Extending the Reach of the Point-to-Set Principle
aa r X i v : . [ c s . CC ] A p r The Dimensions of Hyperspaces
Jack H. Lutz ∗ Iowa State University Neil LutzIowa State University Elvira Mayordomo † Universidad de Zaragoza
Abstract
We use the theory of computing to prove general hyperspace dimension theorems for threeimportant fractal dimensions. Let X be a separable metric space, and let K ( X ) be the hyperspace of X , i.e., the set of all nonempty compact subsets of X , endowed with the Hausdorff metric.For the lower and upper Minkowski (i.e., box-counting) dimensions, we give precise formulasfor the dimension of K ( E ), where E is any subset of X . For packing dimension, we give a tightbound on the dimension of K ( E ) in terms of the dimension of E , where E is any analytic (i.e., Σ ) subset of X . These results also hold for a large class of gauge families (Hausdorff familiesof gauge functions).The logical structures of our proofs are of particular interest. We first extend two algo-rithmic fractal dimensions—computability-theoretic versions of classical Hausdorff and packingdimensions that assign dimensions dim( x ) and Dim( x ) to individual points x ∈ X —to arbi-trary separable metric spaces and to arbitrary gauge families. We then extend the point-to-setprinciple of J. Lutz and N. Lutz (2018) to arbitrary separable metric spaces and to a largeclass of gauge families. Finally, we use this principle to prove our hyperspace packing dimen-sion theorem. This is one of a handful of cases—all very recent and all using the point-to-setprinciple—in which the theory of computing has been used to prove new theorems in classical geometric measure theory, theorems whose statements do not involve computation or logic. ∗ Research supported in part by National Science Foundation grants 1545028 and 1900716 and partly done duringvisits at the Institute for Mathematical Sciences at the National University of Singapore and the California Instituteof Technology. † Research supported in part by Spanish Ministry of Science, Innovation and Universities grant TIN2016-80347-Rand partly done during a visit to the Institute for Mathematical Sciences at the National University of Singapore.
Introduction
It is rare for the theory of computing to be used to answer open mathematical questions whosestatements do not involve computation or related aspects of logic. The point-to-set principle [22],described below, has enabled several recent developments that do exactly this. This principlehas been used to obtain strengthened lower bounds on the Hausdorff dimensions of generalizedFurstenberg sets [23], extend the fractal intersection formula for Hausdorff dimension from Borel setsto arbitrary sets [24], and prove that Marstrand’s projection theorem for Hausdorff dimension holdsfor any set E whose Hausdorff and packing dimensions coincide, whether or not E is analytic [26]. (See [4, 5] for reviews of these developments.) These applications of the point-to-set principle allconcern fractal geometry in Euclidean spaces R n . This paper extends the reach of the point-to-set principle beyond Euclidean spaces. To explainthis, we first review the point-to-set principle to date. (All quantities defined in this intuitivediscussion are defined precisely later in the paper.) The two best-behaved classical fractal di-mensions, Hausdorff dimension and packing dimension, assign to every subset E of a Euclideanspace R n dimensions dim H ( E ) and dim P ( E ), respectively. When E is a “smooth” set that in-tuitively has some integral dimension between 0 and n , the Hausdorff and packing dimensionsagree with this intuition, but more complex sets E may have any real-valued dimensions satisfy-ing 0 ≤ dim H ( E ) ≤ dim P ( E ) ≤ n . Hausdorff and packing dimensions have many applications ininformation theory, dynamical systems, and other areas of science [2, 8, 15, 34].Early in this century, algorithmic versions of Hausdorff and packing dimensions were devel-oped to quantify the information densities of various types of data. The computational resourcesallotted to these algorithmic dimensions range from finite-state to computable enumerability andbeyond, but the point-to-set principle concerns the computably enumerable algorithmic dimensionsintroduced in [21, 1]. These assign to each individual point x in a Euclidean space R n an algo-rithmic dimension dim( x ) and a strong algorithmic dimension Dim( x ). The point-to-set principleof [22] is a complete characterization of the classical Hausdorff and packing dimensions in terms oforacle relativizations of these very non-classical dimensions of individual points. Specifically, thepoint-to-set principle says that, for every set E in a Euclidean space R n ,dim H ( E ) = min A ⊆ N sup x ∈ E dim A ( x ) (1.1)and dim P ( E ) = min A ⊆ N sup x ∈ E Dim A ( x ) . (1.2)The point-to-set principle is so named because it enables one to infer a lower bound on the classicaldimension of a set E from a lower bound on the relativized algorithmic dimension of a single,judiciously chosen point x ∈ E .The classical Hausdorff and packing dimensions work not only in Euclidean spaces, but in arbi-trary metric spaces. In contrast, nearly all work on algorithmic dimensions to date (the