aa r X i v : . [ c s . CC ] J a n Optimal Oracles for Point-to-Set Principles
D. M. StullDepartment of Computer Science, Iowa State UniversityAmes, IA 50011, USA [email protected]
Abstract
The point-to-set principle [14] characterizes the Hausdorff dimensionof a subset E ⊆ R n by the effective dimension of its individual points.This characterization has been used to prove several results in classical,i.e., without any computability requirements, analysis. Recent work hasshown that algorithmic techniques can be fruitfully applied to Marstrand’sprojection theorem, a fundamental result in fractal geometry.In this paper, we introduce an extension of point-to-set principle -the notion of optimal oracles for subsets E ⊆ R n . One of the primarymotivations of this definition is that, if E has optimal oracles, then theconclusion of Marstrand’s projection theorem holds for E . We show thatevery analytic set has optimal oracles. We also prove that if the Hausdorffand packing dimensions of E agree, then E has optimal oracles. Thus,the existence of optimal oracles subsume the currently known sufficientconditions for Marstrand’s theorem to hold.Under certain assumptions, every set has optimal oracles. However,assuming the axiom of choice and the continuum hypothesis, we constructsets which do not have optimal oracles. This construction naturally leadsto a new, algorithmic, proof of Davies theorem on projections. Effective, i.e., algorithmic, dimensions were introduced [12, 1] to study the ran-domness of points in Euclidean space. The effective dimension, dim( x ) andeffective strong dimension, Dim( x ), are real values which measure the asymp-totic density of information of an individual point x . The connection betweeneffective dimensions and the classical Hausdorff and packing dimension is givenby the point-to-set principle of J. Lutz and N. Lutz [14]: For any E ⊆ R n ,dim H ( E ) = min A ⊆ N sup x ∈ E dim A ( x ) , and (1)dim P ( E ) = min A ⊆ N sup x ∈ E Dim A ( x ) . (2)Call an oracle A satisfying (1) a Hausdorff oracle for E . Similarly, we call anoracle A satisfying (2) a packing oracle for E . Thus, the point-to-set principle1hows that the classical notion of Hausdorff or packing dimension is completelycharacterized by the effective dimension of its individual points, relative to aHausdorff or packing oracle, respectively.Recent work as shown that algorithmic dimensions are not only useful ineffective settings, but can be fruitfully used to solve problems in geometricmeasure theory [15, 17, 18, 29]. It is important to note that the point-to-setprinciple allows one to prove theorems, which do not assume computabilityrestrictions, using algorithmic techniques.In this paper we introduce the notion of optimal oracles for a set E ⊆ R n .An optimal oracle for a set E is a Hausdorff oracle which is strongly restricted ina certain sense. We now give an informal description of this restriction. Thereare, for any set E , infinitely many Hausdorff oracles for E . A natural questionis how the effective dimension, relative to A , are affected by the presence of anadditional oracle B ? The point-to-set principle ensures that, for ”most“ points, B does not change the limit inferior of the density of information. However,for certain applications, we would like the addition of B does not change thedensity whatsoever. Thus, an optimal oracle for E is a Hausdorff oracle A suchthat, for any oracle B , most points are unaffected by the additional informationcontained in B .A priori, it is not clear that any set E has optimal oracles. However, weshow that two natural classes of sets E ⊆ R n do have optimal oracles. We firstshow that every analytic, and therefore Borel, set E has optimal oracles. Wealso show that every E whose Hausdorff and packing dimensions agree also haveoptimal oracles.One of the primary motivations of this definition is that the existence ofoptimal oracles for a set E is strongly connected to Marstrand’s projectiontheorem, a fundamental result of fractal geometry.Marstrand, in his landmark paper [20], was the first to study how the di-mension of a set is changed when projected onto a line. He showed that, forany analytic set E ∈ R , for almost every angle θ ∈ [0 , π ),,dim H ( p θ E ) = min { dim H ( E ) , } , (3)where p θ ( x, y ) = x cos θ + y sin θ . This result was later generalized to R n , forarbitrary n , as well as extended to hyperspaces of dimension m , for any 1 ≤ m ≤ n (see e.g. [21, 22, 23]). The study of projections has since become acentral theme in fractal geometry (see [8] or [24] for a more detailed survey ofthis development).The statement of Marstrands theorem begs the question of whether the an-alytic requirement be dropped. It is known that without further conditions, itcannot. Davies [5] showed that, assuming the continuum hypothesis, there arenon-analytic sets for which Marstrands conclusion fails. However, the problemof classifying the sets for which Marstrands theorem does hold is still open. Veryrecently, Lutz and Stull [19] used the point-to-set principle to prove that the pro-jection theorem holds for sets for which the Hausdorff and packing dimensions2gree . This expanded the reach of Marstrand’s theorem, as this assumption isindependent of analyticity.We prove that Marstrand’s theorem holds for every set E which has optimaloracles. Thus, we show that the existence of optimal oracles encapsulates theknown conditions sufficient for Marstrand’s theorem.We also show that the notion of optimal oracles gives insight to sets forwhich Marstrand’s theorem does not hold. Assuming the axiom of choice andthe continuum hypothesis, we construct sets which do not have optimal oracles.This construction, with minor adjustments, gives a new proof of Davies theorem- existence of sets for which (3) does not hold.The structure of the paper is as follows. In Section 2.1 we review the conceptsof measure theory needed, and the (classical) definition of Hausdorff dimension.In Section 2.2 we review algorithmic information theory, including