On cardinal characteristics associated with the strong measure zero ideal
OON CARDINAL CHARACTERISTICS ASSOCIATED WITH THESTRONG MEASURE ZERO IDEAL
MIGUEL A. CARDONA
Abstract.
Let SN be the strong measure zero σ -ideal. We prove a result providingbounds for cof( SN ) which implies Yorioka’s characterization of the cofinality of thestrong measure zero. In addition, we use forcing matrix iterations to construct a modelof ZFC that satisfies add( SN ) = cov( SN ) < non( SN ) < cof( SN ). Introduction
In this paper we continue the study of [CMRM] on the cardinal characteristics of thecontinuum associated with the ideal of strong measure zero sets. In general these cardinalsare defined as follows. Let I be an ideal on P ( X ) containing all the finite subsets of X .Define the cardinal characteristics associated with I by:add( I ) := min {|J | : J ⊆ I and (cid:91) J / ∈ I} the additivity of I ;cov( I ) := min {|J | : J ⊆ I and (cid:91) J = X } the covering of I ;non( I ) := min {| A | : A ⊆ X and A / ∈ I} the uniformity of I ;cof( I ) := min {|J | : J ⊆ I is cofinal in (cid:104)I , ⊆(cid:105)} the cofinality of I . Figure 1 shows the “trivial” inequalities between the cardinal characteristics associatedwith I . b b bb bb b ℵ add( I ) cov( I )non( I ) cof( I ) | X | | X | Figure 1.
Cardinal characteristics associated with I . An arrow x → y means that (provably) x ≤ y .Classical examples are the cardinal characteristics in Cicho´n’s diagram (see Figure 2),which is composed by the cardinal characteristics associated with M , N , K and C , where M is the family of meager subsets of R , N is the family of Lebesgue measure zero subsetsof R , K is the σ -ideal generated by the subsets of R whose intersection with Q ∗ (the setof irrational numbers) is compact in Q ∗ , and C is the σ -ideal of countable subsets of R . Itis known that add( K ) = non( K ) = b , add( C ) = non( C ) = ℵ , cov( K ) = cof( K ) = d , and Mathematics Subject Classification.
Key words and phrases.
Strong measure zero sets, cardinal invariants, matrix iteration.This work was supported by the Austrian Science Fund (FWF) P30666. Recipent of a DOC Fellowshipof the Austrian Academy of Sciences at the Institute of Discrete Mathematics and Geometry, TU Wien. a r X i v : . [ m a t h . L O ] M a r MIGUEL A. CARDONA b b b b bb bb b b b b ℵ add( N ) add( M ) cov( M ) non( N ) b d cov( N ) non( M ) cof( M ) cof( N ) c Figure 2.
Cicho´n’s diagram. x → y means that (provably) x ≤ y , and thedashed arrows indicate that add( M ) = min { b , cov( M ) } and cof( M ) =max { d , non( N ) } .cov( C ) = cof( C ) = c , where b , d and c are the bounding number, dominating number andthe size of R , respectively.Borel [Bor19] introduced the notion of strong measure zero sets (see Definition 3.1).Borel [Bor19] conjectured that each subset of the real line that has strong measure zerois countable, which is known as Borel’s Conjecture (BC). Sierpi´nski [Sie28] showed thatthe Continuum Hypothesis implies the existence of an uncountable set of real numbersof strong measure zero, and Laver [Lav76] proved the consistency of BC with ZFC byforcing, i.e, BC cannot be proven nor refuted in ZFC.The cardinal characteristics associated with the ideal of strong measure zero sets havebeen interesting objects of research, in particular when related to the cardinals in Cicho´n’sdiagram. Denote by SN the ideal of strong measure zero subsets of R .The following holds in ZFC:(S1) (Carlson [Car93]) add( N ) ≤ add( SN ),(S2) cov( N ) ≤ cov( SN ) ≤ c ,(S3) (Miller [Mil81]) cov( M ) ≤ non( SN ) ≤ non( N ) and add( M ) = min { b , non( SN ) } ,(S4) ([Osu08]) cof( SN ) ≤ d .On the other hand, the following inequalities are consistent with ZFC :(C1) (Goldstern, Judah and Shelah [GJS93]) cof( M ) < add( SN ) ,(C2) (Pawlikowski [Paw90]) cov( SN ) < add( M ),(C3) c < cof( SN ) (follows from CH),(C4) ([Yor02]) cof( SN ) < c ,(C5) ([CMRM]) non( SN ) < cov( SN ) < cof( SN ).(C6) ([CMRM]) cof( N ) < cov( SN ).Yorioka introduced a characterization of SN in terms of σ -ideals I f , parametrized byfunctions f ∈ ω ω , which we call Yorioka ideals (see Definition 3.2). More concretely, SN = (cid:84) {I f | f ∈ ω ω increasing } and I f ⊆ N . Figure 3 summarizes the relationship be-tween the cardinal invariants associated with Yorioka ideals and the cardinals in Cicho´n’sdiagram, see e.g. [KO08, Osu08, CM19].Yorioka also gave a important description of cof( SN ), namely Theorem 1.1 ([Yor02, Thm. 2.6]) . If add( I f ) = cof( I f ) = κ for all increasing f ∈ ω ω then cof( SN ) = d κ (the dominating number of κ κ ) . Original statement abreviated thanks to Figure 3
