Hausdorff Dimension Regularity Properties and Games
aa r X i v : . [ m a t h . L O ] M a r HAUSDORFF DIMENSION REGULARITY PROPERTIES ANDGAMES
LOGAN CRONE, LIOR FISHMAN, AND STEPHEN JACKSON
Abstract.
The Hausdorff δ -dimension game was introduced in [2] and shownto characterize sets in R d having Hausdorff dimension ≤ δ . We introduce avariation of this game which also characterizes Hausdorff dimension and forwhich we are able to prove an unfolding result similar to the basic unfoldingproperty for the Banach-Mazur game for category. We use this to derive anumber of consequences for Hausdorff dimension. We show that under AD any wellordered union of sets each of which has Hausdorff dimension ≤ δ hasdimension ≤ δ . We establish a continuous uniformization result for Hausdorffdimension. The unfolded game also provides a new proof that every Σ set ofHausdorff dimension ≥ δ contains a compact subset of dimension ≥ δ ′ for any δ ′ < δ , and this result generalizes to arbitrary sets under AD . Introduction
Category, measure, and Hausdorff dimension are three fundamental notions oflargeness for sets in a Polish space (for Hausdorff dimension one most commonlyrestricts to subsets of R d ). In the case of category the notion is connected to a well-known game, the Banach-Mazur or ∗∗ -game (see for example [4] for a discussionof the game and related notions; we note that for the Banach-Mazur game G ∗∗ ( A )for a set A it is conventional to have player II being the player trying to get intothe set A ). For example, in any Polish space X , a set A ⊆ X is comeager iff II has a winning strategy in the Banach-Mazur game G ∗∗ ( A ), and I has a winningstrategy iff there is a neighborhood on which A is meager. An important aspect ofthis game is that it permits an unfolding . By this we mean that if A = dom( R )where R ⊆ X × Y , then if I has a winning strategy in the game G ∗∗ ( R ) then I has a winning strategy in the game G ∗∗ ( A ). Assuming the game is determined,this says that if II can win the game G ∗∗ ( A ), then II can actually win the game G ∗∗ ( R ) in which II is not only produces an x ∈ A but also a pair ( x, y ) ∈ R , thatis, where y “witnesses” that x ∈ A .This unfolding phenomenon for the ∗∗ -game has many applications to category.For example, since Σ sets A ⊆ X are projections of closed sets F ⊆ X × ω ω ,this reduces the Banach-Mazur game for Σ sets to the game for closed sets, whichare determined in ZF . This gives a proof of the fact that every Σ set in a Polishspace has the Baire property. Another application of the unfolding is to showcontinuous uniformizations on comeager sets. Namely, suppose R ⊆ X × Y and A = dom( R ) is comeager. If we assume AD , then there is a comeager set C ⊆ A anda continuous function f : C → Y which uniformizes R , that is, for all x ∈ A we have R ( x, f ( x )). Working just in ZF we get that if R is Σ then there is a continuousuniformization on a comeager set (this requires unfolding the game on R to a closedset F ⊆ X × Y × ω ω ). Yet another application of unfolding is to establish the full additivity of category under AD . By this we mean the statement that a wellorderedunion on meager sets is meager. The most common proof given for this uses theanalog of Fubini’s theorem for category, the Kuratowski-Ulam theorem. However,a different proof can be given using the unfolded game. This is important as thereis no Fubini theorem for Hausdorff δ -dimension measure, and we wish to establishthis additivity result for Hausdorff dimension (Theorem 6).In [5] (see also [7]) the Measure game was introduced which was shown to char-acterize Lebesgue measure in a manner similar to how the Banach-Mazur gamecharacterizes category. In [1] a variation of this game was introduced and an un-folding result for it was proved. This game analysis had several applications. Asidefrom giving new proofs of some classical results such as the Borel-Cantelli lemma,a strong form of the R´enyi-Lamperti lemma of probability theory was shown usingthe game.In [2] a game, the Hausdorff δ -dimension game was introduced, and it was shownthat this game characterizes when a set A ⊆ R d has Hausdorff dimension HD( A ) ≤ δ . More precisely, if I wins the game then HD( A ) ≥ δ and if II wins the game thenHD( A ) ≤ δ . In this paper we introduce a variation of this