the formaldefinitions of effective dimensions. We then introduce and study the notion ofoptimal oracles in Section 3. In particular, we give a general condition for theexistence of optimal oracles in Section 3.1. We use this condition to prove thatanalytic sets have optimal oracles in Section 3.2. We conclude in Section 3.3with an example, assuming the axiom of choice and the continuum hypothesis,of a set without optimal oracles. The connection between Marstrands projectiontheorem and optimal oracles is explored in Section 4. In this section, we provethat Marstrands theorem holds for every set with optimal oracles. In Section4.1, we use the construction of a set without optimal oracles to give a new,algorithmic, proof of Davies theorem. A set function µ : P ( R n ) → [0 , ∞ ] is called an outer measure on R n if1. µ ( ∅ ) = 0,2. if A ⊆ B then µ ( A ) ≤ µ ( B ), and3. for any sequence A , A , . . . of subsets, µ ( S i A i ) ≤ P i µ ( A i ).If µ is an outer measure, we say that a subset A is µ -measurable if µ ( A ∩ B ) + µ ( B − A ) = µ ( B ),for every subset B ⊆ R n .An outer measure µ is called a metric outer measure if every Borel subset is µ -measurable and Orponen [28] has recently given another proof of Lutz and Stull’s result using more clas-sical tools. ( A ∪ B ) = µ ( A ) + µ ( B ),for every pair of subsets A, B which have positive Hausdorff distance. That is,inf {k x − y k | x ∈ A, y ∈ B } > s -dimensional Haus-dorff measure. For every E ⊆ [0 , n , define the s -dimensional Hausdorff contentat precision r by h sr ( E ) = inf { P d ( Q ) s | { Q covers E and d ( Q ) ≤ − r } ,where d ( Q ) is the diameter of ball Q . We define the s -dimensional Hausdorffmeasure of E by H s ( E ) = lim r →∞ h sr ( E ). Remark.
It is well-known that H s is a metric outer measure for every s .Let E ⊆ R n and 0 ≤ s < n . We say that E is an s-set if0 < H s ( E ) < ∞ .In order to prove that every analytic set has optimal oracles, we will makeuse of the following well-known facts of geometric measure theory (see, e.g., [7],[2]). Theorem 1.
The following are true.1. Suppose E ⊆ R n is compact and satisfies H s ( E ) > . Then there is acompact subset F ⊆ E such that < H s ( F ) < ∞ .2. Every analytic set E ⊆ R n has a Σ subset F ⊆ E such that dim H ( F ) =dim H ( E ) . Finally, for two outer measures µ and ν , µ is said to be absolutely continuouswith respect to ν , denoted µ ≪ ν , if µ ( A ) = 0 for every set A for which ν ( A ) = 0. The conditional Kolmogorov complexity of a binary string σ ∈ { , } ∗ givenbinary string τ ∈ { , } ∗ is K ( σ | τ ) = min π ∈{ , } ∗ { ℓ ( π ) : U ( π, τ ) = σ } , where U is a fixed universal prefix-free Turing machine and ℓ ( π ) is the lengthof π . The Kolmogorov complexity of σ is K ( σ ) = K ( σ | λ ), where λ is the emptystring. An important fact is that the choice of universal machine effects theKolmogorov complexity by at most an additive constant (which, especially forour purposes, can be safely ignored). See [11, 27, 6] for a more comprehensiveoverview of Kolmogorov complexity. 4e can naturally extend these definitions to Euclidean spaces by introducing“precision” parameters [16, 14]. Let x ∈ R m , and r, s ∈ N . The Kolmogorovcomplexity of x at precision r is K r ( x ) = min { K ( p ) : p ∈ B − r ( x ) ∩ Q m } . The conditional Kolmogorov complexity of x at precision r given q ∈ Q m isˆ K r ( x | q ) = min { K ( p ) : p ∈ B − r ( x ) ∩ Q m } . The conditional Kolmogorov complexity of x at precision r given y ∈ R n atprecision s is K r,s ( x | y ) = max (cid:8) ˆ K r ( x | q ) : q ∈ B − r ( y ) ∩ Q n (cid:9) . We abbreviate K r,r ( x | y ) by K r ( x | y ).The effective Hausdorff dimension and effective packing dimension of apoint x ∈ R n aredim( x ) = lim inf r →∞ K r ( x ) r and Dim( x ) = lim sup r →∞ K r ( x ) r . Intuitively, these dimensions measure the density of algorithmic information inthe point x .By letting the underlying fixed prefix-free Turing machine U be a universal oracle machine, we may relativize the definition in this section to an arbitraryoracle set A ⊆ N . The definitions of K Ar ( x ), dim A ( x ), Dim A ( x ), etc. are then allidentical to their unrelativized versions, except that U is given oracle access to A . Note that taking oracles as subsets of the naturals is quite general. We can,and frequently do, encode a point y into an oracle, and consider the complexityof a point relative to y . In these cases, we typically forgo explicitly referring tothis encoding, and write e.g. K yr ( x ).As mentioned in the introduction, the connection between effective dimen-sions and the classical Hausdorff and packing dimensions shown by the point-to-set principle introduced by J. Lutz and N. Lutz [14]. Theorem 2 (Point-to-set principle) . Let n ∈ N and E ⊆ R n . Then dim H ( E ) = min A ⊆ N sup x ∈ E dim A ( x ) , and dim P ( E ) = min A ⊆ N sup x ∈ E Dim A ( x ) . Although effective Hausdorff was originally defined by J. Lutz [13] using martingales, itwas later shown by Mayordomo [25] that the definition used here is equivalent. For more detailson the history of connections between Hausdorff dimension and Kolmogorov complexity, see [6,26]. Optimal Oracles
There are, for any set E , infinitely many Hausdorff oracles for E . A naturalquestion is how the dimension of points in E are affected by the presence of anadditional oracle B ? That is, what can be said about K A,Br ( x ),for all points x ∈ E ? The point-to-set principle ensures that, for most points,dim A ( x ) > dim A,B ( x ),meaning that additional information contained in B does not change the min-imum value of the density of information for most x . However, for certainapplications, we would like the addition of B does not change the density at any precision.Let E ⊆ R n and A ⊆ N . We say that A is optimal for E if the followingconditions are satisfied.1. dim H ( E ) = sup x ∈ E dim A ( x ).2. For every B ⊆ N and every ǫ > x ∈ E such thatdim A,B ( x ) ≥ dim H ( E ) − ǫ and for almost every r ∈ N K A,Br ( x ) ≥ K Ar ( x ) − ǫr . Remark.
