hence the notion of a completely I-nonmeasurable set generalizes the notion of a Bernstein set.

While constructing completely I-nonmeasurable sets having interesting covering properties we will concentrate on σ-ideals including all unit spheres. Let us observe that classical σ-ideals such as the σ-ideal of null sets N and the σ-ideal of meager sets M have this property.

The following notion of a tiny set is very useful in recursive constructions of completely I-nonmeasurable sets.

Definition 2. Let us fix a family A ⊆ I. We say that a perfect set P ∈ Perf(X) is a tiny set with respect to A if

(1) (∀t ∈ X)(∀A ∈ A) |(P + t) ∩ A| ≤ ω,

(2) (∀B ∈ Borel(X) \ I)(∃t ∈ X) |(P + t) ∩ B| = 𝔠.

In [10] Rałowski proved the following useful lemma.

Lemma 1.1. Let A ⊆ I. If there exists a perfect set P ∈ Perf(X), which is tiny with respect to A then

min{|B| : B ⊆ A ∧ (∃B ∈ Borel(X) \ I)(B ⊆ ⋃ B)} = 𝔠.

Definition 3. We say that the σ-ideal I has the Steinhaus property if for every set A ∈ P(X) \ I and B ∈ Borel(X) \ I the set A − B = {a − b : a ∈ A ∧ b ∈ B} contains a nonempty open set.

It is known that the σ-ideal of null sets and the σ-ideal of meager sets have the Steinhaus property (even in more general context – cf. [2], [6]).

Let observe that the following fact holds.

Fact 1.2. Let Q ⊆ X be any dense countable subgroup of X. If the σ-ideal I has the Steinhaus property then for any set B ∈ Borel(X) \ I we have (B + Q)^{c} ∈ I.

Proof. Let us fix B ∈ Borel(X) \ I and let B^{∗} = B + Q. Suppose that (B^{∗})^{c} ∉ I. Then by the Steinhaus property there exists a nonempty open set U ⊆ X such that U ⊆ (B^{∗})^{c} − B^{∗}. Hence there exist some q ∈ Q and b ∈ B^{∗} such that q + b ∈ (B^{∗})^{c}. Since Q + B^{∗} = B^{∗}, we get q + b ∈ B^{∗} ∩ (B^{∗})^{c} which is a contradiction. □

Now we will focus our attention on σ-ideals N and M. The next lemma is probably folklore, but for the reader’s convenience we present its proof.

Lemma 1.3. Let I = N or I = M. Then

(∀B ∈ Borel(X) \ I)(∀P ∈ Perf(X))(∃t ∈ X) |(t + P) ∩ B| = 𝔠.

Proof. (Cichoń) Firstly, let us assume that cov(I) > ω_{1} and choose any subset T ∈ [P]^{ω_{1}} of a perfect set P. Let B^{∗} = B + Q, where Q is a dense countable subgroup of X. From Fact 1.2 we deduce that ⋃_{t∈T} (t + B^{∗})^{c} ≠ X. Let us fix y ∈ ⋂_{t∈T} (t + B^{∗}). Then T ⊆ −y + B^{∗}. Thus there exist x ∈ X and S ∈ [T]^{ω_{1}} such that S ⊆ x + B. But S ⊆ P and P is perfect, so |(x + B) ∩ P | = 𝔠.

Now let V be a model of ZFC. There exists a generic extension V [G] fulfilling condition MA + 𝔠 ≠ ω_{1}. Consequently, V [G] ⊧ cov(I) > ω_{1}. But the following formula

(∀P ∈ Perf(X))(∀B ∈ Borel(X) \ I)(∃x ∈ X) |(x + P) ∩ B| = 𝔠

is Π_{3}^{1}. So it holds also in the ground model V because by Shoenfield’s absoluteness theorem (cf. [12]) Π_{3}^{1} formulas are downward absolute. □

Remark 1. Another proof for the measure case was given by Ryll-Nardzewski. His proof was based on convolution measures. Yet another proof is due to Morayne, where density points of measure are used.

Remark 2. Let us observe that Lemma 1.3 remains true for any σ-ideal I having the Steinhaus property such that it is consistent that cov(I) > ω_{1} and Borel codes for sets from the ideal I are absolute between transitive models of ZFC.

Question 1. Is there any nontrivial example of a σ-ideal, other than M and N, fulfilling conditions mentioned in Remark 2?

Lemma 1.3 gives us a simpler characterization of a tiny set in case I = N or I = M.

Corollary 1.4. If I = N or I = M then a perfect set P is a tiny set with respect to a family A ⊆ I if

(∀t ∈ X)(∀A ∈ A) |(P + t) ∩ A| ≤ ω.

