Stability and measurability of the modified lower dimension
aa r X i v : . [ m a t h . C A ] M a r STABILITY AND MEASURABILITY OF THE MODIFIEDLOWER DIMENSION
RICH ´ARD BALKA, M ´ARTON ELEKES, AND VIKTOR KISS
Abstract.
The lower dimension dim L is the dual concept of the Assouaddimension. As it fails to be monotonic, Fraser and Yu introduced the modifiedlower dimension dim ML by making the lower dimension monotonic with thesimple formula dim ML X = sup { dim L E : E ⊂ X } .As our first result we prove that the modified lower dimension is finitelystable in any metric space, answering a question of Fraser and Yu.We prove a new, simple characterization for the modified lower dimension.For a metric space X let K ( X ) denote the metric space of the non-emptycompact subsets of X endowed with the Hausdorff metric. As an applicationof our characterization, we show that the map dim ML : K ( X ) → [0 , ∞ ] is Borelmeasurable. More precisely, it is of Baire class 2, but in general not of Baireclass 1. This answers another question of Fraser and Yu.Finally, we prove that the modified lower dimension is not Borel measurabledefined on the closed sets of ℓ endowed with the Effros Borel structure. Introduction
The concept of lower dimension was introduced by Larman [9] under the nameminimal dimension. While the Assouad dimension helps to understand the ‘thickest’part of a set, the lower dimension identifies its ‘thinnest’ part.
Definition 1.1.
The lower dimension of a metric space X is defined asdim L X = sup n α : there is a C > < r < R ≤ diam X and for all x ∈ X we have N r ( B ( x, R )) ≥ C (cid:18) Rr (cid:19) α ) , where diam denotes the diameter, B ( x, r ) is the closed ball of radius r centered at x , and N r ( E ) is the minimal number of sets of diameter at most r required to cover E . We will adopt the convention dim L ∅ = 0.For more information on the lower dimension and applications the reader canconsult Fraser’s monograph [2] and [1, 5, 6, 7, 10, 11]. It is easy to see thatdim L [0 ,
1] = 1 and E = [0 , ∪ { } satisfies dim L E = 0, so the lower dimension isneither monotone nor contains any information about the ‘thicker’ part of E . Fraserand Yu [4] introduced the following notion which overcomes these shortcomings. Mathematics Subject Classification.
Primary 28A75, 28A20.
Key words and phrases. modified lower dimension, finite stability, measurability, Baire class.The first author was supported by the MTA Premium Postdoctoral Research Program andthe National Research, Development and Innovation Office – NKFIH, grant no. 124749. Thesecond author was supported by the National Research, Development and Innovation Office –NKFIH, grants no. 124749 and 129211. The third author was supported by the National Research,Development and Innovation Office – NKFIH, grants no. 124749, 129211, and 128273.
Definition 1.2.
We define the modified lower dimension of a metric space X asdim ML X = sup { dim L E : E ⊂ X } . Although the modified lower dimension is a new concept, it has already foundseveral applications. First, the modified lower dimension can be used to obtainthe best known lower bound on the Assouad dimension of product sets, see [4,Proposition 4.6]. Second, the modified lower dimension yields optimal lower boundsin certain intersection theorems, where one considers the intersection of a generalset with the set of so-called badly approximable numbers, see [2, Theorem 14.2.1]and the paragraph preceding [2, Corollary 14.2.3]. The common theme in bothapplications is to obtain stronger results for sets whose modified lower dimensionis strictly larger than their lower dimension. A well-known family of such sets isthe so-called Bedford-McMullen carpets, which have equal Hausdorff and modifiedlower dimension, while their lower dimension is strictly smaller in general, see [2,Section 8.3].The first problem we are dealing with in the present paper concerns stability.Fraser and Yu [4, Proposition 4.3] proved finite stability for properly separated sets,and Fraser [2, Lemma 3.4.10] showed it for closed sets.
Theorem 1.3 (Fraser–Yu, Fraser) . Let ( X, ρ ) be a metric space and let E, F ⊂ X .We have dim ML ( E ∪ F ) = max { dim ML E, dim ML F } if either(1) dist( E, F ) = inf { ρ ( x, y ) : x ∈ E, y ∈ F } > , or(2) X = R d and E, F are closed.