game which we show alsocharacterizes Hausdorff dimension in this manner, and for which we are able to provean unfolding result (Theorem 4). As with measure and category, this has a numberof consequences. This gives a new proof of the basic regularity result that every Σ set A ⊆ R d with HD( A ) ≥ δ , if δ ′ < δ then A contains a contains a compactset K with HD( K ) ≥ δ ′ . The classical proof of this fact uses an “increasing setslemma” for Hausdorff δ -measure (see Theorems 47 and 48 of [6]). Moreover, thisresult extends to other pointclasses assuming the determinacy of the correspondinggames. For example, assuming Π -determinacy we get the same regularity result for Σ sets. We are able to prove continuous uniformization theorems, (see Theorem 5and the following remarks) again assuming the determinacy of the relevant games.Finally, using the unfolded game we are able to show that under AD we have fulladditivity for Hausdorff dimension ≤ δ sets. That is, any wellordered union (of anylength) of sets each of which has Hausdorff dimension ≤ δ has Hausdorff dimension ≤ δ . This complements the corresponding results for category and measure, whichare known theorems from AD .Throughout, we will be working in a Euclidean space R d . We let H s denote s -dimensional Hausdorff measure on R d . For A ⊆ R d we let HD( A ) denote theHausdorff dimension of A . This is defined for all sets A ⊆ R d , and 0 ≤ HD( A ) ≤ d .We recall that H s is a Borel measure on R d , but it is not σ -finite (unless s = d ).We let ω = N denote the natural numbers and ω ω denote the Baire space (set ofsequences of natural numbers) with the usual product of the discrete topologies on ω . The following theorem is a well-known tool in the theory of Hausdorff dimension,and is also central to our arguments. We include a proof partly for the sake ofcompleteness, and also because we wish to be able to use our results in models ofdeterminacy where AC fails. In the following proof we show that only countablechoice AC ω is needed, and thus we can in particular use this result in any model of ZF + DC . Theorem 1 (Rogers-Taylor-Tricot [6]). ( ZF + AC ω ) . Let µ be a Borel probabilitymeasure on R d . AUSDORFF DIMENSION REGULARITY PROPERTIES AND GAMES 3 (i) If A ⊆ R d and lim sup r → µ ( B ( x, r )) r s < m for every x ∈ A , then H s ( A ) ≥ m − µ ∗ ( A ) . (ii) If A ⊆ R d and lim sup r → µ ( B ( x, r )) r s > m for every x ∈ A , then H s ( A ) ≤ c d m − µ ∗ ( A ) where c d is a constant depending only on d .In these statements, H s ( A ) refers to the Hausdorff s -dimensional outer measure of A , and µ ∗ ( A ) refers to the outer µ -measure of A .Proof. (i) Let A ǫ = n x ∈ A : sup
For such Q , we have r ( Q ) / < diam( Q ), thus we know that the side length of Q is m ( x ( Q )) ≥ r ( Q )2 √ d . Let N d = l √ d m and note that if k is the side length of Q , then kN d ≥ r ( Q ) √ d √ d > r ( Q ) and thus B ( x ( Q ) , r ( Q )) ⊆ Q ∗ , where Q ∗ is Q scaled by N d . and thus B ( x ( Q ) , r ( Q )) can be covered by N dd dyadic cubes of thesame side length as Q , of which Q has maximal µ -measure (since we can discardany dyadic cubes from the cover which do not intersect B ( x ( Q ) , r ( Q )).So for each dyadic cube Q = Q x ( Q ) , we have µ ( Q ) ≥ N − dd µ ( B ( x ( Q ) , r ( Q ))) > N − dd mr ( Q ) s and since each such Q is contained in U , we have µ ( U ) ≥ X Q µ ( Q ) ≥ N − dd m X Q r ( Q ) s . Now also the collection of cubes Q ∗ form a cover of A (since if Q = Q x , the enlarged Q ∗ must contain x , even if x = x ( Q )). Since each Q ∗ is covered by N dd translatesof Q , and since diam( Q ) ≤ r ( Q ) < ǫ , we have H sǫ ( A ) ≤ N dd X Q r ( Q ) s ≤ N dd m − µ ( U ) . Thus by taking a sup as ǫ →
0, we have H s ( A ) ≤ N dd m − µ ( U )for any open set U containing A . Thus finally we have H s ( A ) ≤ l √ d m m − µ ∗ ( A ) (cid:3) The revised Hausdorff dimension game
As we mentioned before, the Hausdorff δ -dimension game was introduced in [2].Here we define a variation of the game, the main difference is that we use not asingle β , but a sequence β i which goes to 0 sufficiently slowly. Using a sequence ofthe β i does not affect the fact that the game characterizes Hausdorff dimension (asTheorems 2 and 3 show), but seems important in our argument for the unfolding(Theorem 4). Definition 1.