N THE STRONG MEASURE ZERO IDEAL 3
To prove Theorem 1.1, Yorioka constructed a dominating family (cid:104) f α | α < κ (cid:105) alongwith a matrix (cid:104) A βα : α, β < κ (cid:105) of subsets of the Cantor space 2 ω fulling the followingproperties:(i) ∀ α, β < κ ( A βα ⊆ ω is a dense G δ set and A βα ∈ I f α );(ii) ∀ α, β, β (cid:48) < κ ( β ≤ β (cid:48) → A βα ⊆ A β (cid:48) α );(iii) ∀ α < κ ∀ A ∈ I f α ∃ β < κ ( A ⊆ A βα ); and(iv) ∀ α < κ ∀ B ∈ I f α ( α > → (cid:84) γ<α A γ (cid:114) B (cid:54) = ∅ ).This gives a Tukey isomorphism between SN and (cid:104) κ κ , ≤(cid:105) (where ≤ is interpreted aspointwise).In this paper, the author introduces the notion of dominating system (see Definition3.4), which improves this construction and refines Theorem 1.1 by providing bounds tocof( SN ) without the hypothesis add( I f ) = cof( I f ) for all f . Concretely the author provesthe following main result. Theorem A (Theorems 3.6 and 3.10) . If there is some λ -dominating system on a directedset (cid:104) S, ≤ S (cid:105) then (i) SN (cid:22) T (cid:104) S λ , ≤(cid:105) . (ii) If minnon ≥ λ and (cid:104) S, ≤ S (cid:105) = (cid:104) κ × λ, ≤(cid:105) with κ ≤ λ , then (cid:104) λ λ , ≤(cid:105) (cid:22) T SN . The author with Mej´ıa and Rivera-Madrid [CMRM, Section 5] asks the following ques-tions: Is it consistent with ZFC that(Q1) add( SN ) < min { cov( SN ) , non( SN ) } ?(Q2) add( SN ) < non( SN ) < cov( SN ) < cof( SN )?(Q3) add( SN ) < cov( SN ) < non( SN ) < cof( SN )?Question (Q2) was answered partially by the author with Mej´ıa and Rivera-Madrid[CMRM]. Concretely, they showed that, in Sack’s model,add( SN ) = non( SN ) = ℵ < cov( SN ) = ℵ = c < cof( SN ) . This is the first result where more than two cardinal invariants associated with SN arepairwise different.In this work, we partially answer question (Q3). More concretely, we prove the follow-ing. Theorem B (Theorem 4.3) . Let κ ≤ λ be regular uncountable cardinals where κ <κ = κ and let λ , λ be cardinals such that λ <λ = λ , λ ≤ λ , λ λ = λ and λ ℵ = λ . Then thereis a cofinality preserving poset that forces add( SN ) = cov( SN ) = κ ≤ non( SN ) = λ ≤ cof( SN ) = λ and c = λ This is the second result where more than two cardinal invariants associated with SN are pairwise different.Now, to prove Theorem B, we use the method of matrix forcing iterations. To achievethis, we go through the following steps:(P1) We will force c = λ and d λκ × λ = d λ = λ by generalized Cohen forcing. Thesecardinals are introduced in Section 2.(P2) Afterwards, we construct the matrix. Along the matrix, we will construct a dom-inating family (cid:104) f γ | γ < λ (cid:105) along with a λ -dominating system on (cid:104) κ × λ, ≤(cid:105) .Thanks to Theorem A, the matrix forces cof( SN ) = λ . For the construction, weuse restricted localization forcing. MIGUEL A. CARDONA b bb bb b b b bb b b b bb bb bb b b bb b ℵ add( N ) minaddcov( N ) cov( I id ) cov( I f ) supcov non( M )add( I f )add( I id ) b d cof( M ) supcof cof( N ) c cof( I f )cof( I id )non( N )non( I id )non( I f )minnoncov( M )add( M ) Figure 3.
Extended Cicho´n’s diagram. σ (0) σ (1) σ (2) σ ( n ) . . . . . .ht σ Figure 4.
Functions σ and ht σ .(P3) The constructed matrix forces cov( M ) = cof( N ) = λ and add( N ) = non( M ) = κ ,so κ ≤ add( SN ) and non( SN ) = λ by (S1) and (S3). Since the matrix is obtainedby a FS iteration of length with cofinality κ , cov( SN ) ≤ κ .This paper is structured as follows. We review in Section 2 the basic notation andthe results this paper is based on. The notions of I f directed system and λ -dominatingsystem are introduced in Section 3, as well as the proof of Theorem A. In Section 4 weprove Theorem B. Finally, in Section 5 we present some open questions.2. Preliminaries