Let us notice this characterization is not true in general (as pointed by the referee):

Example 1.5 (given by the referee). Assume that the cofinality of the σ-ideal of meager subsets of ℝ is ω_{1} and 𝔠 > ω_{1}. Let (A_{α} : α < ω_{1}) be a cofinal tower, consisting of meager sets in ℝ. Let X = ℝ × ℝ and let I be the σ-ideal of subsets of X with meager projections on the first coordinate. Let A = {A_{α} × {0} : α < ω_{1}}, P = {0}×ℝ and B = ℝ×{0}. Then P is tiny with respect to A as |P ∩(A_{α}×{0})| ≤ 1 for each α < ω_{1}. However, B ∈ Borel(X) \ I, |B ∩ P | = 1 and B ⊆ ⋃ A, so the conclusion of Lemma 1.1 fails.

In our applications we will concentrate on families of unit spheres in ℝ^{n}.

Lemma 1.6. Let I = N or I = M. Let D be a family of unit spheres of size less than continuum and let B ∈ Borel(ℝ^{n}) \ I. Then

|B \ ⋃ D| = 𝔠.

Proof. Observe that every line is a tiny set with respect to the family of all unit spheres. So according to Lemma 1.1 and Corollary 1.4 the set B cannot be covered by ⋃ D. Hence |B \ ⋃ D| = 𝔠. □

Lemma 1.6 remains true for every σ-ideal mentioned in Remark 2.

2. Coverings on the real line

In [3] Carlson introduced the following definition.

Definition 4. We say that the set A ⊆ ℝ is a κ-covering if for every set B ⊆ ℝ of cardinality κ there exists a real number x ∈ ℝ such that B + x ⊆ A.

Analogously, a set A ⊆ ℝ is a <κ-covering if every set B ⊆ ℝ of cardinality less then κ can be translated into it (cf. [7]). Of course, these definitions are reasonable also for other uncountable Abelian Polish groups.

Nowik in his papers studied partitions of the Cantor space 2^{ω} into regular (Borel) ω-coverings. He constructed such a partition of size continuum ([8]) and a partition into two sets, one F_{σ}, one G_{δ}, having some special property. We present analogous and even stronger results concerning irregular (Bernstein) sets.

First we prove that we can find a partition of the real line into two Bernstein sets having no covering properties.

Theorem 2.1. There exists a partition of the real line ℝ into two sets A, B such that each of them is a Bernstein set and none of them is a 2-covering.

Proof. Let Perf(ℝ) = {P_{α} : α < 𝔠} and ℝ = {r_{α} : α < 𝔠} be fixed enumerations of all perfect subsets of the real line and of the reals, respectively. By transfinite induction we build two increasing sequences (A_{α})_{α<𝔠}, (B_{α})_{α<𝔠} of subsets of ℝ such that for every α < 𝔠 the following conditions are satisfied:

(1) |A_{α}| = |B_{α}| = |α| · ω;

(2) r_{α} ∈ A_{α} ∪ B_{α};
(3) A_{α} ∩ P_{α} ≠ ∅, B_{α} ∩ P_{α} ≠ ∅;
(4) A_{α} ∩ B_{α} = ∅.

Moreover, to ensure that A_{α} and B_{α} are not 2-coverings we want them to satisfy two more conditions:

(5) (∀x ∈ A_{α})({x − 1, x + 1} ⊆ B_{α});

(6) (∀x ∈ B_{α})({x − 1, x + 1} ⊆ A_{α}).

Now, the set {0, 1} cannot be translated neither into A_{α} nor into B_{α}.

We are able to fulfill all these conditions because being at the αth step of our construction we know that | ⋃_{β<α}(A_{β} ∪ B_{β})| < 𝔠 and for every β < α we have (A_{β} ∪ B_{β}) + ℤ = A_{β} ∪ B_{β}.

Finally, we put A = ⋃_{α<𝔠} A_{α} and B = ⋃_{α<𝔠} B_{α}. These sets are Bernstein sets because of (3), form a partition of ℝ because of (2) and (4) and are not 2-coverings as neither are sets A_{α} and B_{α}. □

The next theorem is in contrast with the previous one.

Theorem 2.2. There is a partition {B_{ξ} : ξ < 𝔠} of the real line into Bernstein sets such that for every ξ < 𝔠 the set B_{ξ} is a <cf(𝔠)-covering.