exceptionbeing [28]) has been in Euclidean spaces or in spaces of infinite sequences over finite alphabets. Ourobjective here is to significantly reduce this gap by extending the theory of algorithmic dimensions, These very non-classical proofs of new classical theorems have provoked new work in the fractal geometry com-munity. Orponen [33] has very recently used a discretized potential-theoretic method of Kaufman [17] and tools ofKatz and Tao [16] to give a new, classical proof of the two main theorems of [26]. Applications of the theory of computing—specifically Kolmogorov complexity—to discrete mathematics are morenumerous and are surveyed in [20]. Other applications to “continuous” mathematics, not involving the point-to-setprinciple, include theorems in descriptive set theory [31, 13, 18], Riemannian moduli space [41], and Banach spaces [19]. These have also been called “constructive” dimensions and “effective” dimensions by various authors. X is separable if it has a countable subset D that is dense in the sense that every point in X has pointsin D arbitrarily close to it.)In parallel with extending algorithmic dimensions to separable metric spaces, we also extendthem to arbitrary gauge families. It was already explicit in Hausdorff’s original paper [Haus19]that his dimension could be defined via various “lenses” that we now call gauge functions . In fact,one often uses, as we do here, a gauge family ϕ , which is a one-parameter family of gauge functions ϕ s for s ∈ (0 , ∞ ). For each separable metric space X , each gauge family ϕ , and each set E ⊆ X ,the classical ϕ -gauged Hausdorff dimension dim ϕ H ( E ) and ϕ -gauged packing dimension dim ϕ P ( E )are thus well-defined. In this paper, for each separable metric space X , each gauge family ϕ , andeach point x ∈ X , we define the ϕ -gauged algorithmic dimension dim ϕ ( x ) and the ϕ -gauged strongalgorithmic dimension Dim ϕ ( x ) of the point x . We should mention here that there is a particulargauge family θ that gives the “un-gauged” dimensions in the sense that the identitiesdim θ H ( E ) = dim H ( E ) , dim θ P ( E ) = dim P ( E ) , dim θ ( x ) = dim( x ) , Dim θ ( x ) = Dim( x )always hold.We generalize the point-to-set principle to arbitrary separable metric spaces and arbitrary gaugefamilies, proving that, for every separable metric space X , every gauge family ϕ that satisfies astandard doubling condition and admits a precision family (a mild condition defined in Section 2),and every set E ⊆ X , dim ϕ H ( E ) = min A ⊆ N sup x ∈ E dim ϕ,A ( x ) (1.3)and dim ϕ P ( E ) = min A ⊆ N sup x ∈ E Dim ϕ,A ( x ) . (1.4)Various nontrivial modifications to both machinery and proofs are necessary in getting from (1.1)and (1.2) to (1.3) and (1.4).As an illustration of the power of our approach, we investigate the dimensions of hyperspaces.The hyperspace K ( X ) of a metric space X is the set of all nonempty compact subsets of X , equippedwith the Hausdorff metric [42]. The hyperspace of a separable metric space is itself a separablemetric space, and the hyperspace is typically infinite-dimensional, even when the underlying metricspace is finite-dimensional. One use of gauge families is reducing such infinite dimensions to enablequantitative comparisons. For example, McClure [29] proved that, for every self-similar subset E of a separable metric space X , dim ψ H ( K ( E )) = dim H ( E ) , (1.5)where ψ is a particular gauge family. Here we are interested in the dimensions of hyperspaces K ( E )for more general gauge families and, especially, for more general sets E ⊆ X .Our main results are hyperspace dimension theorems for three important fractal dimensions.For each gauge family ϕ we define a jump e ϕ that is also a gauge family. For the lower and upperMinkowski (i.e., box-counting) dimensions dim M and dim M we prove that, for every separablemetric space X , every gauge family ϕ , and every E ⊆ X ,dim e ϕ M ( K ( E )) = dim ϕ M ( E ) (1.6) McClure’s gauge family ψ is exactly the jump of the canonical gauge family θ , so his result (1.5) says that, forall self-similar sets E , dim e θ ( K ( E )) = dim H ( E ). e ϕ M ( K ( E )) = dim ϕ M ( E ) . (1.7)We also prove that, for every separable metric space, every “well-behaved” gauge family ϕ , andevery compact set E ⊆ X dim e ϕ P ( K ( E )) = dim e ϕ M ( K ( E )) . (1.8)Finally, we use the point-to-set principle (1.2), the identities (1.7) and (1.8), and some additionalmachinery to prove our main theorem, the hyperspace packing dimension theorem (Theorem 5.4),which says that, for every separable metric space X , every well-behaved gauge family ϕ , and every analytic (i.e., Σ , an analog of NP that Sipser famously investigated [36, 37, 38]) set E ⊆ X ,dim e ϕ P ( K ( E )) ≥ dim ϕ P ( E ) . (1.9)At the time of this writing it is an open question whether there is a hyperspace dimensiontheorem for Hausdorff dimension.David Hilbert famously wrote the following [11].The final test of every new theory is its success in answering preexistent questions thatthe theory was not specifically created to answer.The theory of algorithmic dimensions passed Hilbert’s final test when the point-to-set principlegave us the the results in the first paragraph of this introduction. We hope that the machinerydeveloped here will lead to further such successes in the wider arena of separable metric spaces. We review the definitions of gauged Hausdorff, packing, and Minkowski dimensions. We refer thereader to [8, 27] for a complete introduction and motivation.Let (