Fraser and Yu [4, Question 9.3] asked the following, see also [2, Question 17.1.1].
Question 1.4 (Fraser–Yu) . Let X be a metric space and E, F ⊂ X , is it true that dim ML ( E ∪ F ) = max { dim ML E, dim ML F } ?In Section 2 we answer the above question in the affirmative. Theorem 2.2.
Let X be a metric space and let E, F ⊂ X . Then dim ML ( E ∪ F ) = max { dim ML E, dim ML F } . The main ingredient of the proof is the next lemma about the lower dimension,which might be interesting in its own right.
Lemma 2.1.
Let ( X, ρ ) be a metric space and let U ⊂ X be a non-empty open set.Then there exists an open set V ⊂ U such that dim L V ≥ dim L X . This immediately implies that the supremum in the definition of modified lowerdimension can be taken for bounded sets only.
Corollary 1.5.
For any metric space X we have dim ML X = sup { dim L E : E ⊂ X is bounded } . Remark 1.6.
Some authors define the lower dimension by replacing R ≤ diam X with R ≤ TABILITY AND MEASURABILITY OF THE MODIFIED LOWER DIMENSION 3
In Section 3 we prove two simple characterization theorems for the modifiedlower dimension. Besides interesting in their own rights, they will play a crucialrole in the proof of Theorem 4.3 as well. Recall that a natural number l is identifiedwith the set { , , . . . , l − } , the symbol l n stands for the set of functions from n into l , and l <ω = ∪ n ∈ N l n is the set of finite sequences in l . For s, t ∈ l <ω let s ⌢ t denote the concatenation of s and t . Definition 1.7.
Let (
X, ρ ) be a metric space and let k, l ≥ y s ∈ X for all s ∈ l <ω . We say that { y s } s ∈ l <ω is a ( k, l ) -regular set if(i) ρ ( y s , y t ) ≤ − kn − if s ∈ l n , t ∈ l n +1 and t extends s ,(ii) ρ ( y s , y t ) ≥ − kn +2 if s, t ∈ l n and s = t .We say that the ( k, l )-regular set { y s } s ∈ l <ω is a strongly ( k, l ) -regular set if inaddition to conditions (i) and (ii) it satisfies(iii) y s ⌢ = y s for all s ∈ l <ω .We obtain the following useful characterization of the modified lower dimension. Theorem 3.1.
Let ( X, ρ ) be a metric space and let α ≥ . The following statementsare equivalent:(1) dim ML X > α ;(2) X contains a strongly ( k, l ) -regular set with log lk log 2 > α . The word ‘strongly’ cannot be removed from the above theorem even in separablemetric spaces, see Example 3.2. However, for complete metric spaces this simplercharacterization holds.
Theorem 3.3.
Let ( X, ρ ) be a complete metric space and let α ≥ . The followingstatements are equivalent:(1) dim ML X > α ;(2) X contains a ( k, l ) -regular set with log lk log 2 > α . In Section 4 we consider the measurability of the modified lower dimension.Recall that K ( X ) is the space of non-empty compact sets of X endowed with theHausdorff metric, see Definition 4.2. Fraser and Yu [4, Question 9.3] asked the nextquestion, see also [2, Question 17.1.1]. Question 1.8 (Fraser–Yu) . Let X be a compact metric space. Is the mapping dim ML : K ( X ) → [0 , ∞ ] Borel measurable, and, if so, which Baire classes does itbelong to?
We answer the above question based on Theorem 3.3.
Theorem 4.3.
Let X be a metric space. Then dim ML : K ( X ) → [0 , ∞ ] is Borelmeasurable. More precisely, it is of Baire class . Remark 1.9.