Proof. Let κ = cf(𝔠) and let (c_{α})_{α<κ} be a cofinal increasing sequence of elements of 𝔠. Let us fix an increasing sequence (R_{α})_{α<κ} of subsets of ℝ and a sequence (P_{α})_{α<κ} of families of perfect subsets of ℝ such that

ℝ = ⋃_{α<κ} R_{α}, Perf(ℝ) = ⋃_{α<κ} P_{α}

and |R_{α}| = |P_{α}| = |c_{α}|.

By transfinite induction we build a sequence of families ({B_{ξ}^{α} : ξ < c_{α}})_{α<κ} satisfying the following conditions:

(1) for every α < κ and for every ξ < c_{α} we have |B_{ξ}^{α}| = |c_{α}|;

(2) for every α < κ sets from the family {B_{ξ}^{α} : ξ < c_{α}} are pairwise disjoint;

(3) for every ξ < 𝔠 and every α_{1} < α_{2} < κ such that ξ < c_{α_{1}} we have B_{ξ}^{α_{1}} ⊆ B_{ξ}^{α_{2}};

(4) for every α < κ the intersection B_{ξ}^{α} ∩ P is nonempty for every ξ < c_{α} and every perfect set P from the family P_{α};

(5) for every α < κ and every ξ < c_{α} there exists x ∈ ℝ such that x+R_{α} ⊆ B_{ξ}^{α}. We obtain such a sequence as follows. Assume that we are at the αth step of the construction, so we have already built families {B_{ξ}^{β} : ξ < c_{β}} for β < α. One can observe that the cardinality of the union of all sets B_{ξ}^{β} constructed so far (let us denote this sum by S) is small:

|S| = | ⋃_{β<α} ⋃_{ξ<c_{β}} B_{ξ}^{β}| ≤ |c_{α}| · |c_{α}| · |α| = |c_{α}| < 𝔠.

For every ξ < c_{α} let us put

B_{ξ}^{<α} = ⋃_{β<α} B_{ξ}^{β}

(the set B_{ξ}^{<α} is empty for ⋃_{β<α} c_{β} ≤ ξ < c_{α}). Let us notice that there are at most c_{α} many real numbers x such that (x + R_{α}) ∩ S ≠ ∅. Hence we can recursively enlarge every set B_{ξ}^{<α} adding to it a set x_{ξ} + R_{α} for some x_{ξ} ∈ ℝ and keeping all enlarged sets pairwise disjoint – it is enough to fulfill (5). To fulfill (4) we have to enlarge our sets once more adding recursively to each of them one point from every set P ∈ P_{α}. Again, we can do this without losing disjointness. As a result we obtain a family {B_{ξ}^{α} : ξ < c_{α}} which fulfills conditions (2)–(5). But the condition (1) is also fulfilled because constructing every set B_{ξ}^{α} we have added |c_{α}| many new points.

Finally, we put

B_{ξ} = ⋃_{α<κ} B_{ξ}^{α}

(assuming that B_{ξ}^{α} = ∅ for α < min{η : ξ < c_{η}}).

Thanks to (2) the family {B_{ξ} : ξ < 𝔠} consists of pairwise disjoint sets and without problems we can extend them to get a partition of ℝ. By (4) every set B_{ξ} is a Bernstein set. Moreover, the condition (5) is enough to ensure that every set B_{ξ} is a < κ-covering. It is because every subset of the real line of cardinality smaller than κ is a subset of one of the R_{α}’s. □

On the other hand, as the only 𝔠-covering subset of the real line is the set ℝ itself, we have the following fact.

Fact 2.3. Assume CH. Then there is no Bernstein set which is an ω_{1}-covering.

Now, one can pose the following question.

Question 2. Assume 𝔠 > ω_{1} = cf(𝔠). Is it true that there exists an ω_{1}-covering which is a Bernstein set?

It is worth mentioning that in the proof of Theorem 2.2 we have succeeded in constructing relevant ω_{1}-coverings because we have been able to cover every set of size ω_{1} by a set of size smaller then continuum, taken from the fixed family of size at most continuum. Let us notice that it is not possible to answer Question 2 using the similar method as in the proof of Theorem 2.2 since we have the following observation which is a special case of the fact that if λ is singular then cov(λ, λ, cf(λ)^{+}, 2) > λ.

Fact 2.4 (see [11]). Assume that 𝔠 = ω_{ω_{1}}. Then there is no family B ⊆ [ℝ]^{<𝔠} of size continuum such that every subset of ℝ of size ω_{1} is covered by some set from the family B.

If we deal with completely I-nonmeasurable sets instead of Bernstein sets then we can construct even a <𝔠-covering on condition the σ-ideal I has the Steinhaus property and its uniformity is not too big.