X, ρ ) be a metric space where ρ is the metric. (From now on we will omit ρ when referringto the space ( X, ρ ).) X is separable if there exists a countable set D ⊆ X that is dense in X ,meaning that for every x ∈ X and δ >
0, there is a d ∈ D such that ρ ( x, d ) < δ . The diameter of aset E ⊆ X is diam( E ) = sup { ρ ( x, y ) | x, y ∈ E } ; notice that the diameter of a set can be infinite.A cover of E ⊆ X is a collection U ⊆ P ( X ) such that E ⊆ S U ∈U U , and a δ -cover of E is a cover U of E such that diam( U ) ≤ δ for all U ∈ U . Definition (gauge functions and families) . A gauge function is a continuous, nondecreasing func-tion from [0 , ∞ ) to [0 , ∞ ) that vanishes only at 0 [10, 35]. A gauge family is a one-parameter family ϕ = { ϕ s | s ∈ (0 , ∞ ) } of gauge functions ϕ s satisfying ϕ s ( δ ) = o ( ϕ t ( δ )) as δ → + whenever s > t .The canonical gauge family is θ = { θ s | s ∈ (0 , ∞ ) } , defined by θ s ( δ ) = δ s . “Un-gauged”or “ordinary” Hausdorff, packing, and Minkowski dimensions are special cases of the followingdefinitions, using ϕ = θ .Some of our gauged dimension results will require the existence of a “precision family” for thegauge family. Some authors require only that the function is right-continuous when working with Hausdorff dimension and left-continuous when working with packing dimension. Indeed, left continuity is sufficient for our hyperspace dimensiontheorem. efinition (precision family) . A precision sequence for a gauge function ϕ is a function α : N → Q + that vanishes as r → ∞ and satisfies ϕ ( α ( r )) = O ( ϕ ( α ( r + 1))) as r → ∞ . A precision family fora gauge family ϕ = { ϕ s | s ∈ (0 , ∞ ) } is a one-parameter family α = { α s | s ∈ (0 , ∞ ) } of precisionsequences satisfying X r ∈ N ϕ t ( α s ( r )) ϕ s ( α s ( r )) < ∞ whenever s < t . Observation 2.1. α s ( r ) = 2 − sr defines a precision family for the canonical gauge family θ . Definition (gauged Hausdorff measure and dimension) . For every metric space X , set E ⊆ X ,and gauge function ϕ , the ϕ -gauged Hausdorff measure of E is H ϕ ( E ) = lim δ → + inf ( X U ∈U ϕ (diam( U )) (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) U is a countable δ -cover of E ) . For every gauge family ϕ = { ϕ s | s ∈ (0 , ∞ ) } , the ϕ -gauged Hausdorff dimension of E isdim ϕ H ( E ) = inf { s ∈ (0 , ∞ ) | H ϕ s ( E ) = 0 } . Definition (gauged packing measure and dimension) . For every metric space X , set E ⊆ X , and δ ∈ (0 , ∞ ), let V δ ( E ) be the set of all countable collections of disjoint open balls with centers in E and diameters at most δ . For every gauge function ϕ and δ >
0, define the quantity P ϕδ ( E ) = sup U∈V δ ( E ) X U ∈U ϕ (diam( U )) . Then the ϕ -gauged packing pre-measure of E is P ϕ ( E ) = lim δ → + P ϕδ ( E ) , and the ϕ -gauged packing measure of E is P ϕ ( E ) = inf ( X U ∈U P ϕ ( U ) (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) U is a countable cover of E ) . For every gauge family ϕ = { ϕ s | s ∈ (0 , ∞ ) } , the ϕ -gauged packing dimension of E isdim ϕ P ( E ) = inf { s ∈ (0 , ∞ ) | P ϕ s ( E ) = 0 } . Definition (gauged Minkowski dimensions) . For every metric space X , E ⊆ X , and δ ∈ (0 , ∞ ),let N ( E, δ ) = min ( | F | (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) F ⊆ X and E ⊆ [ x ∈ F B δ ( x ) ) , where B δ ( x ) is the open ball of radius δ centered at x . Then for every gauge family ϕ = { ϕ s } s ∈ (0 , ∞ ) the ϕ -gauged lower and upper Minkowski dimension of E aredim ϕ M ( E ) = inf (cid:26) s (cid:12)(cid:12)(cid:12)(cid:12) lim inf δ → + N ( E, δ ) ϕ s ( δ ) = 0 (cid:27) and dim ϕ M ( E ) = inf (cid:26) s (cid:12)(cid:12)(cid:12)(cid:12) lim sup δ → + N ( E, δ ) ϕ s ( δ ) = 0 (cid:27) , respectively. 4hen X is separable, it is sometimes useful to require that the balls covering E have centersin the countable dense set D . For all E ⊆ X and δ ∈ (0 , ∞ ), letˆ N ( E, δ ) = min ( | F | (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) F ⊆ D and E ⊆ [ x ∈ F B δ ( x ) ) . Observation 2.2. If X is a separable metric space and ϕ = { ϕ s } s ∈ (0 , ∞ ) is a gauge family, thenfor all E ⊆ X ,1. dim ϕ M ( E ) = inf (cid:26) s (cid:12)(cid:12)(cid:12)(cid:12) lim inf δ → + ˆ N ( E, δ ) ϕ s ( δ ) = 0 (cid:27) .2. dim ϕ M ( E ) = inf (cid:26) s (cid:12)(cid:12)(cid:12)(cid:12) lim sup δ → + ˆ N ( E, δ ) ϕ s ( δ ) = 0 (cid:27) . The following relationship between upper Minkowski dimension and packing dimension waspreviously known to hold for the canonical gauge family θ , a result that is essentially due toTricot [40]. Our proof of this gauged generalization, which is in the appendix, is adapted from thepresentation by Bishop and Peres [3] of the un-gauged proof. Lemma 2.3 (generalizing Tricot [40]) . Let X be any metric space, E ⊆ X , and ϕ a gauge family.1. If ϕ t (2 δ ) = O ( ϕ s ( δ )) as δ → + for all s < t , then dim ϕ P ( E ) ≥ inf ( sup i ∈ N dim ϕ M ( E i ) (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) E ⊆ [ i ∈ N E i ) .
