aa r X i v : . [ c s . CC ] J a n Dimension Spectra of Lines
Neil Lutz ∗ Department of Computer Science, Rutgers UniversityPiscataway, NJ 08854, USA [email protected]
D. M. Stull † Department of Computer Science, Iowa State UniversityAmes, IA 50011, USA [email protected]
September 1, 2018
Abstract
This paper investigates the algorithmic dimension spectra of lines inthe Euclidean plane. Given any line L with slope a and vertical intercept b , the dimension spectrum sp( L ) is the set of all effective Hausdorff dimen-sions of individual points on L . We draw on Kolmogorov complexity andgeometrical arguments to show that if the effective Hausdorff dimensiondim( a, b ) is equal to the effective packing dimension Dim( a, b ), then sp( L )contains a unit interval. We also show that, if the dimension dim( a, b )is at least one, then sp( L ) is infinite. Together with previous work, thisimplies that the dimension spectrum of any line is infinite. Algorithmic dimensions refine notions of algorithmic randomness to quantify thedensity of algorithmic information of individual points in continuous spaces. Themost well-studied algorithmic dimensions for a point x ∈ R n are the effectiveHausdorff dimension , dim( x ), and its dual, the effective packing dimension ,Dim( x ) [8, 1]. These dimensions are both algorithmically and geometricallymeaningful [4]. In particular, the quantities sup x ∈ E dim( x ) and sup x ∈ E Dim( x )are closely related to classical Hausdorff and packing dimensions of a set E ⊆ R n [6, 9], and this relationship has been used to prove nontrivial results inclassical fractal geometry using algorithmic information theory [13, 9, 11]. ∗ Research supported in part by National Science Foundation Grant 1445755. † Research supported in part by National Science Foundation Grants 1247051 and 1545028. x ∈ E dim( x ) but the entire (effectiveHausdorff ) dimension spectrum of a set E ⊆ R n , i.e., the setsp( E ) = { dim( x ) : x ∈ E } . The dimension spectra of several classes of sets have been previously investi-gated. Gu, et al. studied the dimension spectra of randomly selected subfractalsof self-similar fractals [5]. Dougherty, et al. focused on the dimension spectraof random translations of Cantor sets [3]. In the context of symbolic dynamics,Westrick has studied the dimension spectra of subshifts [15].This work concerns the dimension spectra of lines in the Euclidean plane R . Given a line L a,b with slope a and vertical intercept b , we ask what sp( L a,b )might be. It was shown by Turetsky that, for every n ≥
2, the set of all pointsin R n with effective Hausdorff 1 is connected, guaranteeing that 1 ∈ sp( L a,b ). Inrecent work [11], we showed that the dimension spectrum of a line in R cannotbe a singleton. By proving a general lower bound on dim( x, ax + b ), which ispresented as Theorem 5 here, we demonstrated thatmin { , dim( a, b ) } + 1 ∈ sp( L a,b ) . Together with the fact that dim( a, b ) = dim( a, a + b ) ∈ sp( L a,b ) and Turetsky’sresult, this implies that the dimension spectrum of L a,b contains both endpointsof the unit interval [min { , dim( a, b ) } , min { , dim( a, b ) } + 1].Here we build on that work with two main theorems on the dimension spec-trum of a line. Our first theorem gives conditions under which the entire unitinterval must be contained in the spectrum. We refine the techniques of [11] toshow in our main theorem (Theorem 8) that, whenever dim( a, b ) = Dim( a, b ),we have [min { , dim( a, b ) } , min { , dim( a, b ) } + 1] ⊆ sp( L a,b ) . Given any value s ∈ [0 , x ∈ R such that dim( x, ax + b ) = s + min { dim( a, b ) , } . Our secondmain theorem shows that the dimension spectrum sp( L a,b ) is infinite for everyline such that dim( a, b ) is at least one. Together with Theorem 5, this showsthat the dimension spectrum of any line has infinite cardinality.We begin by reviewing definitions and properties of algorithmic informationin Euclidean spaces in Section 2. In Section 3, we sketch our technical approachand state our main technical lemmas; their proofs are deferred to the appendix.In Section 4 we prove our first main theorem and state our second main theorem,whose proof is deferred to the appendix. We conclude in Section 5 with a briefdiscussion of future directions. 2 Preliminaries
The conditional Kolmogorov complexity of binary string σ ∈ { , } ∗ given abinary string τ ∈ { , } ∗ is the length of the shortest program π that willoutput σ given τ as input. Formally, it 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 . Any π that achieves this minimum is said to testify to, or be a witness to,the value K ( σ | τ ). The Kolmogorov complexity of a binary string σ is K ( σ ) = K ( σ | λ ), where λ is the empty string. These definitions extends naturally toother finite data objects, e.g., vectors in Q n , via standard binary encodings;see [7] for details. The above definitions can also be extended to Euclidean spaces, as we nowdescribe. The
Kolmogorov complexity of a point x ∈ R m at precision r ∈ N isthe length of the shortest program π that outputs a precision - r rational estimatefor x . Formally, it is K r ( x ) = min { K ( p ) : p ∈ B − r ( x ) ∩ Q m } , where B ε ( x ) denotes the open ball of radius ε centered on x . The conditionalKolmogorov complexity of x at precision r given y ∈ R n at precision s ∈ R n is K r,s ( x | y ) = max (cid:8) min { K r ( p | q ) : p ∈ B − r ( x ) ∩ Q m } : q ∈ B − s ( y ) ∩ Q n (cid:9) . When the precisions r and s are equal, we abbreviate K r,r ( x | y ) by K r ( x | y ).As the following lemma shows, these quantities obey a chain rule and are onlylinearly sensitive to their precision parameters. Lemma 1 (J. Lutz and N. Lutz [9], N. Lutz and Stull [11]) . Let x ∈ R m and y ∈ R n . For all r, s ∈ N with r ≥ s ,1. K r ( x, y ) = K r ( x | y ) + K r ( y ) + O (log r ) .2. K r ( x ) = K r,s ( x | x ) + K s ( x ) + O (log r ) . J. Lutz initiated the study of algorithmic dimensions by effectivizing Hausdorffdimension using betting strategies called gales , which generalize martingales.Subsequently, Athreya, et al., defined effective packing dimension, also usinggales [1]. Mayordomo showed that effective Hausdorff dimension can be charac-terized using Kolmogorov complexity [12], and Mayordomo and J. Lutz showed3hat effective packing dimension can also be characterized in this way [10]. Inthis paper, we use these characterizations as definitions. The effective Hausdorffdimension and effective packing dimension of a point 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 . Guided by the information-theoretic nature of these characteriza-tions, J. Lutz and N. Lutz [9] defined the lower and upper conditional dimension of x ∈ R m given y ∈ R n asdim( x | y ) = lim inf r →∞ K r ( x | y ) r and Dim( x | y ) = lim sup r →∞ K r ( x | y ) r . By letting the underlying fixed prefix-free Turing machine U be a universal ora-cle machine, we may relativize the definition in this section to an arbitrary ora-cle set A ⊆ N . The definitions of K A ( σ | τ ), K A ( σ ), K Ar ( x ), K Ar ( x | y ), dim A ( x ),Dim A ( x ) dim A ( x | y ), and Dim A ( x | y ) are then all identical to their unrelativizedversions, except that U is given oracle access to A .We will frequently consider the complexity of a point x ∈ R n relative to apoint y ∈ R m , i.e., relative to a set A y that encodes the binary expansion of y isa standard way. We then write K yr ( x ) for K A y r ( x ). J. Lutz and N. Lutz showedthat K yr ( x ) ≤ K r,t ( x | y ) + K ( t ) + O (1) [9]. In this section we describe the basic ideas behind our investigation of dimensionspectra of lines. We briefly discuss some of our earlier work on this subject, andwe present two technical lemmas needed for the proof our main theorems.The dimension of a point on a line in R has the following trivial bound. Observation 2.
For all a, b, x ∈ R , dim( x, ax + b ) ≤ dim( x, a, b ) . In this work, our goal is to find values of x for which the approximate conversedim( x, ax + b ) ≥ dim a,b ( x ) + dim( a, b ) (1)holds. There exist oracles, at least, relative to which (1) does not always hold.This follows from the point-to-set principle of J. Lutz and N. Lutz [9] and theexistence of Furstenberg sets with parameter α and Hausdorff dimension lessthan 1 + α (attributed by Wolff [16] to Furstenberg and Katznelson “in allprobability”). The argument is simple and very similar to our proof in [11] of alower bound on the dimension of generalized Furstenberg sets.Specifically, for every s ∈ [0 , x of effective Hausdorffdimension s such that (1) holds. Note that equality in Observation 2 implies(1). 4 bservation 3. Suppose ax + b = ux + v and u = a . Then dim( u, v ) ≥ dim a,b ( u, v ) ≥ dim a,b (cid:18) b − vu − a (cid:19) = dim a,b ( x ) . This observation suggests an approach, whenever dim a,b ( x ) > dim( a, b ),for showing that dim( x, ax + b ) ≥ dim( x, a, b ). Since ( a, b ) is, in this case,the unique low-dimensional pair such that ( x, ax + b ) lies on L a,b , one mightna¨ıvely hope to use this fact to derive an estimate of ( x, a, b ) from an estimate of( x, ax + b ). Unfortunately, the dimension of a point is not even semicomputable,so algorithmically distinguishing ( a, b ) requires a more refined statement. The following lemma, which is essentially geometrical, is such a statement.
Lemma 4 (N. Lutz and Stull [11]) . Let a, b, x ∈ R . For all ( u, v ) ∈ R suchthat ux + v = ax + b and t = − log k ( a, b ) − ( u, v ) k ∈ (0 , r ] , K r ( u, v ) ≥ K t ( a, b ) + K a,br − t ( x ) − O (log r ) . Roughly, if dim( a, b ) < dim a,b ( x ), then Lemma 4 tells us that K r ( u, v ) >K r ( a, b ) unless ( u, v ) is very close to ( a, b ). As K r ( u, v ) is upper semicomputable,this is algorithmically useful: We can enumerate all pairs ( u, v ) whose precision- r complexity falls below a certain threshold. If one of these pairs satisfies,approximately, ux + v = ax + b , then we know that ( u, v ) is close to ( a, b ).Thus, an estimate for ( x, ax + b ) algorithmically yields an estimate for ( x, a, b ).In our previous work [11], we used an argument of this type to prove ageneral lower bound on the dimension of points on lines in R : Theorem 5 (N. Lutz and Stull [11]) . For all a, b, x ∈ R , dim( x, ax + b ) ≥ dim a,b ( x ) + min { dim( a, b ) , dim a,b ( x ) } . The strategy in that work is to use oracles to artificially lower K r ( a, b )when necessary, to essentially force dim( a, b ) < dim a,b ( x ). This enables theabove argument structure to be used, but lowering the complexity of ( a, b ) alsoweakens the conclusion, leading to the minimum in Theorem 5. In the present work, we circumvent this limitation and achieve inequality (1)by controlling the choice of x and placing a condition on ( a, b ). Adapting theabove argument to the case where dim( a, b ) > dim a,b ( x ) requires refining thetechniques of [11]. In particular, we use the following two technical lemmas,which strengthen results from that work. Lemma 6 weakens the conditionsneeded to compute an estimate of ( x, a, b ) from an estimate of ( x, ax + b ).5 emma 6. Let a, b, x ∈ R , k ∈ N , and r = 1 . Suppose that r , . . . , r k ∈ N , δ ∈ R + , and ε, η ∈ Q + satisfy the following conditions for every ≤ i ≤ k .1. r i ≥ log(2 | a | + | x | + 6) + r i − .2. K r i ( a, b ) ≤ ( η + ε ) r i .3. For every ( u, v ) ∈ R such that t = − log k ( a, b ) − ( u, v ) k ∈ ( r i − , r i ] and ux + v = ax + b , K r i ( u, v ) ≥ ( η − ε ) r i + δ · ( r i − t ) .Then for every oracle set A ⊆ N , K Ar k ( a, b, x | x, ax + b ) ≤ k (cid:18) K ( ε ) + K ( η ) + 4 εδ r k + O (log r k ) (cid:19) . Lemma 7 strengthens the oracle construction of [11], allowing us to controlcomplexity at multiple levels of precision.