As both finite sets and finite unions of closed balls are dense in K ( R d ), the map dim ML : K ( R d ) → [0 , d ] attains both the values 0 and d on densesets, therefore it has no point of continuity. However, it is well known that K ( R d ) isa Polish space [8, Theorem 4.25], and real-valued Baire class 1 functions on Polishspaces have points of continuity [8, Theorem 24.15]. Hence dim ML is not of Baireclass 1, so Theorem 4.3 is sharp in general. RICH´ARD BALKA, M´ARTON ELEKES, AND VIKTOR KISS
In Section 5 we consider ℓ = { x ∈ R ω : P ∞ n =0 | x ( n ) | < ∞} equipped with thenorm k x k = ∞ X n =0 | x ( n ) | . Let F ( ℓ ) be the set of closed subsets of ℓ . We endow F ( ℓ ) with the σ -algebra B generated by the sets(1.1) { F ∈ F ( ℓ ) : F ∩ U = ∅} , where U runs over the open subsets of ℓ . By [8, Theorem 12.6] the measurablespace ( F ( ℓ ) , B ) is standard Borel , that is, there is a Polish topology on F ( ℓ ) suchthat its family of Borel sets coincides with B . We prove the following theorem. Theorem 5.1.
The map dim ML : F ( ℓ ) → [0 , ∞ ] is not Borel measurable. Finite stability of the modified lower dimension
The goal of this section is to prove Theorem 2.2. First we need the followinglemma.
Lemma 2.1.
Let ( Y, ρ ) be a metric space and let U ⊂ Y be a non-empty open set.Then there exists an open set V ⊂ U such that dim L V ≥ dim L Y . In particular, dim ML U ≥ dim L Y .Proof. For y ∈ Y and r > U ( y, r ) be the open ball of radius r centered at y ,and for A ⊂ Y let U ( A, r ) = S { U ( y, r ) : y ∈ A } . Fix y ∈ U and 0 < ε < / Y \ U, { y } ) > ε . Set V = ∅ and V = U ( y , ε ). If V n is already definedfor some n ≥ V n +1 = U ( V n , ε n +1 ). Define V = ∞ [ n =1 V n . Clearly, V is open, and P ∞ n =1 ε n < ε yields V ⊂ U ( y , ε ) ⊂ U . Therefore, it isenough to prove that dim L V ≥ dim L Y . Let 0 < R < ε and y ∈ V be arbitrarilygiven, it is enough to show that there exists z ∈ V such that(2.1) U ( z, ( ε/ R ) ⊂ V ∩ U ( y, R ) . Let k, m be the unique positive integers such that y ∈ V k \ V k − and(2.2) 3 ε m +1 ≤ R < ε m . If m ≥ k then U ( y, ( ε/ R ) ⊂ U ( y, ε k +1 ) ⊂ V k +1 ⊂ V , so z = y satisfies (2.1).Finally, assume that k > m . Define y , . . . , y k ∈ V such that y k = y and y n ∈ V n ∩ U ( y n − , ε n ) for all 1 ≤ n ≤ k . Let z = y m . Then (2.2) and y m ∈ V m imply that(2.3) U ( z, ( ε/ R ) ⊂ U ( y m , ε m +1 ) ⊂ V m +1 ⊂ V. Inequalities ρ ( y, y m ) < k X n = m +1 ε n < ε m +1 and (2.2) yield(2.4) U ( z, ( ε/ R ) ⊂ U ( y m , ε m +1 ) ⊂ U ( y, ε m +1 ) ⊂ U ( y, R ) . Then (2.3) and (2.4) imply (2.1), and the proof is complete. (cid:3)
TABILITY AND MEASURABILITY OF THE MODIFIED LOWER DIMENSION 5
Theorem 2.2.
Let X be a metric space and let E, F ⊂ X . Then dim ML ( E ∪ F ) = max { dim ML E, dim ML F } . Proof.