Proposition 2.5. Assume that I ⊆ P(ℝ) is a σ-ideal having the Steinhaus property and such that non(I) < 𝔠. Then there exists a <𝔠-covering which is completely Inonmeasurable.

Proof. Let us fix a set N ∉ I such that |N| = non(I) and put C = (N + ℚ)^{c}. Suppose now that B ∈ Borel(ℝ) \ I. Then from the Steinhaus property of I we obtain that there exists a rational q ∈ ℚ such that q ∈ C^{c} − B. Hence B ∩ C^{c} ≠ ∅. As |C^{c}| < 𝔠 we have also B ∩ C ≠ ∅, so the set C is completely I-nonmeasurable.

Moreover, the set C is a <𝔠-covering. Indeed, suppose that there exists a set A ∈ [ℝ]^{<𝔠} such that for every x ∈ ℝ we obtain (A + x) ∩ C^{c} ≠ ∅. For every x ∈ ℝ let us fix a_{x} ∈ A such that a_{x} + x ∈ C^{c}. Then there exists c ∈ C^{c} such that |{x ∈ ℝ : a_{x} + x = c}| > |A|. But all reals c − x = a_{x} ∈ A are different and we have got a contradiction. □

3. S-coverings

We can interpret κ-coverings in terms of coloring sets. Namely, we can treat a κ-covering as set which can color every set of size κ monochromatically. From this point of view we may ask about a family of sets which can color every set of size κ in such a way that different points in the given set have different colors. This leads us to the following definition.

Definition 5. A family A of pairwise disjoint subsets of the real line is called a κ-S-covering if |A| = κ and

(∀F ∈ [ℝ]^{κ})(∃t ∈ ℝ)(F + t ⊆ ⋃ A ∧ (∀A ∈ A)|(F + t) ∩ A| = 1).

This definition is reasonable also for other uncountable Abelian Polish groups. First we prove a relation between 2-S-coverings and 2-coverings.

Theorem 3.1. Assume that {A_{0}, A_{1}} is a partition of the real line and a 2-S-covering. Then at least one of the sets A_{0}, A_{1} is a 2-covering.

Proof. Assume that none of the sets A_{0}, A_{1} is a 2-covering. It means that there are positive reals a, b such that for every x, y ∈ A_{0} we have x − y ≠ a and for every x, y ∈ A_{1} we have x − y ≠ b. We will show that the set {0, a + b} cannot be S-covered by {A_{0}, A_{1}}.

Indeed, let us fix any x ∈ A_{0}. Then x+a ∈ A_{1} and, consequently, x+a+b ∈ A_{0}. Analogously, if x ∈ A_{1} then x + b + a ∈ A_{1}, which ends the proof. □

Now we focus our attention on constructing κ-S-coverings which consist of Bernstein sets or completely I-nonmeasurable sets and such that none of their elements is a κ-covering (which is opposite to the situation from Theorem 3.1).

Theorem 3.2. Let κ be a cardinal number such that 2 < κ < 𝔠. If 2^{κ} ≤ 𝔠 then there exists a partition {B_{ξ} : ξ < κ} of the real line such that

(1) (∀ξ < κ) B_{ξ} is a Bernstein set,

(2) (∀ξ < κ) B_{ξ} is not a 2-covering,
(3) {B_{ξ} : ξ < κ} is a κ-S-covering.

Proof. Let Perf(ℝ) = {P_{α} : α < 𝔠} and ℝ = {r_{α} : α < 𝔠} be fixed enumerations of all perfect subsets of the real line and of the reals, respectively. Let us also enumerate the set [ℝ]^{κ} = {F_{α} : α < 𝔠}. By transfinite induction we build a sequence ({A_{ξ}^{α} : ξ < κ})_{α<𝔠} of families of subsets of ℝ of size less than continuum such that for every α < 𝔠 the following conditions are fulfilled:

(1) for every different ξ_{1}, ξ_{2} < κ the sets A_{ξ_{1}}^{α} and A_{ξ_{2}}^{α} are disjoint;

(2) for every ξ < κ the intersection A_{ξ}^{α} ∩ P_{α} is nonempty;

(3) there exists t_{α} ∈ ℝ such that t_{α} + F_{α} ⊆ ⋃_{ξ<κ} A_{ξ}^{α} and for every ξ < κ we

have |(t_{α} + F_{α}) ∩ A_{ξ}^{α}| = 1;
(4) there exists ξ < κ such that r_{α} ∈ A_{ξ}^{α};
(5) for every ξ < κ and every β < α we have A_{ξ}^{β} ⊆ A_{ξ}^{α};
(6) for every ξ < κ and every x, y ∈ A_{ξ}^{α} we have |x − y| ≠ 1;
(7) for every ξ < κ we have |A_{ξ}^{α}| ≤ |α| · ω.