2. If there is a precision family for ϕ , then dim ϕ P ( E ) ≤ inf ( sup i ∈ N dim ϕ M ( E i ) (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) E ⊆ [ i ∈ N E i ) . In this section we formulate algorithmic dimensions in arbitrary separable metric spaces and witharbitrary gauge families.Throughout this section, let X = ( X, ρ ) be a separable metric space, and fix a function f : { , } ∗ → X such that the set D = range( f ) is dense in X . The metric space X is computable if there is a computable function g : ( { , } ∗ ) × Q + → Q that approximates ρ on D in the sensethat, for all v, w ∈ { , } ∗ and δ ∈ Q + . | g ( v, w, δ ) − ρ ( f ( v ) , f ( w )) | ≤ δ. Our results here hold for all separable metric spaces, whether or not they are computable, but ourmethods make explicit use of the function f .Following standard practice [32, 6, 20], fix a universal oracle prefix Turing machine U , anddefine the conditional Kolmogorov complexity of a string w ∈ { , } ∗ given a string v ∈ { , } ∗ relative to an oracle A ⊂ N to beK A ( w | v ) = min (cid:8) | π | (cid:12)(cid:12) π ∈ { , } ∗ and U A ( π, v ) = w (cid:9) , U to output w when it has access to the oracle A . The conditional Kolmogorov complexity of w given v is thenK( w | v ) = K ∅ ( w | v ) , and the Kolmogorov complexity of w is K( w ) = K( w | λ ) , where λ is the empty string.We define the Kolmogorov complexity of a point q ∈ D to beK( q ) = min { K( w ) | w ∈ { , } ∗ and f ( w ) = q } , noting that this depends on the enumeration f of D that we have fixed.The Kolmogorov complexity of a point x ∈ X at a precision δ ∈ Q + = Q ∩ (0 , ∞ ) isK δ ( x ) = min { K( q ) | q ∈ D and ρ ( q, x ) ≤ δ } , and the precision-conditioned Kolmogorov complexity of x at precision δ isK δ ( x | δ ) = min { K ( q | δ ) | q ∈ D and ρ ( q, x ) ≤ δ } . The algorithmic dimension of a point x ∈ X isdim( x ) = lim inf δ → + K δ ( x | δ )log(1 /δ ) , (3.1)and the strong algorithmic dimension of x isDim( x ) = lim sup δ → + K δ ( x | δ )log(1 /δ ) . (3.2)These two dimensions have been extensively investigated in the special cases where X is a Eu-clidean space R n or a sequence space Σ ω [25, 6].Having generalized algorithmic dimensions to arbitrary separable metric spaces, we now gener-alize them to arbitrary gauge families.If ϕ = { ϕ s | s ∈ (0 , ∞ ) } is a gauge family, then the ϕ -gauged algorithmic dimension of a point x ∈ X is dim ϕ ( x ) = inf (cid:26) s (cid:12)(cid:12)(cid:12)(cid:12) lim inf δ → + K δ ( x | δ ) ϕ s ( δ ) = 0 (cid:27) , (3.3)and the ϕ -gauged strong algorithmic dimension of x isDim ϕ ( x ) = inf (cid:26) s (cid:12)(cid:12)(cid:12)(cid:12) lim sup δ → + K δ ( x | δ ) ϕ s ( δ ) = 0 (cid:27) . (3.4)Gauged algorithmic dimensions dim ϕ ( x ) have been investigated by Staiger [39] in the specialcase where X is a sequence space Σ ω .A routine inspection of (3.1)–(3.4) verifies the following. The definitions given here differ slightly from the standard formulation in which the precision parameter δ belongsto { − r | r ∈ N } and the Kolmogorov complexities are not conditional. The present formulation is equivalent to thestandard one and facilitates our generalization to arbitrary gauged algorithmic dimensions. In particular, conditioningon δ is only needed to accommodate gauge families ϕ in which the convergence of ϕ s to 0 as δ → + is very slow. bservation 3.1. For all x ∈ X , dim θ ( x ) = dim( x ) and Dim θ ( x ) = Dim( x ) , where θ is the canonical gauge family given by θ s ( δ ) = δ s . A specific investigation of algorithmic (or classical) dimensions might call for a particular gaugefunction on family for one of two reasons. First, many gauge functions may assign the samedimension to an object under consideration (because they converge to 0 at somewhat similar ratesas δ → + ) but additional considerations may identify one of these as being the most preciselytuned to the phenomenon of interest. Finding such a gauge function is called finding the “exactdimension” of the object under investigation. This sort of calibration has been studied extensivelyfor classical dimensions [8, 35] and by Staiger [39] for algorithmic dimension.The second reason, and the reason of interest to us here, why specific investigations might callfor particular gauge families is that a given gauge family ϕ may be so completely out of tune with thephenomenon under investigation that the ϕ -gauged dimensions of the objects of interest are eitherall minimum (all 0) or else all maximum (all the same dimension as the space X itself). In sucha circumstance, a gauge family that converges to 0 more quickly or slowly than ϕ may yield moreinformative dimensions. Several such circumstances were investigated in a complexity-theoreticsetting by Hitchcock, J. Lutz, and Mayordomo [12].The following routine observation indicates the direction in which one adjusts a gauge family’sconvergence to 0 in order to adjust the resulting gauged dimensions upward or downward. Observation 3.2. If ϕ and ψ are gauge families with ϕ s ( δ ) = o ( ψ s ( δ )) as δ → + for all s ∈ (0 , ∞ ) ,then, for all x ∈ X , dim ϕ ( x ) ≤ dim ψ ( x ) and Dim ϕ ( x ) ≤ Dim ψ ( x ) . We now define an operation on gauge families that is implicit in earlier work [29] and is explicitlyused in our main theorems.