We start with the following basic notions. Let κ be an infinite cardinal. Denote byFn <κ ( I, J ) the poset of partial functions from I into J with domain of size < κ , orderedby ⊇ . If z is an ordered pair, z and z denotes the first and second component of z respectively. Set ω ↑ ω := { d ∈ ω ω : d (0) = 0 and d is increasing } . For any set A , id A denotes the identity function on A . For each σ ∈ (2 <ω ) ω define ht σ ∈ ω ω by ht σ ( i ) := | σ ( i ) | (see Figure 4).Typically, cardinal invariants of the continuum are defined through relational systemsas follows. A relational system is a triplet R = (cid:104) X, Y, R (cid:105) where R is a relation contained in N THE STRONG MEASURE ZERO IDEAL 5 X × Y . For x ∈ X and y ∈ Y , xRy is often read y R - dominates x . A family E ⊆ X is R -bounded if ∃ y ∈ Y ∀ x ∈ E ( xRy ). Dually, D ⊆ Y is R -dominating if ∀ x ∈ X ∃ y ∈ D ( xRy ).Such a relational system has two cardinal invariants associated with it: b ( R ) := min {| E | : E ⊆ X is R -unbounded } , d ( R ) := min {| D | : D ⊆ Y is R -dominating } . Let R (cid:48) := (cid:104) X (cid:48) , Y (cid:48) , R (cid:48) (cid:105) be another relational system. If there are maps Ψ : X → X (cid:48) and Ψ : Y (cid:48) → Y such that, for any x ∈ X and y (cid:48) ∈ Y (cid:48) , if Ψ ( x ) R (cid:48) y (cid:48) then xR Ψ ( y (cid:48) ),we say that R is Tukey below R (cid:48) , denoted by R (cid:22) T R (cid:48) . Say that R and R (cid:48) are Tukeyequivalent , denoted by R ∼ = T R (cid:48) , if R (cid:22) T R (cid:48) and R (cid:48) (cid:22) T R . Note that R (cid:22) T R (cid:48) implies b ( R (cid:48) ) ≤ b ( R ) and d ( R ) ≤ d ( R (cid:48) ).Let R := (cid:104) X, Y, R (cid:105) and R (cid:48) := (cid:104) X (cid:48) , Y (cid:48) , R (cid:48) (cid:105) be two relational systems. Set R ⊗ R (cid:48) := (cid:104) X × X (cid:48) , Y × Y (cid:48) , R ⊗ (cid:105) , where ( x, x (cid:48) ) R ⊗ ( y, y (cid:48) ) iff xRx (cid:48) and yR (cid:48) y (cid:48) . Fact 2.1 ([Bla10, Thm 4.11]) . max { d ( R ) , d ( R (cid:48) ) } ≤ d ( R ⊗ R (cid:48) ) ≤ d ( R ) · d ( R (cid:48) ) and b ( R ⊗ R (cid:48) ) = min { b ( R ) , b ( R (cid:48) ) } A directed set is a set S with a preorder ≤ S such that every finite subset of S has anupper bound. In other words, for any x and y in S there exists a z in S with x ≤ S z and y ≤ S z .Given a function b with domain ω such that b ( i ) (cid:54) = ∅ for all i < ω , and h ∈ ω ω ,define S ( b, h ) = (cid:81) n<ω [ b ( n )] ≤ h ( n ) . For x ∈ ω ω and ψ ∈ S ( b, h ), say that x ∈ ∗ ϕ iff ∀ ∞ n < ω ( x ( n ) ∈ ϕ ( n )), which is read ϕ localizes x . Example 2.2.
Let κ and λ be non-zero cardinals, and let (cid:104) S, ≤ S (cid:105) be a directed set.(1) Consider the relational system Ed := (cid:104) ω ω , ω ω , (cid:54) = ∗ (cid:105) , where for f, g ∈ ω ω , f (cid:54) = ∗ g iff ∃ n < ω ∀ m ≥ n ( f ( m ) (cid:54) = g ( m )). By [BJ95, Thm. 2.4.1 & Thm. 2.4.7], b ( Ed ) =non( M ) and d ( Ed ) = cov( M ).(2) Define Lc ( ω, h ) := (cid:104) ω ω , S ( ω, h ) , ∈ ∗ (cid:105) (here, ω denotes the constant function with value ω ), which is a relational system. If h ∈ ω ω goes to infinity then add( N ) := b ( Lc ( ω, h ))and cof( N ) := d ( Lc ( ω, h )) (see [BJ95, Thm. 2.3.9]).(3) As a relational system, S is (cid:104) S, S, ≤ S (cid:105) , b ( S ) := b ( (cid:104) S, S, ≤ S (cid:105) ) and d ( S ) := d ( (cid:104) S, S, ≤ S (cid:105) ). Note that, if S has no maximum, then b ( S ) is regular and b ( S ) ≤ d ( S ). Evenmore, cf( d ( S )) ≥ b ( S ).(4) Consider the relational system D λS := (cid:104) S λ , S λ , ≤(cid:105) where X ≤ Y iff ∀ α < λ ( X ( α ) ≤ S Y ( α )). Define b λS := b ( D λS ) and d λS := d ( D λS ).(5) Denote b λκ × λ := b ( D λκ × λ ) and d λκ × λ := d ( D λκ × λ ) where λ × κ is ordered by ( α, β ) ≤ ( α (cid:48) , β (cid:48) ) iff α ≤ α (cid:48) and β ≤ β (cid:48) .(6) Assume that λ is infinite. Consider the relational system D λS ( ≤ ∗ ) := (cid:104) S λ , S λ , ≤ ∗ (cid:105) where X ≤ ∗ Y iff ∃ α < λ ∀ β ∈ [ α, λ )( X ( β ) ≤ Y ( β )). Set b λS ( ≤ ∗ ) := b ( D λS ( ≤ ∗ )) and d λS ( ≤ ∗ ) := d ( D λS ( ≤ ∗ )).(7) When κ is infinite, define b κ := b κκ ( ≤ ∗ ) and d κ := d κκ ( ≤ ∗ ) (this is a particular case of D λS ( ≤ ∗ ) with S = λ = κ ). These are the well known unbounding number of κ κ and dominating number of κ κ respectively.The following result follows from (3). Corollary 2.3.
Let S be a directed partial order and let λ be a non-zero cardinal. If S has no maximum then ℵ ≤ cf( b λS ) = b λS ≤ cf( d λS ) ≤ d λS ≤ | S | λ . We prove some results about the cardinal invariants associated with D λS and D λS ( ≤ ∗ ). MIGUEL A. CARDONA
Lemma 2.4.