Lemma 7.
Let z ∈ R n , η ∈ Q ∩ [0 , dim( z )] , and k ∈ N . For all r , . . . , r k ∈ N ,there is an oracle D = D ( r , . . . , r k , z, η ) such that1. For every t ≤ r , K Dt ( z ) = min { ηr , K t ( z ) } + O (log r k )
2. For every ≤ i ≤ k , K Dr i ( z ) = ηr + i X j =2 min { η ( r j − r j − ) , K r j ,r j − ( z | z ) } + O (log r k ) .
3. For every t ∈ N and x ∈ R , K z,Dt ( x ) = K zt ( x ) + O (log r k ) . We are now prepared to prove our two main theorems. We first show that,for lines L a,b such that dim( a, b ) = Dim( a, b ), the dimension spectrum sp( L a,b )contains the unit interval. Theorem 8.
Let a, b ∈ R satisfy dim( a, b ) = Dim( a, b ) . Then for every s ∈ [0 , there is a point x ∈ R such that dim( x, ax + b ) = s + min { dim( a, b ) , } .Proof. Every line contains a point of effective Hausdorff dimension 1 [14], andby the preservation of effective dimensions under computable bi-Lipschitz func-tions, dim( a, a + b ) = dim( a, b ), so the theorem holds for s = 0.Now let s ∈ (0 ,
1] and d = dim( a, b ) = Dim( a, b ). Let y ∈ R be randomrelative to ( a, b ). That is, there is some constant c ∈ N such that for all r ∈ N , K a,br ( y ) ≥ r − c . Define sequence of natural numbers { h j } j ∈ N inductively asfollows. Define h = 1. For every j >
0, define h j = min (cid:26) h ≥ h j − : K h ( a, b ) ≤ (cid:18) d + 1 j (cid:19) h (cid:27) . h j always exists. For every r ∈ N , let x [ r ] = ( rh j ∈ ( s,
1] for some j ∈ N y [ r ] otherwiseDefine x ∈ R to be the real number with this binary expansion. Then K sh j ( x ) = sh j + O (log sh j ).We first show that dim( x, ax + b ) ≤ s + min { d, } . For every j ∈ N , K h j ( x, ax + b ) = K h j ( x ) + K h j ( ax + b | x ) + O (log h j )= K sh j ( x ) + K h j ( ax + b | x ) + O (log h j )= K sh j ( y ) + K h j ( ax + b | x ) + O (log h j ) ≤ sh j + min { d, } · h j + o ( h j ) . Therefore, dim( x, ax + b ) = lim inf r →∞ K r ( x, ax + b ) r ≤ lim inf j →∞ K h j ( x, ax + b ) h j ≤ lim inf j →∞ sh j + min { d, } h j + o ( h j ) h j = s + min { d, } . If 1 ≥ s ≥ d , then by Theorem 5 we also havedim( x, ax + b ) ≥ dim( x | a, b ) + dim( a, b )= dim( x ) + d = lim inf r →∞ K r ( x ) r + d = lim inf j →∞ K h j ( x ) h j + d = s + min { d, } . Hence, we may assume that s < d .Let H = Q ∩ ( s, min { d, } ). We now show that for every η ∈ H and ε ∈ Q + ,dim( x, ax + b ) ≥ s + η − αε , where α is some constant independent of η and ε .Let η ∈ H , δ = 1 − η >
0, and ε ∈ Q + . Let j ∈ N and m = s − η − . We firstshow that K r ( x, ax + b ) ≥ K r ( x ) + ηr − c εδ r − o ( r ) , (2)for every r ∈ ( sh j , mh j ]. Let r ∈ ( sh j , mh j ]. Set k = rsh j , and define r i = ish j for all 1 ≤ i ≤ k . Note that k is bounded by a constant depending only on s and7 . Therefore o ( r k ) is sublinear for all r i . Let D r = D ( r , . . . , r k , ( a, b ) , η ) be theoracle defined in Lemma 7. We first note that, since dim( a, b ) = Dim( a, b ), K r i ,r i − ( a, b | a, b ) = K r i ( a, b ) − K r i − ( a, b ) − O (log r i )= dim( a, b ) r i − o ( r i ) − dim( a, b ) r i − − o ( r i − ) − O (log r i )= dim( a, b )( r i − r i − ) − o ( r i ) ≥ η ( r i − r i − ) − o ( r i ) . Hence, by property 2 of Lemma 7, for every 1 ≤ i ≤ k , | K D r r i ( a, b ) − ηr i | ≤ o ( r k ) . (3)We now show that the conditions of Lemma 6 are satisfied. By inequality (3),for every 1 ≤ i ≤ k , K D r r i ( a, b ) ≤ ηr i + o ( r k ) , and so K D r r i ( a, b ) ≤ ( η + ε ) r i , for sufficiently large j . Hence, condition 2 ofLemma 6 is satisfied.To see that condition 3 is satisfied for i = 1, let ( u, v ) ∈ B ( a, b ) such that ux + v = ax + b and t = − log k ( a, b ) − ( u, v ) k ≤ r . Then, by Lemmas 4 and 7,and our construction of x , K D r r ( u, v ) ≥ K D r t ( a, b ) + K D r r − t,r ( x | a, b ) − O (log r ) ≥ min { ηr , K t ( a, b ) } + K r − t ( x ) − o ( r k ) ≥ min { ηr , dt − o ( t ) } + ( η + δ )( r − t ) − o ( r k ) ≥ min { ηr , ηt − o ( t ) } + ( η + δ )( r − t ) − o ( r k ) ≥ ηt − o ( t ) + ( η + δ )( r − t ) − o ( r k ) . We conclude that K D r r ( u, v ) ≥ ( η − ε ) r + δ ( r − t ), for all sufficiently large j .To see that that condition 3 is satisfied for 1 < i ≤ k , let ( u, v ) ∈ B − ri − ( a, b )such that ux + v = ax + b and t = − log k ( a, b ) − ( u, v ) k ≤ r i . Since ( u, v ) ∈ B − ri − ( a, b ), r i − t ≤ r i − r i − = ish j − ( i − sh j ≤ sh j + 1 ≤ r + 1 . Therefore, by Lemma 4, inequality (3), and our construction of x , K D r r i ( u, v ) ≥ K D r t ( a, b ) + K D r r i − t,r i ( x | a, b ) − O (log r i ) ≥ min { ηr i , K t ( a, b ) } + K r i − t ( x ) − o ( r i ) ≥ min { ηr i , dt − o ( t ) } + ( η + δ )( r i − t ) − o ( r i ) ≥ min { ηr i , ηt − o ( t ) } + ( η + δ )( r i − t ) − o ( r i ) ≥ ηt − o ( t ) + ( η + δ )( r i − t ) − o ( r i ) ,
8e conclude that K D r r i ( u, v ) ≥ ( η − ε ) r i + δ ( r i − t ), for all sufficiently large j .Hence the conditions of Lemma 6 are satisfied, and we have K r ( x, ax + b ) ≥ K D r r ( x, ax + b ) − O (1) ≥ K D r r ( a, b, x ) − k (cid:18) K ( ε ) + K ( η ) + 4 εδ r + O (log r ) (cid:19) = K D r r ( a, b ) + K D r r ( x | a, b ) − k (cid:18) K ( ε ) + K ( η ) + 4 εδ r + O (log r ) (cid:19) ≥ sr + ηr − k (cid:18) K ( ε ) + K ( η ) + 4 εδ r + O (log r ) (cid:19) . Thus, for every r ∈ ( sh j , mh j ], K r ( x, ax + b ) ≥ sr + ηr − αεδ r − o ( r ) , where α is a fixed constant, not depending on η and ε .To complete the proof, we show that (2) holds for every r ∈ [ mh j , sh j +1 ).By Lemma 1 and our construction of x , K r ( x ) = K r,h j ( x | x ) + K h j ( x ) + o ( r )= r − h j + sh j + o ( r ) ≥ ηr + o ( r ) . The proof of Theorem 5 gives K r ( x, ax + b ) ≥ K r ( x ) + dim( x ) r − o ( r ), and so K r ( x, ax + b ) ≥ r ( s + η ).Therefore, equation (2) holds for every r ∈ [ sh j , sh j +1 ), for all sufficientlylarge j . Hence,dim( x, ax + b ) = lim inf r →∞ K r ( x, ax + b ) r ≥ lim inf r →∞ K r ( x ) + ηr − αεδ r − o ( r ) r ≥ lim inf r →∞ K r ( x ) r + η − αεδ = s + η − αεδ . Since η and ε were chosen arbitrarily, the conclusion follows. Theorem 9.
Let a, b ∈ R such that dim( a, b ) ≥ . Then for every s ∈ [ , there is a point x ∈ R such that dim( x, ax + b ) ∈ (cid:2) + s − s , s + 1 (cid:3) . Corollary 10.
Let L a,b be any line in R . Then the dimension spectrum sp( L a,b ) is infinite. roof. Let ( a, b ) ∈ R . If dim( a, b ) <
1, then by Theorem 5 and Observa-tion 2, the spectrum sp( L a,b ) contains the interval [dim( a, b ) , a, b ) ≥
1. By Theorem 9, for every s ∈ [ , x suchthat dim( x, ax + b ) ∈ [ + s − s , s + 1]. Since these intervals are disjoint for s n = n − n , the dimension spectrum sp( L a,b ) is infinite. We have made progress in the broader program of describing the dimensionspectra of lines in Euclidean spaces. We highlight three specific directions forfurther progress. First, it is natural to ask whether the condition on ( a, b ) maybe dropped from the statement our main theorem:
Does Theorem 8 hold forarbitrary a, b ∈ R ?Second, the dimension spectrum of a line L a,b ⊆ R may properly contain theunit interval described in our main theorem, even when dim( a, b ) = Dim( a, b ).If a ∈ R is random and b = 0, for example, then sp( L a,b ) = { } ∪ [1 , L a,b ) might itself contain aninterval, or even be infinite. How large (in the sense of cardinality, dimension,or measure) may sp( L a,b ) ∩ (cid:2) , min { , dim( a, b ) } (cid:1) be? Finally, any non-trivial statement about the dimension spectra of lines inhigher-dimensional Euclidean spaces would be very interesting. Indeed, an n -dimensional version of Theorem 5 (i.e., one in which a, b ∈ R n − , for all n ≥ 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] Adam Case and Jack H. Lutz. Mutual dimension.