Suppose that we have already constructed the sequence ({A_{ξ}^{β} : ξ < κ})_{β<α} for some α < 𝔠. Let A_{ξ} = ⋃_{β<α} A_{ξ}^{β} and A = ⋃_{ξ<κ} A_{ξ}. We can observe that there are not many ”bad translations” of the set F_{α}, namely the set

T = {t ∈ ℝ : (∃x ∈ F_{α})(∃a ∈ A) |t + x − a| = 1 ∨ t + x = a}

has the cardinality less then 𝔠. Thus we can choose a real t_{α} ∉ T. Next we choose a subset Y ⊆ P_{α} of size κ such that

(Y + {0, 1, −1}) ∩ ((t_{α} + F_{α}) ∪ A) = ∅.

Let {a_{ξ} : ξ < κ} and {b_{ξ} : ξ < κ} be enumerations of sets t_{α} + F_{α} and Y, respectively, and let Â_{ξ}^{α} = A_{ξ} ∪ {a_{ξ}, b_{ξ}} for ξ < κ. Finally, if r_{α} ∉ Y ∪ (t_{α} + F_{α}) ∪ A then we fix ξ_{0} < κ such that Â_{ξ_{0}}^{α} ∩ {r_{α} − 1, r_{α} + 1} = ∅ and put A_{ξ_{0}}^{α} = Â_{ξ_{0}}^{α} ∪ {r_{α}}. In all other cases we put A_{ξ}^{α} = Â_{ξ}^{α} and our construction is completed.

Let B_{ξ} = ⋃_{α<𝔠} A_{ξ}^{α} for ξ < κ. Then B_{ξ} is a Bernstein set thanks to the condition (2) and is not a 2-covering thanks to the conditions (5) and (6). The conditions (1) and (4) ensure us that the family {B_{ξ} : ξ < κ} is a partition of ℝ and the condition (3) makes this family a κ-S-covering. □

Remark 3. Let us observe that if κ is countable then the condition 2^{κ} ≤ 𝔠 is fulfilled. In general we need extra set theoretic assumptions. For example it is enough to assume Martin’s Axiom, which implies that 2^{κ} = 𝔠 for ω ≤ κ < 𝔠 (see [5]).

In more general situation, constructing S-coverings consisting of completely Inonmeasurable subsets of a given Polish group, none of which is a 2-covering is a bit more complicated. That is why we need some additional assumptions about a σ-ideal I.

Theorem 3.3. Let (X, +) be an uncountable Abelian Polish group with a complete metric d. Let I ⊆ P(X) be a σ-ideal such that

(∀B ∈ Borel(X) \ I)(∀D ∈ [I]^{<𝔠}) |B \ ⋃ D| = 𝔠

and there exists a ∈ range(d), a ≠ 0 such that

(∀x ∈ X) {y ∈ X : d(x, y) = a} ∈ I.

If κ is a cardinal number such that 2^{κ} = 𝔠, then there exists a family {B_{ξ} : ξ < κ} of pairwise disjoint subsets of X such that

(1) (∀ξ < κ) B_{ξ} is a completely I-nonmeasurable set,

(2) (∀ξ < κ) B_{ξ} is not a 2-covering,

(3) {B_{ξ} : ξ < κ} is a κ-S-covering.

Proof. Without loss of generality we can assume that a = 1. Let Borel(X) \ I = {P_{α} : α < 𝔠} be an enumeration of all I-positive Borel subsets of X. Let us also enumerate the set [X]^{κ} = {F_{α} : α < 𝔠}. We proceed similarly as in the proof of Theorem 3.2, constructing a sequence ({A_{ξ}^{α} : ξ < κ})_{α<𝔠} of families of subsets of X of size less than continuum such that for every α < 𝔠 the following conditions are fulfilled:

(1) for every different ξ_{1}, ξ_{2} < κ the sets A_{ξ_{1}}^{α} and A_{ξ_{2}}^{α} are disjoint;

(2) for every ξ < κ the intersection A_{ξ}^{α} ∩ P_{α} is nonempty and we have |A_{ξ}^{α}| ≤

|α| · ω;

(3) there exists t_{α} ∈ X such that t_{α} + F_{α} ⊆ ⋃_{ξ<κ} A_{ξ}^{α} and for every ξ < κ we

have |(t_{α} + F_{α}) ∩ A_{ξ}^{α}| = 1;
(4) for every ξ < κ and every β < α we have A_{ξ}^{β} ⊆ A_{ξ}^{α};
(5) for every ξ < κ and every x, y ∈ A_{ξ}^{α} we have d(x, y) ≠ 1.