Let S be a directed partial order and let λ be a non-zero cardinal. If S hasno maximum then (i) b ( S ) = b λS ≤ d ( S ) ≤ d λS ≤ d ( S ) λ ≤ | S | λ . Even more, S (cid:22) T D λS . (ii) If λ < b ( S ) then D λS ∼ = T S . (iii) If λ ≤ λ (cid:48) are non-zero cardinals, then D λS (cid:22) T D λ (cid:48) S . In particular, d λS ≤ d λ (cid:48) S . (iv) If λ is infinite then d λS ( ≤ ∗ ) ≤ d λS and b ( S ) ≤ b λS ( ≤ ∗ ) .Proof. (i) Clearly d λS ≤ d ( S ) λ ≤ | S | λ and b ( S ) ≤ d ( S ). It remains to prove that b ( S ) = b λS and d ( S ) ≤ d λS . To see b ( S ) ≤ b λS , let B ⊆ S λ with | B | < b ( S ). For ζ < λ , define Γ ζ := { f ( ζ ) | f ∈ B } . Since | Γ ζ | < b ( S ), choose i ζ ∈ S such that f ( ζ ) ≤ S i ζ for all f ∈ B . Define g ∈ S λ by g ( ζ ) := i ζ . Then g bounds B .For the converse inequality, it suffices to prove that S (cid:22) T D λS . For f ∈ S λ putΨ ( f ) := f (0). On the other hand, for i ∈ S define f i ∈ S λ by f i ( ξ ) := i foreach ξ < λ , so put Ψ ( i ) := f i . It is clear that if f i ≤ f then i ≤ S f (0). Hence, d ( S ) ≤ d λS and b λS ≤ b ( S ).(ii) By (i), it is enough to show that D λS (cid:22) T S . To this end let f ∈ S λ . Define D := { f ( β ) : β < λ } . Since | D | < λ , choose i f in S such that f ( β ) ≤ S i f for each β < λ and put Ψ ( f ) := i f .Finally, put Ψ ( j ) := f j for j ∈ S as in (i). It remains to check that, if i f ≤ S j then f ≤ f j . Fix β < λ . Then f ( β ) ≤ S i f ≤ S j = f j ( β ).(iii) Define Ψ : S λ (cid:48) → S λ as follows: For f ∈ S λ (cid:48) set Ψ ( f ) := f (cid:22) λ .To define Ψ : S λ → S λ (cid:48) , for g ∈ S λ define g ∗ ∈ S λ (cid:48) by setting, for any ξ < λ (cid:48) , g ∗ ( ξ ) := g ( ξ ) if ξ < λ , and g ∗ ( ξ ) = 0 otherwise. Put Ψ ( g ) := g ∗ . It is clear thatif g ∗ ≤ f then g ≤ f (cid:22) λ .(iv) Obvious because D λS ( ≤ ∗ ) (cid:22) T D λS . (cid:3) Lemma 2.5. If λ is an infinite cardinal and S has no maximum, then d λS ( ≤ ∗ ) > λ .Proof. Let F := { f ξ | ξ < λ } ⊆ S λ be a family of size λ , and let K be a bijection from λ onto λ × λ . Define f ∈ S λ as follows: for any β < λ we can choose s β > f K ( β ) ( β ) (such s β exists because S has no maximum) and put f ( β ) := s β . For each ξ, η < λ set β ξ,η := K − ( ξ, η ), so K ( β ξ,η ) = ξ and f ( β ξ,η ) > f ξ ( β ξ,η ). Then |{ β < λ | f ( β ) > f ξ ( β ) }| = λ . (cid:3) In the next theorem we give a characterization of d λS . Theorem 2.6. If λ is an infinite cardinal and S has no maximum, then d λS = d λS ( ≤ ∗ ) · sup κ<λ { d κS } . Proof.
Clearly d λS ( ≤ ∗ ) · sup ξ<λ { d | ξ | S } ≤ d λS because d | ξ | S ≤ d λS (Lemma 2.4(ii)) and d λS ( ≤ ∗ ) ≤ d λS .For ξ < λ , choose D ξ ⊆ S ξ ≤ -dominating with | D ξ | = d | ξ | S . Take a ≤ ∗ -dominatingfamily D ⊆ S λ . For g ∈ D and h ∈ D ξ with ξ < λ define the function f g,h ∈ S λ by f g,h ( β ) := (cid:40) g ( β ) if β ≥ ξ , h ( β ) if β < ξ . Since |{ f g,h | g ∈ D ∧ ∃ ξ < λ ( h ∈ D ξ ) }| ≤ d λS ( ≤ ∗ ) · sup ξ<λ { d | ξ | S } · λ = d λS ( ≤ ∗ ) · sup ξ<λ { d | ξ | S } by Lemma 2.5, it sufficies to prove that this family is ≤ -dominating. To this end let f ∈ S λ . Find g ∈ D and ξ < λ such that f ( β ) ≤ g ( β ) for all β ≥ ξ . Then, for β < ξ set h ξ ( β ) := max { f ( β ) , g ( β ) } ∈ S ξ , so there is some h ∈ D ξ such that h ξ ≤ h . Therefore, f g,h dominates f everywhere. (cid:3) N THE STRONG MEASURE ZERO IDEAL 7
It is known that d λλ = d λ when λ is regular, even more, this follows from Theorem 2.6because d κλ = λ when κ < λ . However, d λλ = d λ cf ( λ ) in general. More details about d λκ canbe found in [Bre19]. Lemma 2.7. D λκ × λ ∼ = T D λκ ⊗ D λλ . In particular, d λκ × λ = max { d λκ , d λλ } .Proof. To define Ψ : ( κ × λ ) λ → κ λ × λ λ , for F ∈ ( κ × λ ) λ define f F ∈ κ λ and g F ∈ λ λ by setting f F ( ξ ) := F ( ξ ) and g F ( ξ ) := F ( ξ ) . Put Ψ ( F ) := ( f F , g F ). Now, defineΨ : κ λ × λ λ → ( κ × λ ) λ as follows: For f ∈ κ λ and g ∈ λ λ set F f,g ∈ ( κ × λ ) λ bysetting, for any ζ < λ , F f,g ( ζ ) = ( f ( ζ ) , g ( ζ )). Put Ψ ( f, g ) := F f,g . It is clear that if( f F , g F ) ≤ ⊗ ( f, g ) then F ≤ κ × λ F f,g . Also, if F f,g ≤ κ × λ F then ( f, g ) ≤ ⊗ ( f F , g F ). (cid:3) Definition 2.8.