Definition (jump) . The jump of a gauge family ϕ is the family e ϕ given by e ϕ s ( δ ) = 2 − /ϕ s ( δ ) . Observation 3.3.
The jump of a gauge family is a gauge family.
We now note that the jump of a gauge family always converges to 0 more quickly than theoriginal gauge family.
Lemma 3.4.
For all gauge families ϕ and all s ∈ (0 , ∞ ) , e ϕ s ( δ ) = o ( ϕ s ( δ )) as δ → + . Observation 3.3 and Lemma 3.4 immediately imply the following.
Corollary 3.5.
For all gauge families ϕ and all x ∈ X , dim e ϕ ( x ) ≤ dim ϕ ( x ) and Dim e ϕ ( x ) ≤ Dim ϕ ( x ) . The definitions and results of this section relativize to arbitrary oracles A ⊆ N in the obvi-ous manner, so the Kolmogorov complexities K A ( q ) and K Aδ ( x | δ ) and the dimensions dim A ( x ),Dim A ( x ), dim ϕ,A ( x ), and Dim ϕ,A ( x ) are all well-defined and behave as indicated.7 bservation 3.6. For all gauge families ϕ , all x ∈ X , and all s > , log (cid:0) K δ ( x | δ ) e ϕ s ( δ ) (cid:1) = K δ ( x | δ ) ϕ s ( δ ) − ϕ s ( δ ) . The e ϕ -gauged algorithmic dimensions admit the following characterizations, the second of whichis used in the proof of our hyperspace packing dimension theorem. Theorem 3.7.
For all gauge families ϕ and all x ∈ X , the following identities hold.1. dim e ϕ ( x ) = inf (cid:26) s (cid:12)(cid:12)(cid:12)(cid:12) lim inf δ → + K δ ( x | δ ) ϕ s ( δ ) = 0 (cid:27) .2. Dim e ϕ ( x ) = inf (cid:26) s (cid:12)(cid:12)(cid:12)(cid:12) lim sup δ → + K δ ( x | δ ) ϕ s ( δ ) = 0 (cid:27) . We now show that the point-to-set principle of J. Lutz and N. Lutz [22] holds in arbitrary separablemetric spaces and for gauged dimensions. The proofs of these theorems, which can be found in theappendix, are more delicate and involved than those in [22]. This is partially due to the fact thatthe metric spaces here need not be finite-dimensional, and to the weak restrictions we place on thegauge family.
Theorem 4.1 (general point-to-set principle for Hausdorff dimension) . For every separable metricspace X , every gauge family ϕ , and every set E ⊆ X , dim ϕ H ( E ) ≥ min A ⊆ N sup x ∈ E dim ϕ,A ( x ) . Equality holds if there is a precision family for ϕ . Theorem 4.2 (general point-to-set principle for packing dimension) . Let X be any separable metricspace, E ⊆ X , and ϕ a gauge family.1. If ϕ t (2 δ ) = O ( ϕ s ( δ )) as δ → + for all s < t , then dim ϕ P ( E ) ≥ min A ⊆ N sup x ∈ E Dim ϕ,A ( x ) .
2. If there is a precision family for ϕ , then dim ϕ P ( E ) ≤ min A ⊆ N sup x ∈ E Dim ϕ,A ( x ) . The two assertions of Theorem 4.2 correspond directly to the two assertions of Lemma 2.3.
This section presents our main theorems. 8s before, let X = ( X, ρ ) be a separable metric space. The hyperspace of X is the metric space K ( X ) = ( K ( X ) , ρ H ), where K ( X ) is the set of all nonempty compact subsets of X and ρ H is the Hausdorff metric [9] on K ( X ) defined by ρ H ( E, F ) = max ( sup x ∈ E ρ ( x, F ) , sup y ∈ F ρ ( E, y ) ) , where ρ ( x, F ) = inf y ∈ F ρ ( x, y ) and ρ ( E, y ) = inf x ∈ E ρ ( x, y ) . Let f : { , } ∗ → X and D = range( f ) be fixed as at the beginning of section 3, so that D is dense in X . Let D be the set of all nonempty, finite subsets of D . It is well known and easyto show that D is a countable dense subset of K ( X ), and it is routine to define from f a function e f : { , } ∗ → K ( X ) such that range( e f ) = D . Thus K ( X ) is a separable metric space, and theresults in section 4 hold for K ( X ).It is important to note the distinction between the classical Hausdorff and packing dimensionsdim H ( E ) and dim P ( E ) of a nonempty compact subset E of X and the algorithmic dimensionsdim( E ) and Dim( E ) of this same set when it is regarded as a point in K ( X ). In the appendix, weconstruct an example of a set E with Hausdorff and packing dimensions dim H ( E ) = dim P ( E ) =log(2) / log(4) ≈ .