An optimal oracle for a set E is necessarily (by the first condition) aHausdorff oracle for E . The second condition ensures that an optimal oracleessentially minimizes the complexity of “most” points. More concretely, anoptimal oracle minimizes the complexity of a subset of full dimension. Thus,any additional information given by an arbitrary oracle cannot infinitely oftendecrease the complexity of every point in E .We now give some basic closure properties of the class of sets with optimaloracles. Observation 3.
Suppose F ⊆ E such that F has an optimal oracle and that dim H ( F ) = dim H ( E ) . Then E has an optimal oracle. We can also show that having optimal oracles is closed under countableunions.
Lemma 4. If E , E , . . . have optimal oracles, then the union E = ∪ n E n hasan optimal oracle.Proof. We first note that dim H ( E ) = sup n dim H ( E n ).For each n , let A n be an optimal oracle for E n . Let A be the join of A , A , . . . .Let B be an oracle guaranteed by Theorem 2 such that sup x dim B ( x ) = s , where6 = sup n dim H ( E n ).We now claim that ( A, B ) is an optimal oracle for E . Theorem 2 showsthat item (1) of the definition of optimal oracles is satisfied. For item (2), let C ⊆ N be an oracle, and let ǫ >
0. Let n be a number such that dim H ( E n ) > dim H ( E ) − ǫ/
2. Since A n is an optimal oracle for E n , there is a point x ∈ E n such that(i) dim A,B,C ( x ) ≥ dim H ( E n ) − ǫ/ ≥ dim H ( E ) − ǫ , and(ii) for almost every r , K A,B,Cr ( x ) ≥ K A n r ( x ) − ǫr/ r , K A,B,Cr ( x ) = K A n ,A,B,Cr ( x ) ≥ K A n r ( x ) − ǫr/ ≥ K A n ,A,Br ( x ) − ǫr/ K A,Br ( x ) − ǫr/ . In this section we give a sufficient condition for a set to have optimal oracles.Specifically, we prove that if dim H ( E ) = s , and there is a metric outer measure,absolutely continuous with respect to H s , such that 0 < µ ( E ) < ∞ , then E hasoptimal oracles. Although stated in this general form, the main application ofthis result (in Section 3.2) is for the case µ = H s .For every r ∈ N , let Q r be the set of all dyadic cubes at precision r , i.e.,cubes of the form Q = [ m − r , ( m + 1)2 − r ) × [ m − r , ( m + 1)2 − r ),where 0 ≤ m , m ≤ r . For each r , we refer to the 2 r cubes in Q r as Q r, , . . . , Q r, r . We can identify each dyadic cube Q r,i with the unique dyadicrational d r,i at the center of Q r,i .We now associate, to each metric outer measure, a discrete semimeasure onthe dyadic rationals D . Recall that discrete semimeasure on D n is a function p : D n → [0 ,
1] which satisfies Σ r,i p ( d r,i ) < ∞ .Let E ⊆ R n and µ be a metric outer measure such that 0 < µ ( E ) < ∞ .Define the function p µ : D n → [0 ,
1] by p µ,E ( d r,i ) = µ ( E ∩ Q r,i ) r µ ( E ) . Observation 5.
Let µ be a metric outer measure and E ⊆ R n such that <µ ( E ) < ∞ . Then for every r , every dyadic cube Q ∈ Q r , and all r ′ > r , ( E ∩ Q ) = P Q ′ ⊂ QQ ′ ∈Q r ′ µ ( E ∩ Q ′ ) .Proof. By induction, it suffices to show that the conclusion holds for r ′ = r + 1.Let Q , . . . , Q n ∈ Q r +1 be the dyadic cubes contained in Q . Since µ is a metricouter measure, every Borel subset is µ -measurable. Thus µ ( E ∩ Q ) = µ (( E ∩ Q ) ∩ Q ) + µ (( E ∩ Q ) − Q ) , or, equivalently, µ ( E ∩ Q ) = µ ( E ∩ Q ) + µ ( E ∩ ( ∪ i> Q i )).Similarly, we have µ (( E ∩ Q ) − Q ) = µ ( E ∩ Q ) + µ ( E ∩ ( ∪ i> Q i )) . Combining the two equalities shows that µ ( E ∩ Q ) = µ ( E ∩ Q ) + µ ( E ∩ Q ) + µ ( E ∩ ( ∪ i> Q i ) . Continuing in this way up to Q n completes the proof. Lemma 6.