Assume that we are at the αth step of the construction. Let A_{ξ} = ⋃_{β<α} A_{ξ}^{β} and A = ⋃_{ξ<κ} A_{ξ}. Moreover, let C = ⋃_{x∈F_{α}} ⋃_{a∈A}{t ∈ X : d(t + x, a) = 1}. Then the set T = C ∪ (A − F_{α}) is the set of ”bad translations” of the set F_{α}. But C is a collection of less then continuum many unit spheres and |A − F_{α}| < 𝔠 so according to our assumptions the complement of T is of size continuum. Thus we can choose t_{α} ∉ T.

Analogously, we can choose a subset Y ⊆ P_{α} of size κ such that

Y ∩ ((t_{α} + F_{α}) ∪ A ∪ {x ∈ X : (∃a ∈ (t_{α} + F_{α}) ∪ A) d(x, a) = 1}) = ∅.

Finally, we enumerate sets t_{α} + F_{α} = {a_{ξ} : ξ < κ} and Y = {b_{ξ} : ξ < κ}, put A_{ξ}^{α} = A_{ξ} ∪ {a_{ξ}, b_{ξ}} for ξ < κ and we are done.

Let B_{ξ} = ⋃_{α<𝔠} A_{ξ}^{α} for ξ < κ. Then {B_{ξ} : ξ < κ} is the needed family. □

Remark 4. Let us observe that in Theorem 3.3 we can replace the assumption

(∀B ∈ Borel(X) \ I)(∀D ∈ [I]^{<𝔠}) |B \ ⋃ D| = 𝔠

by a stronger, but shorter assumption, namely add(I) = 𝔠.

When our Polish space is simply a Euclidean vector space and we deal with meagre or null sets, we can omit one assumption in Theorem 3.3.

Corollary 3.4. Let I = N or I = M. Then for every cardinal number κ such that 2^{κ} = 𝔠 there exists a family {B_{ξ} : ξ < κ} of pairwise disjoint subsets of X such that (1) (∀ξ < κ) B_{ξ} is a completely I-nonmeasurable set,

(2) (∀ξ < κ) B_{ξ} is not a 2-covering,

(3) {B_{ξ} : ξ < κ} is a κ-S-covering.

Proof. It is enough to observe that we can repeat the proof of Theorem 3.3. Indeed, our choice of Y (and t_{α}) is possible because thanks to Lemma 1.6 after removing less than continuum many unit spheres from an I-positive Borel set we have still continuum many points left. □

Corollary 3.4 remains true for every σ-ideal fulfilling conditions mentioned in Remark 2.

Just as in case of Theorem 3.2, assuming Martin’s Axiom we obtain from Theorem 3.3 a suitable κ-S-covering for every κ < 𝔠. For example, we get a result concerning an S-covering made of Lebesgue completely nonmeasurable sets in ℝ^{n}.

Corollary 3.5. Assume Martin’s Axiom and 𝔠 = ℵ_{2}. Then there exists a family {B_{ξ} : ξ < 𝔠} of pairwise disjoint subsets of ℝ^{n} such that

(1) (∀ξ < 𝔠) λ_{∗}(B_{ξ}) = 0 and λ_{∗}(ℝ^{n} \ B_{ξ}) = 0,
(2) (∀ξ < 𝔠) B_{ξ} is not a 2-covering,
(3) {B_{ξ} : ξ < 𝔠} is a ω_{1}-S-covering,

where λ_{∗} denotes the inner Lebesgue measure in ℝ^{n}.

Proof. Immediate from Theorem 3.3, Corollary 3.4 and Remark 4 together with the fact that under Martin’s Axiom the additivity of the σ-ideal of Lebesgue null sets is equal to continuum. □

Theorem 3.3 gives us a κ-S-covering separately for every κ < 𝔠. It occurs that we can do this uniformly.

Definition 6. A family A of pairwise disjoint subsets of an uncountable Abelian Polish group (X, +) is called a <κ-S-covering

(∀F ∈ [X]^{<κ})(∃t ∈ X)(F + t ⊆ ⋃ A ∧ (∀A ∈ A)|(F + t) ∩ A| ≤ 1).