Clearly, dim ML ( E ∪ F ) ≥ max { dim ML E, dim ML F } holds by the mono-tonicity of modified lower dimension, so it is enough to prove that(2.5) dim ML ( E ∪ F ) ≤ max { dim ML E, dim ML F } . Assume to the contrary that (2.5) does not hold. By shrinking E and F if necessarywe may suppose that(2.6) max { dim ML E, dim ML F } < dim L ( E ∪ F ) . First assume that E is dense in E ∪ F . The lower dimension is stable under takingclosures, see [3, Theorem 2.3]. Therefore, we obtaindim ML E ≥ dim L E = dim L E = dim L E ∪ F = dim L ( E ∪ F ) , which contradicts (2.6). Finally, suppose that E is not dense in E ∪ F . Then thereis an open set U ⊂ X such that U ∩ F = U ∩ ( E ∪ F ) = ∅ . Applying Lemma 2.1 for Y = E ∪ F implies that dim ML ( U ∩ ( E ∪ F ) ≥ dim L ( E ∪ F ). By the monotonicityof modified lower dimension we obtaindim ML F ≥ dim ML ( U ∩ F ) = dim ML ( U ∩ ( E ∪ F )) ≥ dim L ( E ∪ F ) , which contradicts (2.6). The proof is complete. (cid:3) Characterizations of the modified lower dimension
The goal of this section is to prove Theorems 3.1 and 3.3.
Theorem 3.1.
Let ( X, ρ ) be a metric space and let α ≥ . The following statementsare equivalent:(1) dim ML X > α ;(2) X contains a strongly ( k, l ) -regular set with log lk log 2 > α .Proof. First we prove (1) ⇒ (2). We may assume by shrinking X if necessary thatdim L X > α . Choose β so that α < β < dim L X . The definition of lower dimensionimplies that there exists a constant C > < r < R ≤ diam X and x ∈ X we have N r ( B ( x, R )) ≥ C (cid:0) Rr (cid:1) β . If diam X <
1, then by replacing C with C (diam X ) β we obtain that the above property holds for all R ≤
1, that is,(3.1) N r ( B ( x, R )) ≥ C (cid:18) Rr (cid:19) β for all 0 < r < R ≤ x ∈ X. As β > α , we can fix a large enough integer k such that k ≥ l = j C · ( k − β k satisfy log lk log 2 > α, where ⌊·⌋ denotes the integer part. Let us pick any y ∅ ∈ X and suppose that y s isalready defined for some n ∈ N and s ∈ l n . Let c ∈ { , . . . , l − } , we need to define y s ⌢ c ∈ X satisfying the conditions of Definition 1.7. This will give the required( k, l )-regular set.Let R = 2 − kn − and r = 2 − k ( n +1)+3 . By k ≥ < r < R ≤
1. Let Z be a maximal family of points from B ( y s , R ) containing y s such that(3.2) ρ ( z, z ′ ) ≥ − k ( n +1)+2 for distinct points z, z ′ ∈ Z. RICH´ARD BALKA, M´ARTON ELEKES, AND VIKTOR KISS
It is enough to prove that Z ≥ l . Indeed, then we can choose distinct points y s ⌢ c ∈ Z for all c ∈ { , . . . , l − } with y s ⌢ = y s . By Z ⊂ B ( y s , R ) we obtain ρ ( y s , y s ⌢ c ) ≤ − kn − for all c , so Definition 1.7 (i) holds. Inequality (3.2) yieldsthat the points y s ⌢ c satisfy Definition 1.7 (ii), while Definition 1.7 (iii) clearly holdsfor s .Finally, we prove Z ≥ l . Using the maximality of Z we obtain that(3.3) B ( y s , R ) ⊂ [ z ∈ Z B (cid:16) z, − k ( n +1)+2 (cid:17) . As the union in (3.3) is a covering of B ( y s , R ) with sets of diameter at most r , usingalso (3.1) we obtain that Z ≥ N r ( B ( y s , R )) ≥ C (cid:18) Rr (cid:19) β = C · ( k − β ≥ l, and the proof of (1) ⇒ (2) is complete.Now we prove (2) ⇒ (1). Assume that Y = { y s } s ∈ l <ω is a strongly ( k, l )-regularset in X with log lk log 2 > α . By the definition of modified lower dimension it is enoughto show for (1) that(3.4) dim L Y ≥ log lk log 2 . We will check the definition for arbitrarily given 0 < r < R ≤ diam Y ≤ y u ∈ Y for some u ∈ l <ω . Let n ≥ m ≥ − − kn +1 < R ≤ − k ( n − and 2 − k ( n + m +1)+1 ≤ r < − k ( n + m )+1 . Let s ∈ l n with s = u ↾ n if length( u ) ≥ n and s = u ⌢ (0 , . . . ,