Relative to some oracle A , the function p µ,E is a lower semi-computable discrete semimeasure.Proof. We can encode the real numbers p µ,E ( d ) into an oracle A , relative towhich p µ,E is clearly computable.To see that p µ,E is indeed a discrete semimeasure, by Observation 5, X r,i p µ,E ( d r,i ) = X r r X i =1 µ ( E ∩ Q r,i ) r µ ( E )= X r r µ ( E ) r X i =1 µ ( E ∩ Q r,i )= X r µ ( E ) r µ ( E ) < ∞ . In order to connect the existence of such an outer measure µ to the existenceof optimal oracles, we need to relate the semimeasure p µ and Kolmogorov com-plexity. We achieve this using a fundamental result in algorithmic informationtheory.Levin’s optimal lower semicomputable subprobability measure, relative toan oracle A , on the dyadic rationals D is defined by8 A ( d ) = P π : U A ( π )= d −| π | .Case and Lutz [3], generalizing Levin’s coding theorem [9, 10], showed that thereis a constant c such that m A ( d r,i ) ≤ − K A ( d r,i )+ K A ( r )+ c ,for every r ∈ N and d r,i ∈ D n . The optimality of m A ensures that, for everylower semicomputable (relative to A ) discrete semimeasure ν on D n , m A ( d r,i ) ≥ αν ( d r,i ).We condense the previous paragraph into the following. Corollary 7.
There is a constant α > such that m A ( d ) ≥ αp µ,E ( d ) , for every d ∈ D n . The results of this section have dealt with the dyadic rationals. However, weultimately deal with the Kolmogorov complexity of Euclidean points. A resultof Case and Lutz [3] relates the Kolmogorov complexity of Euclidean pointswith the complexity of dyadic rationals.
Lemma 8.
Let x ∈ [0 , , A ⊆ N , and r ∈ N . Let Q r,i be the (unique) dyadiccube at precision r containing x . Then K Ar ( x ) = K A ( d r,i ) − O (log r ) . We now have the machinery in place to prove the main theorem of thissection.
Theorem 9.
Let E ⊆ R n with dim H ( E ) = s . Suppose there is a metric outermeasure µ such that µ ≪ H s and < µ ( E ) < ∞ . Then E has an optimal oracle A .Proof. Let A ⊆ N be a Hausdorff oracle for E such that p µ,E is computablerelative to A . Note that such an oracle exists by the point-to-set principle androutine encoding. We will show that A is optimal for E .For the sake of contradiction, suppose that there is an oracle B and ǫ > x ∈ E either1. dim A,B ( x ) < s − ǫ , or2. there are infinitely many r such that K A,Br ( x ) < K Ar ( x ) − ǫr .Let N be the set of all x for which the first item holds. Then, by thepoint-to-set principle, dim H ( N ) ≤ s − ǫ ,and so H s ( N ) = 0. Thus, by our assumption on µ , µ ( N ) = 0.For every R ∈ N , there is a set C R of dyadic cubes Q r,i satisfying the follow-ing. 9 The cubes in C R cover E − N . • Every Q r,i in C R satisfies r ≥ R . • For every Q r,i ∈ C R , K A,B ( d r,i ) < K A ( d r,i ) − ǫr + O (log r ).Note that the last item holds by Lemma 8.Since the family of cubes in C R covers E − N , by the subadditive propertyof µ , P Q r,i ∈C R µ ( E ∩ Q r,i ) ≥ µ ( E − N ) = µ ( E ).Thus, for every R , by Corollary 7 and Kraft’s inequality,1 ≥ X Q r,i ∈C R − K A,B ( d r,i ) ≥ X Q r,i ∈C R ǫr − K A ( d r,i ) ≥ X Q r,i ∈C R ǫr m A ( d r,i ) ≥ X Q r,i ∈C R ǫr αp µ,E ( d r,i ) ≥ X Q r,i ∈C R ǫr α µ ( E ∩ Q r,i ) r µ ( E ) ≥ ǫR α/R X Q r,i ∈C R µ ( E ∩ Q r,i ) µ ( E ) ≥ ǫR α/R µ ( E − N ) µ ( E )= 2 ǫR α/R . Since R can be arbitrarily large, we have a contradiction. Therefore no such B exists and so A is optimal for E . Recall that E ⊆ [0 , n is called an s -set if 0 < H s ( E ) < ∞ . Since H s is ametric outer measure, and trivially absolutely continuous with respect to itself,we have the following corollary. Corollary 10 (of Theorem 9) . Let E ⊆ [0 , n be an s -set. Then there is anoptimal oracle for E . We now show that every analytic set has optimal oracles.10 emma 11.