Theorem 3.6. Let (X, +) be an uncountable Abelian Polish group with a complete metric d. Let I ⊆ P(X) be a σ-ideal such that

(∀B ∈ Borel(X) \ I)(∀D ∈ [I]^{<𝔠}) |B \ ⋃ D| = 𝔠

and there exists a ∈ range(d), a ≠ 0 such that

(∀x ∈ X) {y ∈ X : d(x, y) = a} ∈ I.

If for every κ < 𝔠 we have 2^{κ} ≤ 𝔠 then there exists a family {B_{ξ} : ξ < 𝔠} of pairwise disjoint subsets of X such that

(1) (∀ξ < 𝔠) B_{ξ} is a completely I-nonmeasurable set,

(2) (∀ξ < 𝔠) B_{ξ} is not a 2-covering,

(3) {B_{ξ} : ξ < 𝔠} is a <𝔠-S-covering.

Proof. The construction is analogous to this from the proof of Theorem 3.3. □

4. I-coverings on the plane

In this chapter we focus our attention on the plane ℝ^{2} treated as a Polish group. According to Definition 4 we can investigate a κ-covering as a subset of the plane such that every planary set of size κ can be translated into it. However, we may also generalize this definition letting sets of size κ to be not only translated but moved by any isometry.

Definition 7. We say that a set A ⊆ ℝ^{2} is a κ-I-covering if

(∀B ∈ [ℝ^{2}]^{κ})(∃φ : ℝ^{2} → ℝ^{2})(φ is an isometry and φ[B] ⊆ A).

It occurs that we cannot partition the plane into two sets none of which is a 2-I-covering.

Theorem 4.1. If {A_{0}, A_{1}} is a partition of ℝ^{2} then one of the sets A_{0}, A_{1} is a 2-I-covering.

Proof. Suppose that A_{0} is not a 2-I-covering. Then there exists a positive real d such that none two points in A_{0} are at a distance of d from each other. Let us fix any a ∈ A_{0} and consider a circle C with a center a and a radius equal to d. Next, let us fix a halfline that starts from a and consider such a sequence (a_{n})_{n<ω} of elements of this halfline that d(a, a_{n}) = (n + 2)d for all n < ω. Then for every real x ∈ [(n + 1)d, (n + 3)d] there exists a point p ∈ C such that d(p, a_{n}) = x.

Observe now that C ⊆ A_{1}. Moreover, at least one of every two consecutive elements of the sequence (a_{n})_{n<ω} belongs to A_{1}. Hence for every x > 0 we can find two elements of A_{1} which are at a distance of x from each other. Consequently, the set A_{1} is a 2-I-covering. □

Next two theorems show that from the point of view of Bernstein sets there is a big difference between 2-I-coverings and 3-I-coverings.

Theorem 4.2. Every Bernstein set is a 2-I-covering.

Proof. Let B ⊆ ℝ^{2} be a Bernstein set. To show that B is also a 2-I-covering let us fix two different points a, b ∈ ℝ^{2}. It is enough to observe that any circle with a center in a fixed point c ∈ B and a radius d(a, b) (where d stands for a standard Euclidean metric) is a perfect set, thus meets B. □

Theorem 4.3. There exists a Bernstein set which is not a 3-I-covering.

Proof. Let Perf(ℝ^{2}) = {P_{α} : α < 𝔠} be a fixed enumeration of all perfect subsets of ℝ^{2}. We build by transfinite induction two sequences (a_{α})_{α<𝔠}, (b_{α})_{α<𝔠} of elements of the plane satisfying the following conditions:

(1) (∀α < 𝔠) a_{α}, b_{α} ∈ P_{α},

(2) {a_{α} : α < 𝔠} ∩ {b_{α} : α < 𝔠} = ∅,
(3) (∀α, β, γ < 𝔠)(d(a_{α}, a_{β}) ≠ 1 ∨ d(a_{α}, a_{γ}) ≠ 1 ∨ d(a_{β}, a_{γ}) ≠ 1).

Suppose that we have already constructed (a_{ξ})_{ξ<α} and (b_{ξ})_{ξ<α} for some α < 𝔠. Since the set A = {(a_{ξ_{1}}, a_{ξ_{2}}) : ξ_{1}, ξ_{2} < α ∧ d(a_{ξ_{1}}, a_{ξ_{2}}) = 1} has at most |α × α| < 𝔠 elements and for every pair (a_{ξ_{1}}, a_{ξ_{2}}) ∈ A there are only two points with distance 1 from both a_{ξ_{1}} and a_{ξ_{2}} we can pick a_{α} ∈ P_{α} \ ({a_{ξ} : ξ < α} ∪ {b_{ξ} : ξ < α}) such that d(a_{α}, a_{ξ_{1}}) ≠ 1 or d(a_{α}, a_{ξ_{2}}) ≠ 1 for all ξ_{1}, ξ_{2} < α. Let b_{α} be any element of P_{α} \ ({a_{ξ} : ξ ≤ α} ∪ {b_{ξ} : ξ < α}).