0) if length( u ) < n ,where length( u ) denotes the number of coordinates of u and u ↾ n is the restrictionof u to its first n coordinates. Note that by Definition 1.7 (i) if t extends s then(3.6) ρ ( y s , y t ) < − kn . Note also that if length( u ) < n then y s = y u , so in either case,(3.7) ρ ( y s , y u ) < − kn . By Definition 1.7 (ii) we obtain ρ ( y t , y t ′ ) ≥ − k ( n + m )+2 for all distinct sequences t, t ′ ∈ l n + m , so each set of diameter r < − k ( n + m )+1 can contain at most one of these points.We claim that(3.8) N r ( B ( y u , R ) ∩ Y ) ≥ l m . The inequality is clear if m = −
1. Let m ≥ t ∈ l n + m that extend s is exactly l m , and each such y t satisfies ρ ( y s , y t ) < − kn by(3.6). Thus (3.7) yields that y t is contained in the ball B ( y u , R ), so (3.8) holds.Hence (3.5) and (3.8) yield that α = log lk log 2 satisfies (cid:18) Rr (cid:19) α ≤ αk ( m +2) = l m ≤ l N r ( B ( y u , R ) ∩ Y ) . Hence applying the definition of the lower dimension with the absolute constant C = l − yields (3.4). This completes the proof of the theorem. (cid:3) TABILITY AND MEASURABILITY OF THE MODIFIED LOWER DIMENSION 7
Example 3.2.
The word ‘strongly’ cannot be removed from Theorem 3.1, even inseparable metric spaces: For all n ∈ N and s ∈ n let us define y s = n − X i =0 (2 s ( i ) − − i − , where we assume that y ∅ = 0. Let X = { y s } s ∈ <ω be a countable metric spacewith the inherited metric from R . It is easy to see that X is a (2 , X is discrete, so every subset of X isof lower dimension 0, hence dim ML X = 0.However, if we assume completeness then we can replace strong regularity byregularity. Theorem 3.3.
Let ( X, ρ ) be a complete metric space and let α ≥ . The followingstatements are equivalent:(1) dim ML X > α ;(2) X contains a ( k, l ) -regular set with log lk log 2 > α .Proof. Implication (1) ⇒ (2) follows from the analogous implication of Theorem3.1. Hence, it remains to prove (2) ⇒ (1). Assume that { y s } s ∈ l <ω is a ( k, l )-regularset in X with log lk log 2 > α . For all n ∈ N and s ∈ l n let D s = B ( y s , − kn ). Let usdefine K = ∞ \ n =0 [ s ∈ l n D s . By the definition of modified lower dimension it is enough to show for (1) that(3.9) dim L K ≥ log lk log 2 . We will check the definition for arbitrarily given 0 < r < R ≤ diam K ≤ x ∈ K . Let n ≥ m ≥ − − kn +1 < R ≤ − k ( n − and 2 − k ( n + m +1)+1 ≤ r < − k ( n + m )+1 . Choose the unique s ∈ l n with x ∈ D s . Then R > − kn +1 yields D s ⊂ B ( x, R ). ByDefinition 1.7 (ii) we obtain dist( D t , D t ′ ) ≥ − k ( n + m )+1 for all distinct sequences t, t ′ ∈ l n + m , so each set of diameter r < − k ( n + m )+1 can intersect at most one ofthese balls. As D t ∩ K = ∅ and the number of sets D t with t ∈ l n + m and D t ⊂ D s is l m , we obtain(3.11) N r ( B ( x, R ) ∩ K ) ≥ l m , which holds for m = − α = log lk log 2 satisfies (cid:18) Rr (cid:19) α ≤ αk ( m +2) = l m ≤ l N r ( B ( x, R ) ∩ K ) . Hence applying the definition of the lower dimension with the absolute constant C = l − yields (3.9), and the proof of the theorem is complete. (cid:3) RICH´ARD BALKA, M´ARTON ELEKES, AND VIKTOR KISS Measurability of the modified lower dimension
The goal of this section is to prove Theorem 4.3 using a characterization fromthe previous section. First we need some preparation.