Every analytic set E has optimal oracles.Proof. We begin by assuming that E is compact, and let s = dim H ( E ). Thenfor every t < s , H t ( E ) >
0. Thus, by Theorem 1(1), there is a sequence ofcompact subsets F , F , . . . of E such thatdim H ( S n F n ) = dim H ( E ),and, for each F n is a s n -set, where s n = s − /n . Therefore, by Theorem 9, eachset F n has optimal oracles. Hence, by Lemma 4, E has optimal oracles and theconclusion follows.We now show that every Σ set has optimal oracles. Suppose E = ∪ n F n is Σ , where each F n is compact. As we have just seen, each F n has optimaloracles. Therefore, by Lemma 4, E has optimal oracles and the conclusionfollows.Finally, let E be analytic. By Theorem 1(2), there is a Σ subset F of thesame Hausdorff dimension as E . We have just seen that F must have an optimaloracle. Since dim H ( F ) = dim H ( E ), by Observation 3 E has optimal oracles,and the proof is completeCrone, Fishman and Jackson [4] have recently shown that, assuming theAxiom of Determinacy (AD) , every subset E has a Borel subset F such thatdim H ( F ) = dim H ( E ). This, combined with Lemma 11, yields the followingcorollary. Corollary 12.
Assuming AD, every subset E has optimal oracles. Lemma 13.
Suppose that E ⊆ R n satisfies dim H ( E ) = dim P ( E ) . Then E hasan optimal oracle. Moreover, the join ( A, B ) is an optimal oracle, where A and B are the Hausdorff and packing oracles, respectively, of E .Proof. By the point-to-set principle, there are oracles
A, B ⊆ N such thatdim H ( E ) = sup x dim A ( x ) anddim P ( E ) = sup x Dim B ( x ) . We claim that that the join (
A, B ) is an optimal oracle for E . By the point-to-set principle, and the fact that extra information cannot increase effectivedimension, dim H ( E ) = sup x ∈ E dim A ( x ) ≥ sup x ∈ E dim A,B ( x ) ≥ dim H ( E ) . Therefore Note that AD is inconsistent with the axiom of choice. H ( E ) = sup x ∈ E dim A,B ( x ),and the first condition of optimal oracles is satisfied.Let C ⊆ N be an oracle and ǫ >
0. By the point-to-set principle,dim H ( E ) ≤ sup x ∈ E dim A,B,C ( x ),so there is an x ∈ E such thatdim H ( E ) − ǫ/ < dim A,B,C ( x ).Let r be sufficiently large. Then, by our choice of B and the fact thatadditional information cannot increase the complexity of a point, K A,Br ( x ) ≤ K Br ( x ) ≤ dim P ( E ) r + ǫr/
4= dim H ( E ) r + ǫr/ < dim A,B,C ( x ) r + ǫr/ ≤ K A,B,Cr ( x ) + ǫr. Since the oracle C and ǫ were arbitrarily chosen, the proof is complete. In the previous section, we gave general conditions for a set E to have optimaloracles. Indeed, we saw that under the axiom of determinacy, every set hasoptimal oracles.However, assuming the axiom of choice (AC) and the continuum hypothesis(CH), we are able to construct sets without optimal oracles. Theorem 14.
Assume AC and CH. Then, for every n ∈ N and s ∈ (0 , ,there is a subset E ⊆ R n with dim H ( E ) = s such that E does not have optimaloracles. Let s ∈ (0 , { a n } and { b n } . Let a = 2, and b = ⌊ /s ⌋ . Inductively define a n +1 = b n and b n +1 = ⌊ a n +1 /s ⌋ . Note that lim n a n /b n = s .Using AC and CH, we order the subsets of the natural numbers such thatevery subset has countably many predecessors. For every ordinal α , let f α : N → { β | β < α } be a function such that each ordinal β strictly less than α is mapped to by infinitely many n . Note that such a function exists, since therange is countable assuming CH. 12e will define real numbers x α , y α via transfinite induction. Let x be areal which is random relative to A . Let y be the real whose binary expansionis given by y [ r ] = ( a n < r ≤ b n for some n ∈ N x [ r ] otherwiseFor the induction step, suppose we have defined our points up to α . Let x α be a real number which is random relative to the join of S β<α ( A β , x β ) and A α .This is possible, as we are assuming that this union is countable. Let y α be thepoint whose binary expansion is given by y α [ r ] = ( x β [ r ] if a n < r ≤ b n , where f α ( n ) = βx α [ r ] otherwiseFinally, we define our set E = { y α } . We now claim that dim H ( E ) = s , andthat E does not have an optimal oracle. Lemma 15.