Let d ≥ ρ >
0, 0 < β i +1 ≤ β i < be sothat lim i →∞ β i = 0 satisfying(1) ∀ η > ∃ n ∀ n ≥ n β n ≥ Y i In the original game of [2], the points which player I plays need notbe rational. It turns out that to prove the two theorems characterizing Hausdorffdimension (Theorems 2 and 3) it suffices to use the rational version above, whichof course is determined from AD . It is not clear that the the rational and realversions of the game are equivalent, however. For sets A for which the games aredetermined, the winning players must agree for δ = HD( A ), but even though thegames are determined, it seems possible that they may disagree on who wins at δ = HD( A ). Theorem 2. If player I has a winning strategy in the δ -Hausdorff dimension game,then there is a compact K ⊆ A with HD( K ) ≥ δ .Proof. Suppose σ is a winning strategy for player I in the δ -Hausdorff dimensiongame. Define a finitely splitting tree T by T = { ( x , . . . , x n ) : ∀ i x i ∈ σ ( x , . . . , x i − ) } Define a map π : [ T ] → R d by π ( x , . . . , x n , . . . ) = lim n →∞ x n , which is clearlycontinuous. Define a probability measure µ on [ T ] by µ ([( x , . . . , x n )]) = Y i If player II has a winning strategy in the δ -Hausdorff dimensiongame, then HD( A ) ≤ δ Proof. Note first that for a ball B ⊆ R d of radius ρ , any 3 βρ separated subset E ⊆ B has size at most | E | ≤ & √ d β ' d . This can be seen by comparing the volumes of a cube of side length 4 ρ (whichcontains B ) and the sums of the volumes of cubes centered on points in E of sidelengths 3 βρ/ √ d , which must be disjoint by hypothesis on E .The actual bound is unimportant, we need that it depends only on d and β .Suppose now that τ is a winning strategy for player II in the δ -Hausdorff di-mension game. Let ρ n , β n etc. be the parameters of the game.Let E n be a maximal ρ n -separated subset of Q d . and let (cid:8) E in (cid:9) ≤ i<ℓ partition E n into 3 ρ n -separated subsets. Note that this can be done with a fixed ℓ whichdoesn’t depend on n . In fact, if each E in is a maximal 3 ρ n -separated subset of E n \ S j k u ,or if j = 0, then u is inappropriate , otherwise, we let p u be the position of length (cid:12)(cid:12) p ( s,t ) (cid:12)(cid:12) + 2 in which player I has played E i | s | ( t, j ) where E i | s | ( t, j ) = ( E i | s | ∩ B ( s,t ) j = k u E i | s | ∩ B ( s,t ) \ S j 12 + 12 β n (cid:19) < ρ n (1 − β n )So that each x n +1 is a legal possibility following x n . Because of this, we canobtain sequences i n and j n so that for every n , x n ∈ E i n n ( j , . . . j n ) and x n = τ ( E i n n ( j , . . . j n )). Since τ is a winning strategy, and x is in player I ’s target set,and each move we made for player I was legal, it must be the case that player I ’scondition on the number of choices offered is violated, i.e. for every constant C lim sup n →∞ (cid:16)Q m ≤ n j m (cid:17) − (cid:16)Q m ≤ n β m (cid:17) δ > C. LOGAN CRONE, LIOR FISHMAN, AND STEPHEN JACKSON Now we can compute, for any γ > δ (1 + ǫ )(1 + η ) > δ (1 + ǫ )lim sup r → µ ( B ( x, r )) r γ ≥ lim sup n →∞ µ ( B ( x, ρ n ))(2 ρ n ) γ (because it is a lim sup) ≥ lim sup n →∞ µ ( B ( x n , ρ n )(2 ρ n ) γ (monotonicity) ≥ lim sup n →∞ µ ( N ( i ...i n ,j ...j n ) )(2 ρ n ) γ (push-forward and monotonicity) ≥ lim sup n →∞ c n +1 Q m ≤ n j − (1+ ǫ ) m (2 ρ Q m 0, and since δ (1 + ǫ )(1 + η ) < γ , we havethat for large enough m , cβ mγ − δ (1+ ǫ )(1+ η ) > n →∞ c n +1 Y m Example 1. Let 0 < δ ≤ K n ⊆ ( n, n + 1) be a compact set with HD( K n ) = δ (cid:16) − n +1 (cid:17) for each n ∈ ω . Then player