Let us put B = {a_{α} : α < 𝔠}. The condition (2) ensures B is a Bernstein set. To show that B is not a 3-I-covering it is enough to observe that there is no equilateral triangle of sides of length 1 with all vertices in B. □

When we replace Bernstein sets by completely I-nonmeasurable sets then it occurs that the theorem analogous to Theorem 4.2 may not be true.

Theorem 4.4. Let I = N or I = M. Then there exists a completely I-nonmeasurable planary set which is not a 2-I-covering.

Proof. Let Borel(X) \ I = {B_{α} : α < 𝔠} be an enumeration of all I-positive Borel subsets of X. We build by transfinite induction two sequences (a_{α})_{α<𝔠}, (b_{α})_{α<𝔠} of elements of the plane satisfying the following conditions:

(1) (∀α < 𝔠) a_{α}, b_{α} ∈ B_{α},

(2) {a_{α} : α < 𝔠} ∩ {b_{α} : α < 𝔠} = ∅,

(3) (∀α, β < 𝔠) d(a_{α}, a_{β}) ≠ 1. Assume that we are at an αth step of the construction. Let

D = B_{α} \ ⋃_{β<α}{a ∈ ℝ^{2} : d(a, a_{β}) = 1}.

From Lemma 1.6 we get |D| = 𝔠. Let us pick a_{α} ∈ D \ {a_{β} : β < α} and let b_{α} ∈ B_{α} \ ({a_{β} : β ≤ α} ∪ {b_{β} : β < α}).

Finally, the set B = {a_{α} : α < 𝔠} is completely I-nonmeasurable and not a 2-I-covering. □

References

[1] T. Bartoszyński, H. Judah, Set Theory: On the structure of the real line (A. K. Peters, Wellesley, Massachusetts, 1995).

[2] A. Beck, H.H. Corson, A.B. Simon, The interior points of the product of two subsets of a locally compact group, Proc. Amer. Math. Soc. 9, 648–652 (1958).

[3] T.J. Carlson, Strong measure zero and strongly meager sets, Proc. Amer. Math. Soc. 118, 577–586 (1993).

[4] J. Cichoń, M. Morayne, R. Rałowski, C. Ryll-Nardzewski, S. Żeberski, On nonmeasurable unions, Topology Appl. 154, 884–893 (2007).

[5] T. Jech, Set theory, The Third Millennium Edition (Springer-Verlag, 2002).

[6] E.J McShane, Images of sets satysfying the condition of Baire, Ann. Math. 51, 380–386 (1950).

[7] K. Muthuvel, Application of covering sets, Colloq. Math. 80, 115–122 (1999).

[8] A. Nowik, Some topological properties of ω-covering sets, Czechoslovak Math. J. 50(125), 865–877 (2000).

[9] A. Nowik, On extended version of ℵ_{0}-covering sets and their applications, Tatra Mt. Math. Publ. 35, 13–23 (2007).

[10] R. Rałowski, Remarks on nonmeasurable unions of big point families, accepted to Math. Log. Quart.

[11] S. Shelah, Cardinal Arithmetic, Chapter IX, Oxford Logic Guides, Vol. 29, (Oxford Univ. Press, London and New York, 1994).

[12] J.R. Shoenfield, The problem of predicativity, in: Essays on the foundations of mathematics

(Magnes Press, Hebrew Univ., Jerusalem, 1961).

E-mail address, Jan Kraszewski: jan.kraszewski@math.uni.wroc.pl

E-mail address, Robert Rałowski: robert.ralowski@pwr.wroc.pl E-mail address, Przemysław Szczepaniak: pszczepaniak@math.uni.opole.pl E-mail address, Szymon Żeberski: szymon.zeberski@pwr.wroc.pl

Jan Kraszewski, Mathematical Institute, University of Wrocław, pl. Grunwaldzki 2/4, 50-384 Wrocław, Poland.

Robert Rałowski and Szymon Żeberski , Institute of Mathematics and Computer Sciences, Wrocław University of Technology, Wybrzeże Wyspiańskiego 27, 50-370 Wrocław, Poland.

Przemysław Szczepaniak , Institute of Mathematics and Computer Sciences, University of Opole, ul. Oleska 48, 45-052 Opole, Poland.