The Hausdorff dimension of E is s .Proof. We first upper bound the dimension. Let A be an oracle encoding x .From our construction, for every element y ∈ E , there are infinitely manyintervals [ a n , b n ] such that y [ a n , b n ] = x [ a n , b n ]. Hence, for every y ∈ E , thereare infinitely many n such that K Ab n ( y ) = K Aa n ( y ) + K Ab n ,a n ( y ) + o ( b n ) ≤ K Aa n ( y ) + o ( b n ) ≤ K A ( y ) + o ( b n ) ≤ a n + o ( b n ) . Therefore, by the point to set principle,dim H ( E ) ≤ sup y dim A ( y )= sup y lim inf r K Ar ( y ) r ≤ sup y lim inf n K Ab n ( y ) b n ≤ sup y lim inf n a n + o ( b n ) b n ≤ sup y lim inf n s = s, and the proof that dim H ( E ) ≤ s is complete.13or the lower bound, let A be a Hausdorff oracle for E , and let α be theordinal corresponding to A . By our construction of y α , for every n , K A α a n ( y α ) ≥ K A α a n ( x α ) − b n − ≥ a n − b n − − o ( a n ) ≥ a n − a n − o ( a n ) . Hence, for every n , and every a n < r ≤ b n , K A α r ( y α ) ≥ K A α a n ( y α ) ≥ a n − a n − o ( a n ) . This implies that K Aαr ( y α ) r ≥ s − o (1),for every n , and every a n < r ≤ b n .We can also conclude that, for every n and every b n < r ≤ a n +1 , K A α r ( y α ) = K A α b n ( y α ) + K A α r,b n ( y α ) − o ( r ) ≥ a n − a n + r − b n − o ( r ) . This implies that K A α r ( y α ) r = 1 + a n r − b n r − o (1)= 1 − a n (1 /s − r − o (1) ≥ − s (1 /s − − o (1)= s − o (1) . for every n , and every a n < r ≤ b n .Together, these inequalities and the point-to-set principle show that dim H ( E ) = sup x dim A ( x ) ≥ dim A ( y α )= lim inf r K A ( y α ) r ≥ lim inf r s − o (1)= s, and the proof is complete. Lemma 16. E does not have optimal oracles. roof. Let A α ⊆ N be an oracle. It suffices to show that A α is not optimal.With this goal in mind, let B be an oracle encoding x α and the set { y β | β < α } .Note that we can encode this information since this set is countable.Let y β ∈ E . First, suppose that β ≤ α . Then by our choice of B ,dim A α ,B ( y β ) = 0. So then suppose that β > α . By our construction thereare infinitely many n such that y β [ a n . . . b n ] = x α [ a n . . . b n ].By our construction, x α and x β are random relative to A α , and so, for any such n , K A α b n ( y β ) ≥ b n − o ( b n ).However, since we can compute x α given B , K A α ,Bb n ( y β ) = K A α ,Ba n ( y β ) ≤ a n − o ( a n )= sb n − o ( a n )= sK A α b n ( y β ) − o ( a n ) . Therefore A α is not optimal, and the claim follows. The following theorem, due to Lutz and Stull [19], gives sufficient conditions forstrong lower bounds on the complexity of projected points.
Theorem 17.
Let z ∈ R , θ ∈ [0 , π ] , C ⊆ N , η ∈ Q ∩ (0 , ∩ (0 , dim( z )) , ε > ,and r ∈ N . Assume the following are satisfied.1. For every s ≤ r , K s ( θ ) ≥ s − log( s ) .2. K C,θr ( z ) ≥ K r ( z ) − εr .Then, K C,θr ( p θ z ) ≥ ηr − εr − ε − η r − O (log r ) . The second condition of this theorem requires the oracle (
C, θ ) to give essen-tially no information about z . The existence of optimal oracles gives a sufficientcondition for this to be true, for all sufficiently large precisions. Thus we areable to show that Marstrands projection theorem holds for any set with optimaloracles. Theorem 18.
Suppose E ⊆ R has an optimal oracle. Then for almost every θ ∈ [0 , π ] , H ( p θ E ) = min { dim H ( E ) , } .Proof. Let A be an optimal oracle for E . Let θ be random relative to A . Let B be oracle testifying to the point-to-set principle for p θ E . It suffices to showthat sup z ∈ E dim A,B ( p θ z ) = min { , dim H ( E ) } .Since E has optimal oracles, for each n ∈ N , we may choose a point z n ∈ E such that • dim A,B,θ ( z n ) ≥ dim H ( E ) − n , and • K A,B,θr ( z n ) ≥ K Ar ( z n ) − r n for almost every r .Fix a sufficiently large n , and let ε = 1 / n . Let η ∈ Q be a rational suchthat min { , dim H ( E ) } − ε / < η < − ε / .We now show that the conditions of Theorem 17 are satisfied for η, ε , relativeto A . By our choice of θ , K Ar ( θ ) ≥ r − O (log r ),for every r ∈ N . By our choice of z n and the optimality of A , K C,θr ( z n ) ≥ K r ( z n ) − εr ,for all sufficiently large r . We may therefore apply Theorem 17, to see that, forall sufficiently large r , K A,B,θr ( p θ z n ) ≥ ηr − εr − ε − η r − O (log r ) . Thus, dim
A,B ( p θ z n ) ≥ dim A,B,θ ( p θ z n )= lim sup r K A,B,θr ( p θ z n ) r ≥ lim sup r ηr − εr − ε − η r − O (log r ) r = lim sup r η − ε − ε − η − o (1) > η − ε − ε / − o (1) > min { , dim H ( E ) } − ε − ε / − o (1) . Hence, lim n dim A,B ( p θ z n ) = min { , dim H ( E ) } ,and the proof is complete. 16 .1 New Proof of Davies Theorem In this section we give a new proof Davies theorem, showing that there are setsfor which Marstrand’s theorem does not hold. While not explicitly mention-ing optimal oracles, the new proof proceeds almost identically as that for theconstruction in Section 3.3.
Theorem 19.
Assuming AC and CH, for every s ∈ (0 , there is a set E suchthat dim H ( E ) = 1 + s but dim H ( p θ E ) = s for every θ ∈ ( π/ , π/ . We will need the following simple observation.