ACM Transactions onComputation Theory , 7(3):12, 2015.[3] Randall Dougherty, Jack Lutz, R Daniel Mauldin, and Jason Teutsch.Translating the Cantor set by a random real.
Transactions of the AmericanMathematical Society , 366(6):3027–3041, 2014.[4] Rod Downey and Denis Hirschfeldt.
Algorithmic Randomness and Com-plexity . Springer-Verlag, 2010.[5] Xiaoyang Gu, Jack H. Lutz, Elvira Mayordomo, and Philippe Moser. Di-mension spectra of random subfractals of self-similar fractals.
Ann. PureAppl. Logic , 165(11):1707–1726, 2014.106] John M. Hitchcock. Correspondence principles for effective dimensions.
Theory of Computing Systems , 38(5):559–571, 2005.[7] Ming Li and Paul M.B. Vit´anyi.
An Introduction to Kolmogorov Complexityand Its Applications . Springer, third edition, 2008.[8] Jack H. Lutz. The dimensions of individual strings and sequences.
Inf.Comput. , 187(1):49–79, 2003.[9] Jack H. Lutz and Neil Lutz. Algorithmic information, plane Kakeya sets,and conditional dimension.
Proceedings of the 34th International Sympo-sium on Theoretical Aspects of Computer Science, STACS 2017, Hannover,Germany , to appear.[10] Jack H. Lutz and Elvira Mayordomo. Dimensions of points in self-similarfractals.
SIAM J. Comput. , 38(3):1080–1112, 2008.[11] Neil Lutz and D. M. Stull. Bounding the dimension of points on a line.
Proceedings of the 14th Annual Conference on Theory and Applications ofModels of Computation, TAMC 2017, Bern, Switzerland , to appear.[12] Elvira Mayordomo. A Kolmogorov complexity characterization of construc-tive Hausdorff dimension.
Inf. Process. Lett. , 84(1):1–3, 2002.[13] Jan Reimann. Effectively closed classes of measures and randomness.
An-nals of Pure and Applied Logic , 156(1), 2008.[14] Daniel Turetsky. Connectedness properties of dimension level sets.
Theor.Comput. Sci. , 412(29):3598–3603, 2011.[15] Linda Brown Westrick.
Computability in Ordinal Ranks and Symbolic Dy-namics . PhD thesis, University of California, Berkeley, 2014.[16] Thomas Wolff. Recent work connected with the Kakeya problem.
Prospectsin Mathematics , pages 129–162, 1999.11
Technical Appendix
A.1 Precursors to Technical Lemmas
The following lemma will be used in the proof of Lemma 6.
Lemma A.1 (Case and J. Lutz [2], J. Lutz and N. Lutz [9]) . Let x ∈ R m and y ∈ R n . For all r, s, r ′ , s ′ ∈ N ,1. K r ′ ( x ) = K r ( x ) + O ( | r ′ − r | ) + O (log r ) .2. K r ′ ,s ′ ( x | y ) = K r,s ( x | y ) + O ( | r ′ − r | + | s ′ − s | ) + O (log rs ) . The following two lemmas from our previous work (stated in slightly differentforms here) are precursors to Lemmas 6 and 7. The proof of Lemma 6 is similarto that of Lemma A.2, and the proof of Lemma 7 is an induction on Lemma A.3.
Lemma A.2 (N. Lutz and Stull [11]) . Suppose that a, b, x ∈ R , r ∈ N , δ ∈ R + ,and ε, η ∈ Q + satisfy the following conditions.1. r ≥ log(2 | a | + | x | + 5) + 1 .2. K r ( a, b ) ≤ ( η + ε ) r .3. For every ( u, v ) ∈ R such that t = − log k ( a, b ) − ( u, v ) k ∈ (0 , r ] and ux + v = ax + b , K r ( u, v ) ≥ ( η − ε ) r + δ · ( r − t ) .Then for every oracle set A ⊆ N , K Ar ( x, ax + b ) ≥ K Ar ( a, b, x ) − εδ r − K ( ε ) − K ( η ) − O (log r ) . Lemma A.3 (N. Lutz and Stull [11]) . Let r ∈ N , z ∈ R n , and η ∈ Q ∩ [0 , dim( z )] . There is an oracle A = A ( r, z, η ) such that1. For every t ≤ r , K At ( z ) = min { ηr, K t ( z ) } + O (log r ) .2. For every t > r , K At ( z ) ≥ ηr + K t,r ( z | z ) + O (log r ) .3. For every t ∈ N and y ∈ R m , K z,At ( y ) = K zt ( y ) + O (log r ) . A.2 Computing a Line Given a Point
For our purposes, we will need the following corollary to Lemma A.2. Informally,that lemma gives conditions under which precision- r estimates for ( x, ax + b )and ( a, b, x ) contain similar amounts of information. This corollary shows that,under the same conditions, those two approximations are furthermore nearly“interchangeable,” in the sense that there is a short program which, given aprecision- r estimate for ( x, ax + b ) as input, will output a precision- r estimatefor ( a, b, x ), and, as we argue in the proof, vice versa.A1 orollary A.4. If the conditions of Lemma A.2 are satisfied, then K Ar ( a, b, x | x, ax + b ) ≤ εδ r + K ( ε ) + K ( η ) + O (log r ) . Proof.