II wins G δ~β ( S n K n ), since this is a deter-mined game, and player I cannot win, since S n K n doesn’t contain any compactsubsets of Hausdorff dimension δ . AUSDORFF DIMENSION REGULARITY PROPERTIES AND GAMES 9 Example 2. Let B ⊆ R be a Bernstein set, then since λ ∗ ( B ) > 0, we must haveHD( B ) = 1. Clearly player I cannot win G δ~β ( B ) for any δ > 0, since B cannotcontain any uncountable closed set, so in particular B cannot contain any compactset with positive Hausdorff dimension. On the other hand, player II cannot havea winning strategy.To see this, suppose player II had a winning strategy τ in G δ~β ( B ), then onecan construct a perfect subset of R \ B by building inductively a perfect set ofruns following τ where at at step n we consider x = τ ( E , . . . , E n − , E ) for somemaximal legal move E , and also x ′ = τ ( E , . . . , E n − , E \ { x } ). Since these movesare maximal, playing them doesn’t violate player I ’s requirement ( I is playingapproximately β n − many sets at round n , so is satisfying the rule for any δ ≤ δ ). This gives a perfectly splitting tree of positions, in which eachlevel corresponds to disjoint closed intervals. And since player I ’s condition is met,all branches through this tree must result in points in R \ B , which is impossible.3. The Unfolded Game In this section, we introduce an unfolded version of the Hausdorff dimensiongame, and show that it is equivalent to the original. This result gives that analyticsets have the property that they can be approximated from the inside by compactsets of the appropriate Hausdorff dimension. This is interesting, as other proofs ofthis property are generally quite involved and require the analysis of the approxi-mations to the Hausdorff outer measure, and a so-called “increasing sets lemma”(again, see Theorems 47, 48 of [6]), and these are completely absent from our proof.First a simple combinatorial lemma. Lemma 1. Suppose A is a finite set with linear orders (cid:22) , (cid:22) , . . . , (cid:22) n . There isan element a ∈ A so that for every i ≤ n |{ b ∈ A : b (cid:22) i a }| ≥ n | A | Proof. Let A i = (cid:26) a ∈ A : |{ b ∈ A : b (cid:22) i a }| < n | A | (cid:27) Suppose the lemma fails, so that S i A i = A . We will proceed by counting: Firstnote that A i is an initial segment of A by (cid:22) i , since if a ∈ A i and b (cid:22) i a , thencertainly { c ∈ A : c (cid:22) i b } ⊆ { c ∈ A : c (cid:22) i a } . So for each i , there is some a i ∈ A i so that A i = { a ∈ A : a (cid:22) i a i } But then since a i ∈ A i , we have that | A i | = |{ a ∈ A : a (cid:22) i a i }| < n | A | and so since A = S i A i we have | A | ≤ X i ≤ n A i < X i ≤ n n | A | = | A | a contradiction. (cid:3) Definition 2. Let d ≥ ρ > 0, 0 < β i +1 ≤ β i < be sothat lim i →∞ β i = 0 satisfying(2) ∀ η > ∃ n ∀ n ≥ n β n ≥ Y i If player II has a winning strategy in the unfolded δ -Hausdorff di-mension game, and s > δ , then player II has a winning strategy in the s -Hausdorffdimension game. We will prove a technical lemma that will be central for the argument. For thispurpose, we need a little notation. Notation 1. Let p = ( F , x , F , x , . . . , F n , x n ) be a position in the Hausdorffdimension game. Let q = ( E , x ′ , E , x ′ , . . . , E n , x ′ n ) be a position of the unfoldedHausdorff dimension game in which the digits of the finite sequence u have beenplayed along with the E i sets (in some subsequence of the rounds). We’ll call q a simulation of p with partial witness u if for each i , E i ⊆ F i and x i = x ′ i .And now we are ready to