356 and dim( E ) = Dim( E ) = ∞ .Our first hyperspace dimension theorem applies to lower and upper Minkowski dimensions. Thistheorem, which is proven using a counting argument, is very general, placing no restrictions on thegauge family ϕ or the separable metric space X . Theorem 5.1 (hyperspace Minkowski dimension theorem) . For every gauge family ϕ and every E ⊆ X , dim e ϕ M ( K ( E )) = dim ϕ M ( E ) and dim e ϕ M ( K ( E )) = dim ϕ M ( E ) . The proof of our hyperspace packing dimension theorem relies on surprising fact that in ahyperspace, packing dimension and upper Minkowski dimension are equivalent for compact sets.
Theorem 5.2.
For every separable metric space X , every compact set E ⊆ X , and every gaugefamily ϕ such that ϕ t (2 δ ) = O ( ϕ s ( δ )) as δ → + for all s < t and there is a precision family for ϕ , dim e ϕ P ( K ( E )) = dim e ϕ M ( K ( E )) . The point-to-set principle is central to our proof of this theorem: We recursively construct asingle compact set L ⊆ E (i.e., a single point in the hyperspace K ( E )) that has high Kolmogorovcomplexity at infinitely many precisions, relative to an appropriate oracle A . We then invokeTheorem 3.7 to show that this L has high ϕ -gauged strong algorithmic dimension relative to A .By the point-to-set principle, then, K ( E ) has high packing dimension. Observation 5.3.
The conclusion of Theorem 5.2 does not hold for arbitrary sets E .Proof. Let E = { /n : n ∈ N } . Then dim θ M ( E ) = 1 /
2, but every compact subset of E is finite, so K ( E ) is countable and dim e θ P ( K ( E )) = 0. Theorem 5.4 (hyperspace packing dimension theorem) . If X is a separable metric space, E ⊆ X is an analytic set, and ϕ is a gauge family such that ϕ s (2 δ ) = O ( ϕ s ( δ )) as δ → + for all s ∈ (0 , ∞ ) and there is a precision family for ϕ , then dim e ϕ P ( K ( E )) ≥ dim ϕ P ( E ) . roof. For compact sets E , Theorem 5.2 and the hyperspace Minkowski dimension theorem (The-orem 5.1) imply dim e ϕ P ( K ( E )) = dim ϕ M ( E ) . A theorem of Joyce and Preiss [14] states that every analytic set with positive (possibly infinite)gauged packing measure contains a compact subset with positive (finite) packing measure in thesame gauge. It follows that if E is analytic, then for all ε > E ε ⊆ E with dim ϕ P ( E ε ) ≥ dim ϕ P ( E ) − ε. Therefore dim e ϕ P ( K ( E ε )) = dim ϕ M ( E ε ) ≥ dim ϕ P ( E ε ) ≥ dim ϕ P ( E ) − ε. Letting ε → L we construct is only guaranteed to have high complexity at infinitely many precisions. Ananalogous proof for Hausdorff dimension would require constructing a set L that has high complexityat all but finitely many precisions. 10 Proofs from Section 2
Observation A.1. If E is separable and < δ < ˆ δ , then ˆ N ( E, ˆ δ ) ≤ N ( E, δ ) .Proof. Let F ⊆ X be a witness to N ( E, δ ). For all x ∈ F , there exists ˆ x ∈ D with 0 < ρ ( x, ˆ x ) < ˆ δ − δ , so B δ ( x ) ⊆ B ˆ δ (ˆ x ). Thus, the set ˆ F = { ˆ x | x ∈ F } ⊆ D satisfies E ⊆ [ x ∈ F B δ ( x ) ⊆ [ ˆ x ∈ ˆ F B ˆ δ (ˆ x ) , so ˆ N ( E, ˆ δ ) ≤ | ˆ F | = | F | = N ( E, δ ). Proof of Observation 2.2.
Since ˆ N ( E, δ ) ≥ N ( E, δ ), it is clear that the left-hand side of each equa-tion is bounded above by the right-hand side. We now prove the other direction. Fix s ∈ (0 , ∞ ).1. Assume that lim inf δ → + N ( E, δ ) ϕ s ( δ ) = 0, and let δ , ε >
0. Then there is some δ < δ suchthat N ( E, δ ) ϕ s ( δ ) < ε/
2. By the (right) continuity of ϕ s , there exists some ˆ δ ∈ ( δ, δ ) suchthat ϕ s (ˆ δ ) < ϕ s ( δ ). By Observation A.1, ˆ N ( E, ˆ δ ) ≤ N ( E, δ ), so we have ˆ N ( E, ˆ δ ) ϕ s (ˆ δ ) < N ( E, δ ) ϕ s ( δ ) < ε .2. Assume that lim sup δ → + N ( E, δ ) ϕ s ( δ ) = 0, and let ε >
0. Then there is some δ > N ( E, δ ) ϕ s ( δ ) < ε/ δ < δ . For every ˆ δ < δ , by the (left) continuity of ϕ s , thereis some δ < ˆ δ such that ϕ s (ˆ δ ) < ϕ s ( δ ). Observation 2.2 tells us that ˆ N ( E, ˆ δ ) ≤ N ( E, δ ), sowe have ˆ N ( E, ˆ δ ) ϕ s (ˆ δ ) < N ( E, δ ) ϕ s ( δ ) < ε . Proof of Lemma 2.3.