It is easy to see that K r ( x, ax + b | a, b, x ) = O (log r ): consider a constant-length program that, given ( u, v, y ) ∈ Q , outputs ( y, uy + v ). If ( u, v, y ) ∈ B − r ( a, b, x ), then ( y, uy + v ) ∈ B c − r ( x, ax + b ), where c is constant in r , so K r − c,r ( ax + b | a, b, x ) = O (1). Thus, by Lemma A.1, K r ( ax + b | a, b, x ) = O (log r ).Now suppose that the conditions of Lemma 6 are satisfied. Then by sym-metry of information and Lemma A.2, K Ar ( a, b, x | x, ax + b ) = K Ar ( a, b, x ) − K Ar ( x, ax + b ) + K Ar ( x, ax + b | a, b, x )= K Ar ( a, b, x ) − K Ar ( x, ax + b ) + O (log r ) ≤ εδ r + K ( ε ) + K ( η ) + O (log r ) . We will also need the following pair of geometric facts.
Observation A.5 (N. Lutz and Stull [11]) . Let a, x, b ∈ R and r ∈ N . Let ( q , q ) ∈ B − r ( x, ax + b ) .1. If ( p , p ) ∈ B − r ( a, b ) , then | p q + p − q | < − r ( | p | + | q | + 3) .2. If | p q + p − q | ≤ − r ( | p | + | q | + 3) , then there is some pair ( u, v ) ∈ B − r (2 | a | + | x | +5) ( p , p ) such that ax + b = ux + v . Lemma 6.
Let a, b, x ∈ R , k ∈ N , and r = 1 . Suppose that r , . . . , r k ∈ N , δ ∈ R + , and ε, η ∈ Q + satisfy the following conditions for every ≤ i ≤ k .1. r i ≥ log(2 | a | + | x | + 6) + r i − .2. K r i ( a, b ) ≤ ( η + ε ) r i .3. For every ( u, v ) ∈ R such that t = − log k ( a, b ) − ( u, v ) k ∈ ( r i − , r i ] and ux + v = ax + b , K r i ( u, v ) ≥ ( η − ε ) r i + δ · ( r i − t ) .Then for every oracle set A ⊆ N , K Ar k ( a, b, x | x, ax + b ) ≤ k (cid:18) K ( ε ) + K ( η ) + 4 εδ r k + O (log r k ) (cid:19) . Proof.
Let a, b, x ∈ R . We proceed by induction on k . By Corollary A.4, theconclusion holds for k = 1. Assume the conclusion holds for all i < k . Let r , . . . , r k , δ , ε , η , and A be as described in the lemma statement.Define an oracle Turing machine M that does the following given oracle A and input π = π π π π π such that U A ( π ) = ( q , q ) ∈ Q , U ( π ) =( s , . . . , s k ) ∈ N k , U ( π ) = ζ ∈ Q , U ( π ) = ι ∈ Q and U A ( π , q , q ) = h ∈ Q A2or every program σ ∈ { , } ∗ with ℓ ( σ ) ≤ ( ι + ζ ) s k , in parallel, M simulates U ( σ ). If one of the simulations halts with some output ( p , p ) ∈ Q ∩ B − rk − ( h )such that | p q + p − q | < − s ( | p | + | q | + 3) , then M halts with output ( p , p , q ). Let c M be a constant for the descriptionof M .Now let π , π , π , π , and π testify to K Ar ( x, ax + b ), K ( r , . . . , r k ), K ( ε ), K ( η ), and K r k − ,r k ( a, b | x, ax + b ) respectively, and let π = π π π π π .By condition 2, there is some (ˆ p , ˆ p ) ∈ B − rk ( a, b ) such that K (ˆ p , ˆ p ) ≤ ( η + ε ) r k , meaning that there is some ˆ σ ∈ { , } ∗ with ℓ (ˆ σ ) ≤ ( η + ε ) r k and U (ˆ σ ) = (ˆ p , ˆ p ). By Observation A.5(1), | ˆ p q + ˆ p − q | < − r k ( | ˆ p | + | q | + 3) , for every ( q , q ) ∈ B − rk ( x, ax + b ), so M is guaranteed to halt on input π .Hence, let ( p , p , q ) = M ( π ). By Observation A.5(2), there is some( u, v ) ∈ B γ − rk ( p , p ) ⊆ B − rk − ( a, b )such that ux + v = ax + b , where γ = log(2 | a | + | x | + 5). We have k ( p , p ) − ( u, v ) k < γ − r k and | q − x | < − r k , so ( p , p , q ) ∈ B γ +1 − rk ( u, v, x ) . It therefore follows that K Ar k − γ − ,r k ( u, v, x | x, ax + b ) ≤ K ( p , p , q ) ≤ ℓ ( π π π π π ) + c M ≤ ℓ ( π ) + K ( r , . . . , r k ) + K ( ε ) + K ( η ) + c M = ℓ ( π ) + K ( ε ) + K ( η ) + O (log r k ) . Applying Lemma A.1 yields K Ar k ( u, v, x | x, ax + b ) ≤ ℓ ( π ) + K ( ε ) + K ( η ) + O (log r k ) . (4)By our inductive hypothesis, we have that ℓ ( π ) = K r k − ,r k ( a, b | x, ax + b )= K r k − ( a, b | x, ax + b ) + O (log r k − ) ≤ k − (cid:18) K ( ε ) + K ( η ) + 4 εδ r k − + O (log r k − ) (cid:19) . (5)To complete the proof, we bound K Ar k ( a, b, x | u, v, x ). If t > r k , then K Ar k ( a, b, x | u, v, x ) ≤ log( r k ) . A3therwise, when t ≤ r k , by our construction of M and Lemma A.1,( η + ε ) r k ≥ K ( p , p ) ≥ K r k − γ ( u, v ) ≥ K r k ( u, v ) − O (log r k ) . Combining this with condition 3 in the lemma statement and simplifying yields r k − t ≤ εδ r k + O (log r k ) . Therefore, by Lemma A.1, we have K r k ( a, b, x | u, v, x ) ≤ r k − t ) + O (log r k ) ≤ εδ r k + O (log r k ) , (6)for every t ∈ N .Combining inequalities (4), (5) and (6) gives K r k ( a, b, x | x, ax + b ) ≤ K r k ( u, v, x | x, ax + b ) + K r k ( a, b, x | u, v, x ) ≤ K r k ( u, v, x | x, ax + b ) + 4 εδ r k + O (log r k ) ≤ ℓ ( π ) + K ( ε ) + K ( η ) + 4 εδ r k + O (log r k ) ≤ k (cid:18) K ( ε ) + K ( η ) + 4 εδ r k + O (log r k ) (cid:19) . A.3 Decreasing Complexity Using an Oracle
Given an oracle D ⊆ N , r ∈ N , z ∈ R n , and η ∈ Q ∩ [0 , dim D ( z )], let A D ( r, z, η )be the oracle guaranteed by applying Lemma A.3 relative to D . For oracles, A, B ⊆ N , let h A, B i ⊆ N be an oracle that combines A and B by interleaving.Note that we treat k as a constant for the purposes of asymptotic notation. Lemma 7.