Observation 20.
Let r ∈ N , s ∈ (0 , , and θ ∈ ( π/ , π/ . Then for everydyadic rectangle R = [ d x − − r , d x + 2 − r ] × [ d y − − sr , d y + 2 − sr ] ,there is a point z ∈ R such that K θr ( p θ z ) ≤ sr + o ( r ) . For every r ∈ N , θ ∈ ( π/ , π/ x of length r and string y oflength sr , let g θ ( x, y ) y ′ be a function such that K θr ( p θ ( x, y ′ )) ≤ sr + o ( r ).Let s ∈ (0 , { a n } and { b n } . Let a = 2, and b = ⌊ /s ⌋ . Inductively define a n +1 = b n and b n +1 = ⌊ a n +1 /s ⌋ . We will also need, for every ordinal α , a function f α : N →{ β | β < α } such that each ordinal β < α is mapped to by infinitely many n .Note that such a function exists, since the range is countable assuming CH.Using AC and CH, we first the subsets of the natural numbers and we orderthe angles θ ∈ ( π/ , π/
4) so that each has at most countably many predeces-sors.We will define real numbers x α , y α and z α inductively. Let x be a realwhich is random relative to A . Let y be a real which is random relative to( A , x ). Define z to be the real whose binary expansion is given by z [ r ] = ( g θ ( x , y )[ r ] if a n < r ≤ b n for some n ∈ N y [ r ] otherwiseFor the induction step, suppose we have defined our points up to ordinal α .Let x α be a real number which is random relative to the join of S β<α ( A β , x β )and A α . Let y α be random relative to the join of S β<α ( A β , x β ), A α and x α .This is possible, as we are assuming CH, and so this union is countable. Let z α be the point whose binary expansion is given by z α [ r ] = ( g θ β ( x α , y α )[ r ] if a n < r ≤ b n , where f α ( n ) = βy α [ r ] otherwiseFinally, we define our set E = { ( x α , z α ) } .17 emma 21. For every θ ∈ ( π/ , π/ , dim H ( p θ E ) ≤ s Proof.
Let θ ∈ ( π/ , π/
4) and α be its corresponding ordinal. Let A be anoracle encoding θ and S β ≤ α ( x β , y β , z β ).Note that, since we assumed CH, this is a countable union, and so the oracle iswell defined.Let z = ( x β , z β ) ∈ E . First assume that β ≤ α . Then, by our constructionof A , all the information of p θ z is already encoded in our oracle, and so K Ar ( p θ z ) = o ( r ).Now assume that β > α . Then by our construction of E , there are infinitelymany n such that f β ( n ) = α . Therefore there are infinitely many n such that z β [ r ] = g θ α ( x β , y β )[ r ],for a n < r ≤ b n . Recalling the definition of g θ α , this means that, for each such n , K θb n ( p θ z ) = sb n + o ( r ).Therefore, by the point-to-set principle,dim H ( E ) ≤ sup z ∈ E dim A ( p θ z ) ≤ sup β>α lim inf n K Ab n ( p θ z ) b n ≤ sup β>α lim inf n sb n b n = s, and the proof is complete. Lemma 22.
The Hausdorff dimension of E is s .Proof. We first give an upper bound on the dimension. Let A be an oracleencoding θ . Let z = ( x α , z α ). By our construction of E , there are infinitelymany n such that f α ( n ) = 1. Therefore there are infinitely many n such that z β [ r ] = g θ ( x β , y β )[ r ],for a n < r ≤ b n . Recalling the definition of g θ , this means that, for each such n , K θ b n ( p θ z ) = sb n + o ( r ).18oreover, K θ b n ( x α , z α ) ≤ K θ b n ( x α ) + K θ b n ( z α | x α ) + o ( r ) ≤ b n + K θ b n ( p θ z ) + o ( r ) ≤ b n + sb n + o ( b n )) . Therefore, by the point-to-set principle,dim H ( E ) ≤ sup z ∈ E dim A ( z ) ≤ sup z ∈ E lim inf n K Ab n ( z ) b n ≤ sup z ∈ E lim inf n (1 + s ) b n + o ( b n ) b n = 1 + s. For the upper bound, let A be a Hausdorff oracle for E , and let α be theordinal corresponding to A . By construction of z = ( x α , z α ), K Ar ( x α ) ≥ r − o ( r ),for all r ∈ N . We also have, for every n , K Aa n ( z α | x α ) ≥ K Aa n ( y α | x α ) − b n − − o ( a n ) ≥ a n − b n − − o ( a n )= a n − a n − o ( a n ) . Hence, for every n and every a n < r ≤ b n , K Ar ( z α | x α ) ≥ K Aa n ( z α | x α ) ≥ a n − a n − o ( a n ) . This implies that K Ar ( x α , z α ) r = K Ar ( x α ) + K Ar ( z α | x α ) r ≥ r + a n − a n − o ( a n ) r = 1 + s − o (1) . We can also conclude that, for every n and every b n < r ≤ a n +1 , K Ar ( z α | x α ) ≥ K Ab n ( z α | x α ) K Aa n ,b n ( z α | x α ) − o ( r ) ≥ a n − a n + r − b n − o ( r ) . K Ar ( x α , z α ) r = K Ar ( x α ) + K Ar ( z α | x α ) r ≥ r + a n − a n + r − b n − o ( r ) r ≥ s − o (1) . These inequalities, combined with the point-to-set principle show thatdim H ( E ) = sup z ∈ E dim A ( z ) ≥ sup z ∈ E lim inf r K Ar ( z ) r ≥ sup z ∈ E lim inf r s = 1 + s, and the proof is complete. I would like to thank Chris Porter, Denis Hirschfeldt and Jack Lutz for their veryvaluable discussions and suggestions. I would also like to thank the participantsof the recent AIM workshop on Algorithmic Randomness.