Let γ, π be ordinals. A simple matrix iteration P = (cid:104) P α,ξ , ˙ Q α,ξ | α ≤ γ, ≤ ξ ≤ π (cid:105) fullfils the following requirements.(i) cof( γ ) > ω ,(ii) ˙ Q α, = P α, = C α := Fn( α × ω, < ξ < π , ∆( ξ ) < γ is non-limit and ˙ Q ξ is a P ∆( ξ ) ,ξ -name of a poset suchthat P γ, ξ forces it to be ccc, and(iv) P α,ξ +1 = P α, ξ ∗ ˙ Q α,ξ , where˙ Q α,ξ := (cid:40) ˙ Q ξ if α ≥ ∆( ξ ), otherwise,(v) for ξ limit, P α,ξ = limdir η<ξ P α,η .As a consequence, α ≤ β ≤ γ and ξ ≤ η ≤ π imply P α,ξ (cid:108) P β,η . Lemma 2.9 (See e.g. [Mej19, Cor. 2.6]) . Assume that P = (cid:104) P α,ξ , ˙ Q α,ξ | α ≤ γ, ξ ≤ π (cid:105) isa simple matrix iteration with cf( γ ) > ω . Then, for any ξ ≤ π ,(a) P γ,ξ is the direct limit of (cid:104) P α,ξ : α < γ (cid:105) , and(b) if ˙ f is a P γ,ξ -name of a function from ω into (cid:83) α<γ V α,ξ then ˙ f is forced to be equalto a P α,ξ -name for some α < γ . In particular, the reals in V γ,ξ are precisely thereals in (cid:83) α<γ V α,ξ . Theorem 2.10 ([Mej13, Thm. 10 & Cor. 1]) . Let P = (cid:104) P α,ξ , ˙ Q α,ξ | α ≤ γ, ξ ≤ π (cid:105) bea simple matrix iteration. If γ has uncountable cofinality, then P γ,π forces non( M ) = b ( Ed ) ≤ cf( γ ) ≤ d ( Ed ) = cov( M ) . To finish this section, we review the followig forcing notion:
Localization forcing is theposet
LOC := { ϕ ∈ S ( ω, id ω ) : ∃ m < ω ∀ i < ω ( | ϕ ( i ) | ≤ m ) } ordered by ϕ (cid:48) ≤ ϕ iff ϕ ( i ) ⊆ ϕ (cid:48) ( i ) for every i < ω . Recall that this poset is σ -linked andthat it adds an slalom ϕ ∗ in S ( ω, id ω ) that localizes all the ground model reals in ω ω , thatis, x ∈ ∗ ϕ ∗ for any x ∈ ω ω in the ground model.3. a connection between SN and d λκ × λ In this section we prove Theorem A.
Definition 3.1.
We say that X ⊆ ω has strong measure zero iff for each f ∈ ω ω thereis some σ ∈ (2 <ω ) ω with ht σ = f such that X ⊆ (cid:83) n<ω [ σ ( n )].Denote SN := { X ⊆ ω | X has strong measure zero } . MIGUEL A. CARDONA
Denote pw k : ω → ω the function defined by pw k ( i ) := i k , and define the relation (cid:28) on ω ω by f (cid:28) g iff ∀ k < ω ( f ◦ pw k ≤ ∗ g ).For σ ∈ (2 <ω ) ω set [ σ ] ∞ : = { x ∈ ω | ∃ ∞ n < ω ( σ ( n ) ⊆ x ) } = (cid:92) n<ω (cid:91) m (cid:62) n [ σ ( m )] Definition 3.2 (Yorioka [Yor02]) . Let f ∈ ω ω be an increasing function. Define I f := { X ⊆ ω | ∃ σ ∈ (2 <ω ) ω ( X ⊆ [ σ ] ∞ and h σ (cid:29) f ) } . Any family of this form is called a
Yorioka ideal .Yorioka [Yor02] has proved that I f is a σ -ideal when f is increasing. Moreover, SN = (cid:84) {I f | f increasing } . Denote minnon := min { non( I f ) | f increasing } . Lemma 3.3 ([Yor02, Lemma 3.7]) . Let A be a perfect subset of ω . Then there some f ∈ ω ω such that A / ∈ I f . The next definition plays a central role in the main results.
Definition 3.4.