Definition 4.1.
Let Y be a metric space and let A ⊂ Y . We say that A is F σ if itis a countable union of closed sets, A is G δ if it is a countable intersection of opensets, and A is G δσ if it is a countable union of G δ sets. We say that f : Y → [0 , ∞ ]is of Baire class { y ∈ Y : f ( y ) > α } and { y ∈ Y : f ( y ) < α } are G δσ for all α ≥ Definition 4.2.
For a metric space (
X, ρ ) let ( K ( X ) , d H ) be the set of non-empty compact subsets of X endowed with the Hausdorff metric , that is, for every K , K ∈ K ( X ) we have d H ( K , K ) = min { r : K ⊂ B ( K , r ) and K ⊂ B ( K , r ) } , where B ( A, r ) = { x ∈ X : ∃ y ∈ A such that ρ ( x, y ) ≤ r } . Theorem 4.3.
Let X be a metric space. Then dim ML : K ( X ) → [0 , ∞ ] is Borelmeasurable. More precisely, it is of Baire class .Proof. Let (
X, ρ ) be a metric space, we prove that dim ML : K ( X ) → [0 , ∞ ] is ofBaire class 2. It is enough to show that { K ∈ K ( X ) : dim ML K > α } is an F σ setfor each α ≥
0. Indeed, this would imply that { K ∈ K ( X ) : dim ML K < α } is a G δσ set for all α ≥
0, so dim ML is of Baire class 2. For integers k, l ≥ F ( k, l ) = { K ∈ K ( X ) : K contains a ( k, l )-regular set } . Fix α ≥
0. Theorem 3.3 implies that { K ∈ K ( X ) : dim ML K > α } = [ (cid:26) F ( k, l ) : k, l ≥ , log lk log 2 > α (cid:27) . Hence it is enough to show that the sets F ( k, l ) are closed. Fix k and l and assumethat K i ∈ F ( k, l ) for each i and K i → K in the Hausdorff metric, it remains toshow that K ∈ F ( k, l ). For all i ∈ N let { y is } s ∈ l <ω be a ( k, l )-regular set in K i . Bysuccessively taking subsequences and considering the diagonal sequence, we maysuppose that y is → y s as i → ∞ for each s ∈ l <ω with some y s ∈ K . Then itis straightforward that { y s } s ∈ l <ω is a ( k, l )-regular set in K , so K ∈ F ( k, l ). Theproof of the theorem is complete. (cid:3) A non-measurability result in ℓ The goal of this section is to prove the following theorem.
Theorem 5.1.
The map dim ML : F ( ℓ ) → [0 , ∞ ] is not Borel measurable.Proof. We denote the space of subtrees of N <ω by Tr, that forms a Polish spacewhen equipped with the subspace topology coming from 2 N <ω , see [8, 4.32]. Recallthe definition of the Borel σ -algebra B on F ( ℓ ) from (1.1). We prove the theoremby constructing a Borel measurable mapping ϕ : Tr → F ( ℓ ) (that is, ϕ − ( B ) willbe a Borel subset of Tr for each B ∈ B ) with the property(5.1) T ∈ IF ⇔ dim ML ϕ ( T ) > , TABILITY AND MEASURABILITY OF THE MODIFIED LOWER DIMENSION 9 where IF = { T ∈ Tr : T has an infinite branch } . Since IF is not a Borel subset ofTr by [8, 27.1], it follows from (5.1) that { F ∈ F ( ℓ ) : dim ML F > } 6∈ B , thus dim ML is not Borel measurable.Thus it is enough to construct a Borel measurable map ϕ : Tr → F ( ℓ ) satisfying(5.1). We do so by first constructing a map ϕ : N <ω → Fin( ℓ ) assigning to eachelement u ∈ N <ω a finite subset ϕ ( u ) of ℓ . We then define ϕ : Tr → F ( ℓ ) by ϕ ( T ) = [ { ϕ ( u ) : u ∈ T } . We first check that the map ϕ is necessarily Borel measurable. It