Let z ∈ R n , η ∈ Q ∩ [0 , dim( z )] , and k ∈ N . For all r , . . . , r k ∈ N ,there is an oracle D = D ( r , . . . , r k , z, η ) such that1. For every t ≤ r , K Dt ( z ) = min { ηr , K t ( z ) } + O (log r k )
2. For every ≤ i ≤ k , K Dr i ( z ) = ηr + i X l =2 min { η ( r l − r l − ) , K r l ,r l − ( z | z ) } + O (log r k ) .
3. For every t ∈ N and x ∈ R , K z,Dt ( x ) = K zt ( x ) + O (log r k ) . A4 roof. We define the sequence of oracles recursively. Let D = A ( r , z, η ), asdefined in Lemma A.3, and for every 1 < i ≤ k , let D i = (cid:26) D i − if K D i − r i ( z ) < ηr i h D i − , A D i − ( r i , z, η ) i otherwise . Notice that, for every 1 ≤ i ≤ k , D i is a finite oracle, so dim D i ( z ) = dim( z ) and η ∈ [0 , dim D k ( z )].We now show via induction on k that the lemma holds for all k ∈ N . For k = 1, all three properties hold by Lemma A.3. Fix j >
1, assume the propertieshold for k = j − k = j . Let t ≤ r . It follows fromthe definition of the oracle D j and Lemma A.3, relative to D j − , that K D j t ( z ) = min { ηr j , K D j − t ( z ) } + O (log r j ) . By the induction hypothesis, K D j − t ( z ) = min { ηr , K t ( z ) } + O (log r j − ). Thus, K D j t ( z ) = min { ηr j , min { ηr , K t ( z ) } + O (log r j − ) } + O (log r j )= min { ηr , K t ( z ) } + O (log r j ) . We now show the property 2 holds for k = j . Suppose that i < j . Then bythe definition of D j , K D j r i ( z ) = min { ηr j , K D j − r i ( z ) } + O (log r j ) , and by the induction hypothesis, K D j − r i ( z ) = ηr + i X l =2 min { η ( r l − r l − ) , K r l ,r l − ( z | z ) } + O (log r j − ) . Since ηr + i X l =2 min { η ( r l − r l − ) , K r l ,r l − ( z | z ) } ≤ ηr i , we have K D j r i ( z ) = ηr + i X l =2 min { η ( r l − r l − ) , K r l ,r l − ( z | z ) } + O (log r j )Now suppose that i = j . If K D j − r j ( z ) < ηr j , then, by our induction hypothesisA5nd Lemma A.3, K D j r i ( z ) = K D j − r i ( z )= K D j − r i − ( z ) + K D j − r i ,r i − ( z | z ) − O (log r j )= ηr + i − X l =2 min { η ( r l − r l − ) , K r l ,r l − ( z | z ) } + O (log r j )+ K r i ,r i − ( z | z ) + O (log r j − = ηr + i X l =2 min { η ( r l − r l − ) , K r l ,r l − ( z | z ) } + O (log r j ) . If instead K D j − r i ( z ) ≥ ηr i , then K D j r i ( z ) = ηr i − O (log r i ) by Lemma A.3, relativeto D j − . Since K D j − r i ( z ) ≥ ηr i implies that K r i ,r i − ( z | z ) ≥ η ( r i − r i − ), K D j r i ( z ) = ηr + P il =2 min { η ( r l − r l − ) , K r l ,r l − ( z | z ) } + O (log r i )Therefore property 2 holds for all 1 ≤ i ≤ k .To complete the proof we show that property 3 is satisfied for k = j . Let t ∈ N and y ∈ R m . By Lemma A.3, relativized to D j − , and our inductionhypothesis, K z,D j t ( y ) = K z,D j − t ( y ) + O (log r j )= K zt ( y ) + O (log r j − ) + O (log r j )= K zt ( y ) + O (log r j ) . Thus, by mathematical induction, the lemma holds for all k ∈ N . A.4 Proof of Second Main Theorem
Theorem 9.