Let S be a directed partial order. For each increasing function f ∈ ω ω ,we say that a family A f = (cid:104) A fi | i ∈ S (cid:105) of subsets of 2 ω is an I f directed system on S itif fulfills the following:(I) ∀ i ∈ S ( A fi ⊆ ω is dense G δ and A fi ∈ I f );(II) ∀ i, j ∈ S ( i ≤ j → A fi ⊆ A fj ) and(III) (cid:104) A fi | i ∈ S (cid:105) is cofinal in I f .Assume from now on that S has a minimun i . If λ is a cardinal and there is somedominating family { f γ | γ < λ } on ω ω such that A f γ = (cid:104) A f γ i | i ∈ S (cid:105) is an I f γ directsystem and ∀ γ < λ (cid:16) (cid:92) η<γ A f η i / ∈ I f γ (cid:17) , then we say that (cid:104) A f γ | γ < λ (cid:105) is a λ dominating system on S . Lemma 3.5.
Let S be a directed partial order and let λ be a uncountable cardinal. Assume cov( M ) = d = λ and that, for any increasing function f ∈ ω ω , there is some I f directedsystem on S . Then there is some λ -dominating system on S .Proof. Fix i := min( S ). Let (cid:104) h γ | γ < λ (cid:105) be a dominating family. For each γ < λ , wedenote A γi := A f γ i . We will construct f γ by recursion on γ < λ . Assume that (cid:104) f η | η < γ (cid:105) has been constructed. Now, let us assume that M is a transitive model for ZFC such that | M | < λ = cov( M ) and A ηi is coded in M for any η < γ .Cohen forcing adds a perfect set P of Cohen reals over M (see [BJ95, Lemma 3.3.2]),so P ⊆ (cid:84) η<γ A ηi . Since P is a perfect set, there is some g ∈ ω ω such that P / ∈ I g byLemma 3.3.Choose f γ ∈ ω ω increasing such that h γ ≤ f γ and g ≤ f γ . Then I f γ ⊆ I g and P / ∈ I f γ .But P ⊆ (cid:84) η<γ A ηi , hence (cid:84) η<γ A ηi / ∈ I g .Clearly, (cid:104) f γ | γ < λ (cid:105) is a dominating family. (cid:3) Below, we shall prove main Theorem A(i).
Theorem 3.6.
Assume that there is some λ -dominating system on S . Then SN (cid:22) T D λS . N THE STRONG MEASURE ZERO IDEAL 9
Proof.
For X ∈ SN , choose Ψ ( X ) := G X ∈ S λ such that X ⊆ (cid:84) γ<λ A γG X ( γ ) by Definition3.4 (III). Let F ∈ S λ . Note that (cid:84) γ<λ A γF ( γ ) ∈ SN because (cid:84) γ<λ A γF ( γ ) ⊆ A γF ( γ ) and A γF ( γ ) ∈ I f γ . Define Ψ ( F ) := (cid:84) γ<λ A γF ( γ ) Now assume that Ψ ( X ) ≤ F . Then G X ( γ ) ≤ F ( γ ) for all γ < λ , so by Definition3.4(II), X ⊆ (cid:84) γ<λ A γG X ( γ ) ⊆ (cid:84) γ<λ A γF ( γ ) . (cid:3) As a consequence:
Corollary 3.7.
If there is an λ -dominating system on S then cof( SN ) ≤ d λS and b ( S ) = b λS ≤ add( SN ) . We conclude this section with the proof of Theorem A(ii), which will be used in thefinal section. To prove it, we need the following lemma.
Lemma 3.8.
Let κ < λ be infinite cardinals. Assume minnon ≥ λ and that there is some λ -dominating system on κ × λ . Then, for any f ∈ λ λ , there exist G ∈ ( κ × λ ) λ and { x γα : γ < λ, α < κ } ⊆ ω such that(i) ∀ γ < λ ( { x γ (cid:48) α | γ (cid:48) ≤ γ, α < κ } ⊆ A γG ( γ ) ) ,(ii) ∀ γ < λ ∀ α < κ ( x γα ∈ (cid:84) γ (cid:48) <γ A γ (cid:48) G ( γ (cid:48) ) (cid:114) A γα,f ( γ ) ) , and(iii) ∀ γ < λ ( f ( γ ) ≤ G ( γ ) ) .Proof. We will recursively construct G ( γ ) ∈ κ × λ and x γα ∈ ω . Assume that we alreadyhave G ( γ (cid:48) ) and x γ (cid:48) α for any γ (cid:48) < γ and α < κ . Set B α := A γα,f ( γ ) ∪ { x γ (cid:48) β | γ (cid:48) < γ, β < κ } .Since { x γ (cid:48) β | γ (cid:48) < γ, β < κ } has size < λ , B α ∈ I f γ because κ < λ ≤ non( I f γ ). Then byDefinition 3.4, there is some x γα ∈ (cid:84) η<γ A η , (cid:114) B α . Note that { x γ (cid:48) α | γ (cid:48) ≤ γ, α < κ } ∈ I f γ .Then there must be a G ( γ ) ∈ κ × λ such that { x γ (cid:48) α | γ (cid:48) ≤ γ, α < κ } ⊆ A γG ( γ ) and f ( γ ) ≤ G ( γ ) . This contruction satisfies the required conditions. (cid:3) Lemma 3.9.
With the same asumptions as Lemma 3.8 and G ∈ ( κ × λ ) λ fulfilling itsconclusion, if β < λ and δ ≤ f ( β ) then (cid:84) γ<λ A γG ( γ ) (cid:54)⊆ A βα,δ for all α < κ .Proof. By Lemma 3.8 (i) and (ii), { x γα | γ < λ, α < κ } ⊆ (cid:84) γ<λ A γG ( γ ) and x βα / ∈ A βα,f ( β ) .Hence x βα / ∈ A βα,δ because δ ≤ f ( β ). (cid:3) Theorem 3.10.
