TWO STARS
JANUSZ PAWLIKOWSKI AND MARCIN SABOK
We investigate an operation ∗ on the subsets of P(R).
It is connected with the strong measure zero sets as well as the
strongly meager sets. We give simple proofs of two theorems of
Solecki. The rst one says that the family of countable sets is a
xed-point of the operation ∗∗ (∗ applied twice). The second one
says that there exists a translation invariant σ -ideal I on R such
that I ∗ is the family of countable sets.
Abstract.
1.
Introduction and notation
We will be investigating subsets of the real line R and families of such
subsets. The family of all countable subsets of R will be of particular
interest, so let us denote it by C . Other important families are the
family of sets of Lebesgue's measure zero, which we shall denote by N ,
and the family of meager sets denoted by M. We will focus on the
algebraic structure of the real line. For A, B ∈ P(R) write A + B =
{a + b : a ∈ A, b ∈ B} and −A = {−a : a ∈ A}.
Denition 1. For A ∈ P(P(R)) let
A∗ = {B ∈ P(R) : ∀A ∈ A A + B 6= R}.
The operation ∗ behaves nicely when applied several times. In particular, the following facts hold.
Fact 1.
Fact 2.
Fact 3.
For
A ∈ P(P(R)) A ⊆ A∗∗ .
For
A, B ∈ P(P(R))
For
if
∗∗∗
A ∈ P(P(R)) A
A⊆B
∗
=A
then
B ∗ ⊆ A∗ .
.
One of the main motivations for investigating the above operation
is connected with the family of strongly null sets and the result of
Galvin, Mycielski and Solovay [2].
Denition 2. We say that A ∈ P(R) is strongly null if for every
sequence hεn : n ∈ ωi of positive real numbers there exists a sequence
hI
Sn : n ∈ ωi of open intervals such that |In | ≤ εn for each n and A ⊆
n<ω In . The family of strongly null sets is denoted by SN .
research was supported by MNiSW grant 1 PO3A 029 28.
1
2
JANUSZ PAWLIKOWSKI AND MARCIN SABOK
Theorem 1 (Galvin, Mycielski, Solovay). M∗ = SN
Strongly null sets are related to the Borel conjecture (BC) which says
that SN = C . By the result of Laver ([3]), BC is independent of ZFC. A
measure counterpart of BC, the Dual Borel Conjecture (DBC), says that
N ∗ = C . It is also independent (Carlson, [1]).
Note however, that if any of these conjectures holds, then the family
of countable sets is closed under the operation ∗∗ . In [4] Seredy«ski
asked if this can be proved directly in ZFC.
Problem 1 (P 1368). Is it true that C ∗∗ = C ?
The armative answer to his question was given by Solecki in [5].
However, his proof is quite sophisticated.
2. C
is closed under ∗∗
Let us present another but astonishingly short solution to Problem 1.
Theorem 2. C ∗∗ = C.
By Fact 1, C ∗∗ ⊇ C . To show the reverse inclusion take any
uncountable U ⊆ R. Showing U 6∈ C ∗∗ amounts to nding A ∈ C ∗ such
that
Proof.
U + A = R.
After shrinking U if neccessary, we may assume that |U | = ω1 and that
there exists a Hamel basis H ⊇ U such that |H \ U | = 2ω . In order
to construct A enumerate all countable subsets of R in a sequence
(Cα : α < 2ω ). Now, inductively construct yα ∈ R such that
(Cα + yα − U ) ∩ (Cβ + yβ − U ) = ∅
S
for each α 6= β . This can be done since β<α Cβ + yα + Cα is expressed
by only |β| · ℵ0 elements of H , so at stage α we may pick yα as the rst
unused element of H \ U . Let
[
A=R\
(Cα + yα )
α<2ω
Claim 1. A ∈ C ∗ .
Proof.
Any Cα can be translated via yα outside of A.
Claim 2. U + A = R.
TWO STARS
3
Proof. Fix t ∈ R. We need to show that U + t meets A. Suppose,
S
towards a contradiction, that U + t ⊆ α<2ω Cα + yα . As yα were
chosen so that Cα + yα − U are disjoint, there is at most one α such
that U + t meets Cα + yα and thus U + t ⊆ Cα + yα is countable. 3.
A translation invariant
σ -ideal
In the paper [5] the previous result is established by showing that
there exists a translation invariant σ -ideal I such that C = I ∗ . Under
either of BC or DBC this indeed is the case. The family C ∗ , however,
fails to be an ideal, namely it is not closed under nite unions because
C ∗ contains the ideals of meager and of null sets, and it is well known
that the real line is a union of a meager set and a null set. So the
result doesn't trivially follow from Theorem 2. Let us present another
construction of such an ideal.