We follow the presentation by Bishop and Peres [3] of the proof for the canon-ical gauge family.1. Fix t > s >
0, and assume there is some constant c > ϕ t (2 δ ) < c · ϕ s ( δ )for all sufficiently small δ . Let F ⊆ E be a set with P ϕ s ( F ) < ∞ . It suffices to showthat lim sup δ → + N ( F, δ ) ϕ t ( δ ) < ∞ . To see this, fix δ > N p ( F, δ ) denote themaximum number of disjoint open balls of diameter δ with centers in F , and observe that N p ( F, δ ) ϕ s ( δ ) ≤ P ϕ s δ ( F ) and N ( F, δ ) ≤ N p ( F, δ ). It follows that N ( F, δ ) ϕ t (2 δ ) /c < ∞ forall sufficiently small δ , which yields the desired bound.2. Suppose that α = { α s } s ∈ (0 , ∞ ) is a precision family for ϕ , fix t > s >
0, and suppose that F ⊆ E is a set with P ϕ t ( F ) >
0. It suffices to show that lim sup δ → + N ( F, δ ) ϕ s ( δ ) >
0, andsince N p ( F, δ ) ≤ N ( F, δ ), it suffices to show that lim sup δ → + N p ( F, δ ) ϕ s ( δ ) > γ > ε > { B δ j / ( x j ) } j ∈ N of disjointballs with diameters δ j ≤ ε , centers x j ∈ F , and P j ∈ N ϕ t ( δ j ) > γ . Fix an ε > r = max { r ∈ N | α s ( r ) > ε } , and for each r ∈ N , let n r = |{ j ∈ N | α s ( r + 1) ≤ δ j < α s ( r ) }| . Then N p ( F, α s ( r + 1)) ≥ n r , so ∞ X r = r N p ( F, α s ( r + 1)) ϕ t ( α s ( r )) ≥ ∞ X r = r n r ϕ t ( α s ( r )) > X j ∈ N ϕ t ( δ j ) > γ. r ∈ N , define a r = N p ( F, α s ( r + 1)) ϕ s ( α s ( r + 1)) ≥ N p ( F, α s ( r )) ϕ s ( α s ( r )) c , where c is the implicit constant in the precision sequence condition for α s . Then ∞ X r = r a r ϕ t ( α s ( r )) ϕ s ( α s ( r )) ≥ c ∞ X r = r N p ( F, α s ( r + 1)) ϕ t ( α s ( r + 1)) > γc . If { a r } r ∈ N were bounded, then the sum would tend to 0 as r → ∞ since α is a precisionfamily. This is a contradiction, so we have lim sup r →∞ N p ( F, α s ( r + 1)) ϕ s ( α s ( r + 1)) >
0, andthe claim holds.
B Proofs from Section 3
Proof of Lemma 3.4.
Letting x = ϕ s ( δ )ln 2 and noting that x → + as δ → + , we have e ϕ s ( δ ) = 2 − /ϕ s ( δ ) = e − /x = o ( x ) = o ( ϕ s ( δ ))as δ → + . Proof of Theorem 3.7.
Let ϕ and x be as given, and let S − and S + be the sets on the right-handsides of 1 and 2, respectively.1. It suffices to show that (cid:0) dim e ϕ , ∞ (cid:1) ⊆ S − ⊆ (cid:2) dim e ϕ , ∞ (cid:1) . To verify the first inclusion, note that, by Observation 3.6, t > s > dim e ϕ ( x ) = ⇒ lim inf δ → + K δ ( x | δ ) e ϕ s ( δ ) = 0 ⇐⇒ lim inf δ → + K δ ( x | δ ) ϕ s ( δ ) − ϕ s ( δ ) = −∞ = ⇒ lim inf δ → + K δ ( x | δ ) ϕ s ( δ ) < ⇒ t ∈ S − . To verify the second inclusion, note that, by Observation 3.6, s ∈ S − ⇐⇒ lim inf δ → + K δ ( x | δ ) ϕ s ( δ ) = 0= ⇒ lim inf δ → + K δ ( x | δ ) ϕ s ( δ ) − ϕ s ( δ ) = −∞⇐⇒ lim inf δ → + log (cid:0) K δ ( x | δ ) e ϕ s ( δ ) (cid:1) = −∞⇐⇒ lim inf δ → + K δ ( x | δ ) e ϕ s ( δ ) = 0= ⇒ s ≥ dim e ϕ ( x ) .
2. It suffices to show that (cid:0)
Dim e ϕ , ∞ (cid:1) ⊆ S + ⊆ (cid:2) Dim e ϕ , ∞ (cid:1) . The proof of this is completely analogous to the proof of part 1 of the theorem.12
Proofs from Section 4
Proof of Theorem 4.1.