state our main technical lemma Lemma 2. Let τ be a strategy in the unfolded Hausdorff dimension game and let p be a position of the Hausdorff dimension game. Suppose we have some finitesequence of partial witnesses u , . . . , u n and a finite sequence of positions q , . . . , q n of the unfolded Hausdorff dimension game so that for each i , q i is a simulation of p with partial witness u i so that q i is consistent with τ .Given (1) any finite sequence v , . . . , v n so that for each i , either v i = u i or v i is anextension of u i by a single extra digit, (2) and any move F for I which is legal at p ,there is some x ∈ F so that for each i , there is an extension q ′ i = q i a E i a x which isa simulation of p a F a x with partial witness v i so that | E i | ≥ n +1 | F | , and so that q ′ i is also consistent with τ . AUSDORFF DIMENSION REGULARITY PROPERTIES AND GAMES 11 Proof. We will apply Lemma 1 more or less directly to obtain this result. Given v i either extending u i or identical to u i , we have that τ at position q i induces a linearorder (cid:22) i on F by τ ’s preference of which point to choose in response to the move F where the extra digit (if any) of v i is offered. More precisely, define for each x ∈ F the rank r i ( x ) by: r i ( x ) = | F | ⇔ x = τ ( q i a ( F, v i )) r i ( x ) = j ⇔ x = τ ( q i a ( F \ { y : r ( y ) > j } , v i ))and the linear ordering (cid:22) i by x (cid:22) i y ⇔ r i ( x ) ≤ r i ( y )Note that by the definition of (cid:22) i , τ will always pick the maximal elementof any (cid:22) i -initial segment offered to it, i.e. τ ( q i a ( F (cid:22) i z , v i )) = z where F (cid:22) i z = { y ∈ F : y (cid:22) i z } . By Lemma 1, there is some x ∈ F so that for each i , τ will pick x in response to the move E i = { y ∈ F : y (cid:22) i x } , and | E i | ≥ n +1 | F | . (cid:3) Note that in Lemma 2, we did not require the u i to be distinct. This will makeour application of the lemma easier, when we choose to split a partial witness u into several extensions, and still keep u itself alive. Proof of Theorem 4. Suppose τ is a winning strategy in the unfolded δ -Hausdorffdimension game, and let s > δ . We first attempt to motivate the proof: We wantto construct a strategy for which every full run has a tree of simulations consistentwith τ for all possible witnesses. The main obstacle is to make sure that along eachbranch of this tree, we’ve offered τ enough choices so that the branch is not winningfor trivial reasons. Then we can use that τ is winning to prove that ( x, y ) F forevery y , thus x A , producing a win in the original Hausdorff dimension game. Inorder to maintain that all the simulations are consistent with τ , we need to playfewer sets when copying I ’s moves, and so we need to be able to absorb the extra n factor in each round that we have n partial witnesses. This is where the fact that β i → ω <ω as { w i : i ∈ ω } so that if w j ⊆ w i , then j ≤ i . For a while, playaccording to τ in the s -Hausdorff dimension game, playing no witness moves at all,until β i gets small enough so that β s − δi < . Now we apply Lemma 2 to the currentposition with witnesses u = u = ∅ = w and v = u , v = w . Continue play inevery round afterwards applying Lemma 2 with u = w = v , u = w = v . Notethat this maintains the hypotheses of Lemma 2, so that we can continue to applyit. We do this until β i is small enough so that β s − δi < . We would like to add the witness w to our list at this point, and we know that w must extend either w or w , andso we apply Lemma 2 to three witnesses