is enough to checkthat for each open set U ⊂ ℓ the inverse image of the set in (1.1) is a Borel subsetof Tr. Let us fix an open set U ⊂ ℓ . Since the closure of a set E intersects U ifand only if E intersects U , we obtain that { T : ϕ ( T ) ∩ U = ∅} = { T : ∃ u ∈ T, ϕ ( u ) ∩ U = ∅} = [ u ∈ N <ω , ϕ ( u ) ∩ U = ∅ { T : u ∈ T } , which is clearly an open subset of Tr.It remains to define the map ϕ : N <ω → Fin( ℓ ) so that (5.1) is satisfied for theresulting ϕ . First, for each u ∈ N <ω choose indices n u , n u ∈ N such that n iu = n jv if ( u, i ) = ( v, j ). Next, we define ϕ recursively. Let ϕ ( ∅ ) = { } , where ∈ ℓ is thezero vector. If ϕ ( u ) is defined for some u ∈ N n and v ∈ N n +1 extends u , then let ϕ ( v ) = (cid:8) x + 2 − n − χ ( n iv ) : x ∈ ϕ ( u ) , i ∈ { , } (cid:9) , where χ ( n ) is the unit vector whose only non-zero coordinate has index n .As the indices n iu are distinct, if the number of coordinates of u and v satisfylength( u ) , length( v ) > k and u ( k ) = v ( k ), then(5.2) k x − y k ≥ − k for all x ∈ ϕ ( u ) and y ∈ ϕ ( v ) . First, we claim that if T does not have an infinite branch, that is, T IF, then S u ∈ T ϕ ( u ) does not have limit points. Let { x n } n ∈ ω such that x n ∈ ϕ ( u n ) forsome u n ∈ T be any sequence of points from S u ∈ T ϕ ( u ). As T does not havean infinite branch, there exists k such that for each N there exist n, m ≥ N such that length( u n ) , length( u m ) > k and u n ( k ) = u m ( k ). Then Inequality (5.2)yields that k x n − x m k ≥ − k , hence { x n } n ∈ ω is not a Cauchy-sequence, showingthat S u ∈ T ϕ ( u ) does not have limit points. Therefore its closure, ϕ ( T ) is count-able. Any non-empty subset of a countable closed set has isolated points, thereforedim ML ϕ ( T ) = 0 in this case.Now suppose that T ∈ IF, and z ∈ N ω is an infinite branch of T . We show thatthe closed set E = { ϕ ( z ↾ n ) : n ∈ ω } contains a (2 , { y s } s ∈ <ω . Then applying Theorem 3.3 for thecomplete metric space E implies thatdim ML ϕ ( T ) ≥ dim ML E > . Let y ∅ = and assume by induction that y s ∈ ϕ ( z ↾ n ) is defined for all s ∈ n forsome n ≥
0. For c ∈ { , } define y s ⌢ c ∈ ϕ ( z ↾ ( n + 1)) as(5.3) y s ⌢ c = y s + 2 − n − χ ( n cz ↾ ( n +1) ) . It remains to check that { y s } s ∈ <ω is indeed a (2 , k y s − y s ⌢ c k = 2 − n − , so Definition 1.7 (i) holds. If s, t ∈ n with s = t then s ( k ) = t ( k ) for some k ≤ n −
1, hence (5.2) implies that k y s − y t k ≥ − k ≥ − n +2 ,so Definition 1.7 (ii) is satisfied, too. This completes the proof of the theorem. (cid:3) Acknowledgments.
We are indebted to Jonathan M. Fraser for some illuminatingconversations.
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Alfr´ed R´enyi Institute of Mathematics, Re´altanoda u. 13–15, H-1053 Budapest, Hun-gary
Email address : [email protected] Alfr´ed R´enyi Institute of Mathematics, Re´altanoda u. 13–15, H-1053 Budapest, Hun-gary AND E¨otv¨os Lor´and University, Institute of Mathematics, P´azm´any P´eter s. 1/c,1117 Budapest, Hungary
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