Assume κ ≤ λ and that there is some λ -dominating system on κ × λ and minnon ≥ λ . Then D λλ (cid:22) T SN .Proof. When κ = λ , this is [Yor02, Thm 3.9] (with A βα := A f β α,α ). Assume κ < λ . For B ∈ SN , choose some F B ∈ ( κ × λ ) λ such that B ⊆ (cid:84) γ<λ A γF B ( γ ) . Define Ψ ( B ) := f B where f B ( γ ) := F B ( γ ) for every γ < λ . Now fix f ∈ λ λ , then by Lemma 3.8 and Lemma3.9 we can find some G f ∈ ( κ × λ ) λ fulfilling that, for each g ∈ λ λ , for each β < λ , if g ( β ) ≤ f ( β ) then (cid:92) γ<λ A γG f ( γ ) (cid:54)⊆ A βα,g ( β ) for all α < κ. Define Ψ ( f ) := (cid:84) γ<λ A γG f ( γ ) .Now assume that f (cid:54)≤ f B . We will show that (cid:84) γ<λ A γG f ( γ ) (cid:54)⊆ B . Since f (cid:54)≤ f B choose ξ < λ such that f ( ξ ) > f B ( ξ ). Then (cid:84) γ<λ A γG f ( γ ) (cid:54)⊆ A ξF B ( ξ ) . Thus (cid:84) γ<λ A γG f ( γ ) (cid:54)⊆ B because B ⊆ (cid:84) γ<λ A γF B ( γ ) . (cid:3) As a consequence, we get:
Corollary 3.11.
With the same hypothesis as in Lemma 3.10, cof( SN ) ≥ d λλ and add( SN ) ≤ b λλ = cof( λ ) . Model for the cardinal invariants associated with SN In this section, we prove Theorem B. But first we need the two following lemmas.The next lemma shows that a cofinal family in I f is produced by a localizating familyand a dominating family. Lemma 4.1 ([CM19, Thm. 3.12]) . Let f ∈ ω ω be an increasing function. Then there issome definable function Ψ f : ω ↑ ω × S ( ω, id) → I f such that, if(i) S ⊆ S ( ω, id) is a localizing family i.e, for any x ∈ ω ω there is some ϕ ∈ S such that x ∈ ∗ ϕ , and(ii) D ⊆ ω ↑ ω is a dominating family,then { Ψ f ( d, ϕ ) | d ∈ D and ϕ ∈ S } is cofinal in I f . The same proof actually yields:
Lemma 4.2.
Let M be a transitive model of ZFC with f ∈ ω ω ∩ M increasing. If d ∈ ω ↑ ω is dominating over M and ϕ ∈ S ( ω, id) is localizing over M , then A ⊆ Ψ f ( d, ϕ ) for allBorel A ∈ I f coded in M . Now, we are ready to prove our main Theorem B.
Theorem 4.3.
Let κ ≤ λ be regular uncountable cardinals where κ <κ = κ and let λ , λ be cardinals such that λ <λ = λ , λ ≤ λ , λ λ = λ and λ ℵ = λ . Then there is a cofinalitypreserving poset that forces (I) add( N ) = non( M ) = κ and cov( M ) = cof( N ) = λ . (II) add( SN ) = cov( SN ) = κ ≤ non( SN ) = λ ≤ cof( SN ) = d λ = d λκ × λ = λ and c = λ . Proof. Step 1.
We start with P := Fn <λ ( λ × λ, λ ). P is λ + -cc and < λ -closed, and thusit preserves cofinalities, and P forces d λ = 2 λ = λ . Step 2. In V P , let P := Fn <κ ( λ × λ, κ ). When κ < λ , P forces d λκ = 2 λ = λ and d λ = λ because P is λ -c.c (see e.g. [Car19, Lemma 2.6]); and if λ = κ , the same is forcedby step 1. Step 3. In V P ∗ P , let P := Fn <ω ( λ × ω, ω ), which forces c = λ and 2 λ = max { λ , λ } .In particular, d λκ × λ = λ because P is ccc and by Lemma 2.7. Step 4.
We work in V , := V P ∗ P ∗ P . We define the simple matrix iteration of height γ := λ and lenght π := λκ where the matrix iteration at each interval of the form[ λρ, λ ( ρ + 1)) for each ρ < κ is defined as follows. For each ε ∈ [ λρ, λ ( ρ + 1)), ε > ε = λρ + ξ for some ρ < κ and ξ < λ , put ∆( ε ) = ξ + 1 and ˙ Q ε := LOC V ∆( ε ) ,ε .Set P := P λ,λκ and V α,ξ := V P α,ξ , . We first prove that P forces κ ≤ add( N ) andcof( N ) ≤ λ . For each 0 < ε < λκ denote by ϕ ε ∈ V ∆( ε ) , ε +1 ∩ S ( ω, id) the generic slalomover V ∆( ε ) ,ε added by ˙ Q ∆( ε ) , ε = ˙ Q λ,ε = LOC V ∆( ε ) ,ε . Hence V λ,λκ | = κ ≤ add( N ) is aconsequence of the following Claim 4.4 (see e.g. [CM19, Claim 5.14]) . In V λ,λκ , each family of reals of size < κ islocalizated by some ϕ ε . N THE STRONG MEASURE ZERO IDEAL 11 bbbbb b bbbbb λρ f λ ( ρ f + 1) ε f = λρ f + ξ f ( ξ f < λ ) λρ f ξ f V , V , V ξ f , V ξ f +1 , V λ, CC V ξ f ,ε f V ,λκ V ,λκ V ξ f ,λκ V ξ f +1 ,λκ f ∈ V ξ f ,ε f ∩ ω ω ˙ Q ε f = LOC V ξf,εf φ ρ f ,ξ f d ρ f ,ξ f A fρ,ξ := Ψ f ( ˙ d fρ,ξ , ˙ φ fρ,ξ )For ρ < κ and ξ < λ define ε fρ,ξ := λ ( ρ f + ρ ) + ξ f + ξ f ∈ V λ,λκ ∩ ω ω Figure 5.