Theorem 3.
that
There exists a translation invariant
σ -ideal I
on
I ∗ = C.
R
such
The group (R, +) is clearly isomorphic to i<2ω Q (via expressions in a Hamel basis). We will substitute Z2 for Q in order to simplify
notation. In the original case the proof is analogous. After this substitution, the group is ([κ]<ω , ⊕), where κ = 2ω and ⊕ is the symmetric
dierence. Since we may as well put κ = 2ω + ω1 , let us carry out the
entire construction in the group ([2ω + ω1 ]<ω , ⊕) (denoted by G).
Let Λ be the set of all injections of ω1 into 2ω . For s ∈ G and λ ∈ Λ
dene
L
Proof.
||s|| = min{α ∈ ω1 : (2ω + α, 2ω + ω1 ) ∩ s = ∅}
||s||λ = min{α ∈ ω1 : λ[(α, ω1 )] ∩ s = ∅}
For Z ∈ NSω1 (nonstationary subsets of ω1 ) let
AZ = {s ∈ G : ||s|| ∈ Z}
and for λ ∈ Λ let
Bλ = {s ∈ G : ||s|| < ||s||λ }.
Now, dene I to be the σ -ideal generated via translations and countable unions by the family of sets AZ (Z ∈ NSω1 ) and Bλ (λ ∈ Λ).
Lemma 1.
I is a proper
σ -ideal.
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JANUSZ PAWLIKOWSKI AND MARCIN SABOK
Proof. For Zi ∈ NSω1 , λi ∈ Λ and si , ti ∈ G (i < ω ) nd α ∈ ω1 such
S
that α 6∈ i Zi and α > ||si ||, ||ti ||, ||ti ||λi (for all i < ω ). We claim that
{2ω + α} 6∈
[
(AZi ⊕ si ) ∪
i<ω
[
(Bλi ⊕ ti ).
i<ω
Indeed, if {2ω + α} ∈ AZi ⊕ si , then {2ω + α} ⊕ si ∈ AZi , contradicting
||{2ω + α} ⊕ si || = α 6∈ Zi . On the other hand, if {2ω + α} ∈ Bλi ⊕ ti ,
then {2ω + α} ⊕ ti ∈ Bλi , contradicting ||{2ω + α} ⊕ ti || = α > ||ti ||λi =
||{2ω + α} ⊕ ti ||λi
Lemma 2. I ∗ = C
Let us take any S ⊆ G of size ω1 and show there is C ∈ I such
that C ⊕S = G. Without loss of generality S is a ∆-system with kernel
∅.
Case 1◦ . Uncountably many s ∈ S meet [2ω , 2ω + ω1 ).
Since any set of size ω1 contains a nonstationary subset of size ω1 , we
may nd such NSω1 3 Z ⊆ {||s|| : s ∈ S}. We claim that AZ ⊕ S = G.
Indeed, for any t ∈ G nd s ∈ S such that ||s|| ∈ Z and ||s|| > ||t||.
Then ||s ⊕ t|| = ||s|| ∈ Z , so s ⊕ t ∈ AZ and t ∈ AZ ⊕ s.
Case 2◦ . Uncountably many s ∈ S miss [2ω , 2ω + ω1 ).
Find λ ∈ Λ such that ∀α < ω1 ∃s ∈ S λ(α) ∈ s. Now, we claim that
Bλ ⊕ S = G. For any t ∈ G let s ∈ S be such that ||s||λ > ||t||, ||t||λ .
Then ||s ⊕ t||λ = ||s||λ > ||t|| = ||s ⊕ t|| (the last equality holds because
s ∩ [2ω , 2ω + ω1 ) = ∅). In that case s ⊕ t ∈ Bλ , so t ∈ Bλ ⊕ s.
Proof.
References
[1] Carlson T. J., Strong measure zero and strongly meager sets, Proceedings of the
American Mathematical Society, Vol. 118, No. 2 (1993), pp. 577-586
[2] Galvin F., Mycielski J., Solovay R. M., Strong measure zero sets, Notices
of the American Mathematical Society Vol. 26, No. 3 (1979), A-280
[3] Laver R., On the consistency of Borel's conjecture, Acta Mathematica, Vol. 137,
No. 3-4 (1976), pp. 151-169
[4] Seredy«ski W., Some operations related with translations, Colloquium Mathematicum Vol. 57 (1989), pp. 203-219
[5] Solecki S., Translation invariant ideals, Israel Journal of Mathematics, Vol. 135,
No. 1 (2003), pp. 93-110
Mathematical Institute, Wroclaw University, pl. Grunwaldzki 2/4,
50-384 Wroclaw, Poland
E-mail address
: [email protected], [email protected]