Let a, b ∈ R such that dim( a, b ) ≥ . Then for every s ∈ [ , there is a point x ∈ R such that dim( x, ax + b ) ∈ [ + s − s , s + 1] .Proof. Let s ∈ [ ,
1] and y ∈ R be random relative to ( a, b ). That is, there issome constant c ∈ N such that for all r ∈ N , K a,br ( y ) ≥ r − c .Define sequence of natural numbers { h j } j ∈ N inductively as follows. Define h =1. For every j >
0, define h j = min (cid:26) h ≥ h j − : K h ( a, b ) ≤ (cid:18) dim( a, b ) + 1 j (cid:19) h (cid:27) . Note that h j always exists. For every r ∈ N , let x [ r ] = ( rh j ∈ (cid:0) , s (cid:3) for some j ∈ N y [ r ] otherwiseA6efine x ∈ R to be the real number with this binary expansion. Then, K h j ( x ) = h j + O (log h j ) . We first show that dim( x, ax + b ) ≤ s + 1. For every j ∈ N , K h j /s ( x, ax + b ) = K h j /s ( x ) + K h j /s ( ax + b | x ) + O (log h j /s )= K h j ( x ) + K h j /s ( ax + b | x ) + O (log h j ) ≤ h j + 1 · h j /s + o ( h j ) . Therefore, dim( x, ax + b ) = lim inf r →∞ K r ( x, ax + b ) r ≤ lim inf j →∞ K h j /s ( x, ax + b ) h j /s ≤ lim inf j →∞ sh j + h j + o ( h j ) h j = s + 1 . Let H = Q ∩ ( s, η ∈ H . Let η ′ ∈ Q ∩ (0 , s ], δ = 1 − η >
0, and ε ∈ Q + . Let j ∈ N . We first show that K r ( x, ax + b ) ≥ sr + ηr − c εδ r − o ( r ) , (7)for every r ∈ ( h j , h j ]. Let r ∈ ( h j , h j ]. Let r = h j , r = r , and D r = D ( r , r , ( a, b ) , η ) be the oracle defined in Lemma 7. We first note that, by ourconstruction of x , K r,r ( a, b | a, b ) = K r ( a, b ) − K r ( a, b ) + O (log r ) ≥ K r ( a, b ) − dim( a, b ) r − h j /j + O (log r ) ≥ dim( a, b ) r − dim( a, b ) r − h j /j + O (log r ) ≥ dim( a, b )( r − r ) − h j /j + O (log r ) > η ( r − r ) − h j /j + O (log r ) . Hence, by property 2 of Lemma 7 ηr − h j /j − O (log r ) ≤ K D r r ( a, b ) ≤ ηr + O (log r ) . (8)We now show that the conditions of Lemma 6 are satisfied. By Lemma 7,for each i ∈ { , } , K D r r i ( a, b ) ≤ ηr i + O (log r ) . Hence, condition 2 of Lemma 6 is satisfied.A7o see that condition 3 is satisfied for i = 1, let ( u, v ) ∈ B ( a, b ) such that ux + v = ax + b and t = − log k ( a, b ) − ( u, v ) k ≤ r . Then, by Lemmas 4 and 7,and our construction of x , K D r r ( u, v ) ≥ K D r t ( a, b ) + K D r r − t,r ( x | a, b ) − O (log r ) ≥ min { ηr , K t ( a, b ) } + K r − t ( x ) − o ( r k ) ≥ min { ηr , dim( a, b ) t − o ( t ) } + ( η + δ )( r − t ) − o ( r k ) ≥ min { ηr , ηt − o ( t ) } + ( η + δ )( r − t ) − o ( r k ) ≥ ηt − o ( t ) + ( η + δ )( r − t ) − o ( r k ) ≥ ( η − ε ) r + δ ( r − t )for all sufficiently large j .To see that that condition 3 is satisfied for i = 2, let ( u, v ) ∈ B − r ( a, b )such that ux + v = ax + b and t = − log k ( a, b ) − ( u, v ) k ≤ r . Since ( u, v ) ∈ B − r ( a, b ), r − t ≤ r − r ≤ r − r = r . Therefore, by Lemmas 4 and 7, inequality (8) and our construction of x , K D r r ( u, v ) ≥ K D r t ( a, b ) + K D r r − t,r ( x | a, b ) − O (log r ) ≥ min { ηr , K t ( a, b ) } + K r − t ( x ) − o ( r ) ≥ min { ηr , ηt − h j /j − o ( t ) } + ( η + δ )( r − t ) − o ( r ) ≥ ηt − h j /j − o ( t ) + ( η + δ )( r − t ) − o ( r )= ηr − h j /j − o ( t ) + δ ( r − t ) − o ( r ) ≥ ηr − r /j − o ( t ) + δ ( r − t ) − o ( r ) ≥ ( η − ε ) r + δ ( r − t ) , for all sufficiently large j . Hence the conditions of Lemma 6 are satisfied, andwe have K r ( x, ax + b ) ≥ K D r r ( x, ax + b ) − O (1) ≥ K D r r ( a, b, x ) − (cid:18) K ( ε ) + K ( η ) + 4 εδ r + O (log r ) (cid:19) = K D r r ( a, b ) + K D r r ( x | a, b ) − (cid:18) K ( ε ) + K ( η ) + 4 εδ r + O (log r ) (cid:19) ≥ sr + ηr − (cid:18) K ( ε ) + K ( η ) + 4 εδ r + O (log r ) (cid:19) . A8ence, for every r ∈ ( h j , h j ], K r ( x, ax + b ) ≥ sr + ηr − αεδ r − o ( r ) ≥ sr + ηr − αεδ r − o ( r )where α is a fixed constant, not depending on η and ε .To complete the proof, it suffices to show that K r ( x, ax + b ) ≥ r ( + s − s − ε ),for every r ∈ (2 h j , h j +1 ]. Let r ∈ (2 h j , h j +1 ]. Then by Lemma 1 and ourconstruction of x , K r ( x ) = K r,h j /s ( x | x ) + K h j /s ( x ) + O (log r )= r − h j /s + h j + O (log r ) . The proof of Theorem 5 shows that K r ( x, ax + b ) ≥ K r ( x ) + η ′ r − o ( r ) ≥ r − h j /s + h j + η ′ r − o ( r ) ≥ r (cid:18)
32 + s − s − ε (cid:19) for sufficiently large j .Since η , η ′ and εε