Matrix iterationOn the other hand, { ϕ ε | < ε < λκ } is a family of slaloms of size ≤ λ and, by Claim4.4, any member of V λ,λκ ∩ ω ω is localizated by some ϕ ε . Hence V λ,λκ | = cof( N ) ≤ λ .On the other hand, P adds κ -cofinally many Cohen reals by Lemma 2.10, so it forcesnon( M ) ≤ κ . By Theorem 2.10, P forces cov( M ) = d ( Ed ) ≥ λ . Therefore, P forces κ = add( N ) = non( M ) and cov( M ) = cof( N ) = λ . Now, P forces: κ ≤ add( SN ) by (S1) from the introduction;cov( SN ) ≤ κ because the lenght of the FS iteration on the top has cofinality κ and itis well known that such cofinality becomes an upper bound of cov( SN ) (see e.g. [BJ95,Lemma 8.2.6]);non( SN ) = λ by (S3) from the introduction;cof( SN ) = λ . Let f ∈ V λ,λκ ∩ ω ω be an increasing function. Then, there are some ξ f < λ and ρ f < κ such that f ∈ V ξ f ,ε f with ε f = λρ f + ξ f > ρ < κ and ξ < λ define ε fρ,ξ := λ ( ρ f + ρ ) + ξ f + ξ . Let ˙ ϕ fρ,ξ be the P ∆( ε fρ,ξ ) ,ε fρ,ξ +1 -nameof the slalom over V ∆( ε fρ,ξ ) , ε fρ,ξ and let ˙ d ρ,ξ be the P ∆( ε fρ,ξ ) , ε fρ,ξ +1 -name of some increasingdominating real over V ∆( ε fρ,ξ ) ,ε fρ,ξ added by ˙ Q ε fρ,ξ . Set A fρ, ξ := Ψ f ( ˙ d fρ,ξ , ˙ ϕ fρ,ξ ) (see Figure 5) . Claim 4.5. (cid:104) A fρ, ξ | ρ < κ and ξ < λ (cid:105) is an I f directed system.Proof. It is clear that (I) and (III) follow by Lemma 4.2. To see (II), note that ˙ ϕ fρ,ξ isan slalom over V ∆( ε fρ,ξ ) , ε fρ,ξ and ˙ d fρ,ξ is an increasing dominating real over V ∆( ε fρ,ξ ) , ε fρ,ξ , so A ⊆ A fρ, ξ for any A ∈ I f coded in V ∆( ε fρ,ξ ) , ε fρ,ξ by Lemma 4.2. In particular, A fρ (cid:48) , ξ (cid:48) ⊆ A fρ, ξ if ( ρ (cid:48) , ξ (cid:48) ) ≤ ( ρ, ξ ). (cid:3) We can choose a λ -dominating system (cid:104) A f γ | γ < λ (cid:105) by Lemma 3.5 because cov( M ) = d = λ . Therefore, in V λ,λκ , cof( SN ) ≤ d λκ × λ = λ by Theorem 3.6, and since minnon = λ ,cof( SN ) ≥ d λ = λ by Theorem 3.10. (cid:3) Open problems
In Theorem 3.10 we prove D λλ (cid:22) T SN assuming the existence of a λ -dominating systemon κ × λ . We ask if we could do the same omitting κ , concretely, Questions 5.1.
Assume that λ is an infinite cardinal and assume that S has no maximum.Do we have D λλ (cid:22) T SN whenever the conditions below are satisfied? (i) λ (cid:22) T S , (ii) there is a λ -dominating system on S , and (iii) minnon ≥ λ More generally,
Questions 5.2.
Assume that λ is an infinite cardinal and assume that S has no maximum.Do we have D λλ (cid:22) T SN whenever the conditions below are satisfied? (i) D λλ (cid:22) T D λS , (ii) there is a λ -dominating system on S , and (iii) minnon ≥ λ Concerning the consistency of the cardinals characteristics associated with SN , thefollowing summarises the current open questions. Questions 5.3.
Is it consistent with ZFC that (I) add( SN ) < min { cov( SN ) , non( SN ) } ? (II) add( SN ) < non( SN ) < cov( SN ) < cof( SN ) ? (III) add( SN ) < cov( SN ) < non( SN ) < cof( SN ) ? Any idea to solve Question (I) in the positive could be used to prove the consistencyof (II) and (III), for example using a matrix iteration construction as in this paper. Asmentioned in the introduction, the author with Mej´ıa and Rivera-Madrid solved Question(II) partially.In Theorem 4.3 (Thereom B) we answered Question (III) partially, but its consistencystill remains open. In this situation, the main issue is that tools to deal with add( SN )are still unknown. Acknowledgments.
The author is very thankful to professor D. Mej´ıa for all the guid-ance and support provided during the research that precedes this paper. He offered a lotof his time for discussions that concluded in the results that are presented in this text.The author is also grateful to professor T. Yorioka for his multiple discussions at the SetTheory Seminar in the Departament of Mathematics at Shizuoka University.
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Institute of Discrete Mathematics and Geometry, TU Wien, Wiedner Hauptstrasse8–10/104 A–1040 Wien, Austria.
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