References [1] Krishna B. Athreya, John M. Hitchcock, Jack H. Lutz, and Elvira Mayor-domo. Effective strong dimension in algorithmic information and compu-tational complexity.
SIAM J. Comput. , 37(3):671–705, 2007.[2] Christopher J. Bishop and Yuval Peres.
Fractals in probability and analysis ,volume 162 of
Cambridge Studies in Advanced Mathematics . CambridgeUniversity Press, Cambridge, 2017.[3] Adam Case and Jack H. Lutz. Mutual dimension.
ACM Transactions onComputation Theory , 7(3):12, 2015.[4] Logan Crone, Lior Fishman, and Stephen Jackson. Hausdorff dimensionregularity properties and games. arXiv preprint arXiv:2003.11578 , 2020.[5] Roy O. Davies. Two counterexamples concerning Hausdorff dimensions ofprojections.
Colloq. Math. , 42:53–58, 1979.[6] Rod Downey and Denis Hirschfeldt.
Algorithmic Randomness and Com-plexity . Springer-Verlag, 2010. 207] Kenneth Falconer.
Fractal Geometry: Mathematical Foundations and Ap-plications . Wiley, third edition, 2014.[8] Kenneth Falconer, Jonathan Fraser, and Xiong Jin. Sixty years of fractalprojections. In
Fractal geometry and stochastics V , pages 3–25. Springer,2015.[9] Leonid A. Levin. On the notion of a random sequence.
Soviet Math Dokl. ,14(5):1413–1416, 1973.[10] Leonid Anatolevich Levin. Laws of information conservation (nongrowth)and aspects of the foundation of probability theory.
Problemy PeredachiInformatsii , 10(3):30–35, 1974.[11] Ming Li and Paul M.B. Vit´anyi.
An Introduction to Kolmogorov Complexityand Its Applications . Springer, third edition, 2008.[12] Jack H. Lutz. Dimension in complexity classes.
SIAM J. Comput. ,32(5):1236–1259, 2003.[13] Jack H. Lutz. The dimensions of individual strings and sequences.
Inf.Comput. , 187(1):49–79, 2003.[14] Jack H. Lutz and Neil Lutz. Algorithmic information, plane Kakeya sets,and conditional dimension.
ACM Trans. Comput. Theory , 10(2):Art. 7, 22,2018.[15] Jack H Lutz and Neil Lutz. Who asked us? how the theory of computinganswers questions about analysis. In
Complexity and Approximation , pages48–56. Springer, 2020.[16] Jack H. Lutz and Elvira Mayordomo. Dimensions of points in self-similarfractals.
SIAM J. Comput. , 38(3):1080–1112, 2008.[17] Neil Lutz. Fractal intersections and products via algorithmic dimension. In , 2017.[18] Neil Lutz and D. M. Stull. Bounding the dimension of points on a line. In
Theory and applications of models of computation , volume 10185 of
LectureNotes in Comput. Sci. , pages 425–439. Springer, Cham, 2017.[19] Neil Lutz and D. M. Stull. Projection theorems using effective dimension. In , 2018.[20] J. M. Marstrand. Some fundamental geometrical properties of plane setsof fractional dimensions.
Proc. London Math. Soc. (3) , 4:257–302, 1954.[21] Pertti Mattila. Hausdorff dimension, orthogonal projections and intersec-tions with planes.
Ann. Acad. Sci. Fenn. Ser. AI Math , 1(2):227–244, 1975.2122] Pertti Mattila.
Geometry of sets and measures in Euclidean spaces: fractalsand rectifiability . Cambridge University Press, 1999.[23] Pertti Mattila. Hausdorff dimension, projections, and the fourier transform.
Publicacions matematiques , pages 3–48, 2004.[24] Pertti Mattila. Hausdorff dimension, projections, intersections, and besi-covitch sets. In
New Trends in Applied Harmonic Analysis, Volume 2 ,pages 129–157. Springer, 2019.[25] Elvira Mayordomo. A Kolmogorov complexity characterization of construc-tive Hausdorff dimension.
Inf. Process. Lett. , 84(1):1–3, 2002.[26] Elvira Mayordomo. Effective fractal dimension in algorithmic informationtheory. In S. Barry Cooper, Benedikt L¨owe, and Andrea Sorbi, editors,
NewComputational Paradigms: Changing Conceptions of What is Computable ,pages 259–285. Springer New York, 2008.[27] Andre Nies.
Computability and Randomness . Oxford University Press, Inc.,New York, NY, USA, 2009.[28] Tuomas Orponen. Combinatorial proofs of two theorems of Lutz and Stull. arXiv preprint arXiv:2002.01743 , 2020.[29] D. M. Stull. Results on the dimension spectra of planar lines. In , volume 117 of