u = w , u = w , u = w ↾ | w | − 1, inwhich the ancestor of w appears twice, with v = u , v = u and v = w . Itis clear that we can continue this algorithm to define a strategy in the s -Hausdorffdimension game. We now demonstrate that it does the job: Suppose x is the result of our strategy, and suppose for the sake of a contradictionthat x was a loss for us. In other words x ∈ A and there exists some c > n →∞ Q i In this section we derive some consequences of the Hausdorff dimension gameas well as the unfolding theorem, Theorem 4. Our first application concerns theexistence of continuous uniformizations. Recall first the situation with regards tomeasure and category. Assuming AD , if R ⊆ X × ω ω and dom( R ) is comeager,then there is a comeager set C ⊆ dom( R ) and a continuous function f : C → ω ω which uniformizes R , that is, for all x ∈ C we have R ( x, f ( x )). This continuousuniformization phenomenon is an important aspect of category arguments and fol-lows from an unfolding argument for category (using the Banach-Mazur game, alsoknown as the ∗∗ -game). There is also a corresponding theorem for measure. Againassuming AD , if R ⊆ X × ω ω and dom( R ) has measure 1 with respect to some Borelprobability measure µ , then for any ǫ > A ⊆ X with µ ( A ) > − ǫ anda continuous f : A → ω ω which uniformizes R . Our unfolding result Theorem 4allows us to get a similar result for Hausdorff dimension. Theorem 5. Assume AD . Suppose R ⊆ R d × ω ω and dom ( R ) has Hausdorffdimension at least δ . Then for any δ ′ < δ there is a B ⊆ dom ( R ) with HD( B ) ≥ δ ′ and a continuous f : B → ω ω which uniformizes R .Proof. Fix δ ′ < δ ′′ < δ and consider the unfolded δ ′ -Hausdorff dimension gameas in Definition 2 for the set A = dom( R ), and using R for the set F . Here weuse a fixed sequence { β i } satisfying the conditions of Definition 2. By AD this AUSDORFF DIMENSION REGULARITY PROPERTIES AND GAMES 13 game is determined. If II had a winning strategy for this unfolded game, thenby Theorem 4 II would have a winning strategy for the (regular non-unfolded) δ ′′ -Hausdorff dimension game. From Theorem 3 we have that HD( A ) ≤ δ ′′ , acontradiction. Thus, I has a winning strategy σ for the unfolded δ ′ -Hausdorffdimension game. Ignoring the witness moves that σ makes, σ gives a strategy ¯ σ for the regular δ ′ -Hausdorff dimension game for A . The proof of Theorem 2 gives acompact set K ⊆ A with HD( K ) ≥ δ ′ . For x ∈ K there is a unique run accordingto σ which produces the point x (that is, lim x n = x ). Let y = ( y , y , . . . ) bethe sequences of witness moves played by σ along this run. Then R ( x, y ) as σ iswinning for I . If for x ∈ K we let f ( x ) be this y , then the function f is continuouson K as y ↾ k is determined by some finite part p of this run by σ , and any x ′ in K which is in the same open set determined by x n (where n is the length of p ) willhave f ( x ′ ) ↾ k = f ( x ) ↾ k . (cid:3) We note that although Theorem 5 is stated under AD as a hypothesis, the deter-minacy assumption is entirely local, we just need the determinacy of the unfoldedgame. So, for example, if R ⊆ R d × ω ω is Σ , then we just need the determinacyof ∆ games, which is theorem of ZF (the condition that each digit y ( i ) is eventu-ally played is a Π condition, and the lim sup condition on the size of I ’s movesis a Σ condition). In particular, projective determinacy PD is enough to get theconclusion of Theorem 5 for all projective relations R .Within the realm of AD , another important result about measure and categoryis the full additivity of these notions. That is, any wellordered union (of any length)of meager sets is meager, and likewise for measure zero sets. These results can beproved ether using the Fubini theorem (or Kuratowski-Ulam theorem in the caseof category) or by an argument using an unfolded game. In the case of Hausdorffmeasure, we do not have an analog of the Fubini theorem. However, our unfoldingtheorem can be used to prove the corresponding result. Theorem 6. Assume AD . Then any wellordered union of subsets of R d , each ofwhich has Hausdorff dimension at most δ , has Hausdorff dimension at most δ .Proof. Let A = S α<θ A α where A α ⊆ R d and HD( A α ) ≤ δ . Suppose HD( A ) > δ .We may assume θ is least so that HD( S α<θ A α ) has Hausdorff dimension greaterthan δ , and thus we may assume that the sequence A α in increasing. Fix any δ ′ with δ < δ ′ < HD( A ). From Theorem 3 it suffices to show that II wins the δ ′ -Hausdorff dimension game for A . Suppose not, and let σ be a winning strategy for I in the δ ′ -Hausdorff dimension game for A .Suppose first that cof( θ ) = ω , and let α n < θ be such that sup n α n = θ . So, A = S n A α n . Since each A α n has Hausdorff dimension ≤ δ , H δ ′ ( A α n ) = 0 for each n . As H δ ′ is a measure, H δ ′ ( A ) = 0, a contradiction.Suppose next that cof( θ ) > ω . Note that θ < Θ as Θ is the supremum of thelengths of the prewellorderings of R (or equivalently, the lengths of the increasingsequences of subsets of R ). A theorem of Steel (see Theorem 1.1 of [3] for the generalstatement and proof) says that, assuming AD , for any θ < Θ with cof( θ ) > ω thereis a ϕ : B → θ (for some set B ⊆ ω ω ) which is onto and such that any Σ set S ⊆ B is bounded in the prewellordering, that is, sup { ϕ ( x ) : x ∈ S } < θ . Fix such a map ϕ for the ordinal θ . Consider the relation R ⊆ R d × ω ω given by R ( x, y ) ↔ ( x ∈ A ) ∧ ( y ∈ B ) ∧ ( x ∈ A ϕ ( y ) ) Clearly dom( R ) = A . Consider the unfolded δ ′ -Hausdorff dimension games for R .From Theorem 4 we have that II cannot win this game as otherwise we wouldhave that HD( A ) ≤ δ ′ (since for every δ ′′ > δ ′ , from theorem 4, II would win theregular δ ′′ game for A = dom( R ), and so HD( A ) ≤ δ ′′ ). As we are assuming AD ,we may fix a winning strategy σ for I in this unfolded game. Let σ be the strategywhich extracts the y = ( y , y , . . . ) moves from the play by σ . Let S = σ [ ω ω ],more precisely, let S collect all of the y ’s which come from a any run of σ in which II has followed the rules of the game. Clearly S is Σ , and so there is an α < θ such that sup { ϕ ( y ) : y ∈ S } < α . This says that I wins the unfolded δ ′ -Hausdorffdimension game for the set A α . Thus, I wins the regular δ ′ -Hausdorff dimensiongame for A α and so HD( A α ) ≥ δ ′ , a contradiction. (cid:3) The arguments of the current paper naturally suggest two questions. First, canwe get a characterization for when player I or II wins G δ~β ( A ) when HD( A ) = δ ? As we noted in Example 1, either player could win in this case (just givenHD( A ) = δ ). Second, to which class of metric spaces can we extend our basicresults (Theorems 2,3, and 4)? References 1. 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Logan Crone, University of North Texas, Department of Mathematics, 1155 UnionCircle E-mail address : [email protected] Lior Fishman, University of North Texas, Department of Mathematics, 1155 UnionCircle E-mail address : [email protected] Stephen Jackson, University of North Texas, Department of Mathematics, 1155Union Circle E-mail address ::