Let X , ϕ , and E be as given. A function f : { , } ∗ → X such that D = range( f ) is dense in X is an implicit oracle in all Kolmogorov complexities and algorithmicdimensions in this proof, but we omit f from the notation.For any s ∈ Q + , the density of D implies that H ϕ s ( E ) = 0 can be witnessed by balls withrational radii and centers in D . Hence, for every s, ε ∈ Q + with s > dim ϕ H ( E ), there exist sequences { x s,εi } i ∈ N ⊆ D and { δ s,εi } i ∈ N ⊆ Q + such that { B δ s,εi ( x s,εi ) } i ∈ N is an ε -cover of E and X i ∈ N ϕ s ( δ s,εi ) < . (C.1)Let h : N × ( Q ∩ (dim ϕ H ( E ) , ∞ )) × Q + → { , } ∗ × Q be such that h ( i, s, ε ) = ( w s,εi , δ s,εi ) , where f ( w s,εi ) = x s,εi , and let g : ( Q + ) → R be the (continuous) function g ( s, t, ε ) = inf δ ≤ ε log s ϕ s ( δ ) ϕ t ( δ ) . Let A be an oracle encoding h and g , and let s, t ∈ Q + such that dim ϕ H ( E ) < s < t . We will showthat for every x ∈ E , dim ϕ,A ( x ) ≤ t .Fix x ∈ E and ε ∈ Q + such that K A ( ε ) ≤ g ( s, t, ε ); such an ε exists because g ( s, t, · ) isunbounded and computable relative to A (cf. Section 3.3 of [20]). Let j be such that x ∈ B δ s,εj ( x s,εj ),and let δ = δ s,εj ≤ ε . Then as δ → + ,K Aδ ( x | δ ) ≤ K Aδ ( x ) + O (1) ≤ K A ( x s,εj ) + O (1) ≤ K A ( j, ε ) + O (1) . By (C.1) there are fewer than 1 /ϕ s ( δ ) values of i for which δ s,εi = δ . ThereforeK A ( j, ε ) ≤ log 1 ϕ s ( δ ) + K A ( ε ) + O (1) ≤ log 1 ϕ s ( δ ) + g ( s, t, ε ) + O (1) ≤ log 1 ϕ s ( δ ) + log s ϕ s ( δ ) ϕ t ( δ ) + O (1) . Since we can choose arbitrarily small ε in the above analysis, we have shownlim inf δ → + K Aδ ( x | δ ) ϕ t ( δ ) ≤ lim inf δ → + log ϕs ( δ ) +log q ϕs ( δ ) ϕt ( δ ) + O (1) ϕ t ( δ ) ≤ lim inf δ → + O (1) · ϕ t ( δ ) ϕ s ( δ ) · s ϕ s ( δ ) ϕ t ( δ ) ! = 0 , ϕ,A ( x ) ≤ t .For the other direction, assume that there is a precision family α = { α s } s ∈ (0 , ∞ ) for ϕ , fix anyoracle A ⊆ N , and let s, t ∈ Q be such thatsup x ∈ E dim ϕ,A ( x ) < s < t. (C.2)For all r ∈ N , let U r = (cid:26) B α s ( r ) ( f ( w )) (cid:12)(cid:12)(cid:12)(cid:12) K A ( w | α s ( r )) ≤ log 1 ϕ s ( α s ( r )) (cid:27) , and notice that |U r | ≤ ϕ s ( α s ( r )) . Now fix any r ∈ N , and let W r = ∞ [ k = r U k . For every x ∈ E , (C.2), together with the fact that ϕ ( α s ( r )) = O ( ϕ ( α s ( r + 1))) as r → ∞ , tells usthat the set n r (cid:12)(cid:12)(cid:12) K Aαs ( r ) ( x | α s ( r )) ϕ s ( α s ( r )) ≤ o = (cid:26) r (cid:12)(cid:12)(cid:12)(cid:12) K Aα s ( r ) ( x | α s ( r )) ≤ log 1 ϕ s ( α s ( r )) (cid:27) is unbounded, as is the set { k | x ∈ U k } , so x ∈ W r . Thus W r is a countable α s ( r )-cover of E with X U ∈W r ϕ t (diam( U )) = ∞ X k = r X U ∈U k ϕ t ( α s ( k )) < ∞ X k = r ϕ s ( α s ( k )) · ϕ t ( α s ( k )) . Since α is a precision family for ϕ , this sum converges. We thus have H ϕ t ( E ) ≤ r →∞ ∞ X k = r ϕ t ( α s ( k )) ϕ s ( α s ( k )) ≤ X r ∈ N ϕ t ( α s ( r )) ϕ s ( α s ( r )) < ∞ , so dim ϕ H ( E ) ≤ t . We conclude that dim ϕ H ( E ) ≤ sup x ∈ E dim ϕ,A ( x ). Proof of Theorem 4.2.
1. Assume that ϕ t (2 δ ) = O ( ϕ s ( δ )) as δ → + for all s < t , and let k ∈ N .By Lemma 2.3, there is a cover { E ki } i ∈ N for E such thatsup i ∈ N dim ϕ M ( E ki ) < dim ϕ P ( E ) + 2 − k . Take s ( k ) ∈ Q with s ( k ) > dim ϕ P ( E ) + 2 − k . Then for each i, k ∈ N , and for all sufficientlysmall δ ∈ Q + , N ( E ki , δ ) < ϕ s ( k ) ( δ ) . i, k ∈ N and δ ∈ Q + , take F ( i, k, δ ) ⊆ D with | F ( i, k, δ ) | = N ( E ki , δ )and E ki ⊆ [ d ∈ F ( i,k,δ ) B δ ( d ) . Let h : N × N × Q + → ( { , } ∗ ) N ( E ki ,δ ) be such that h ( i, k, δ ) = (cid:16) w i,k,δ , . . . , w N ( E ki ,δ ) i,k,δ (cid:17) , where F ( i, k, δ ) = n f (cid:0) w i,k,δ (cid:1) , . . . , f (cid:0) w N ( E ki ,δ ) i,k,δ (cid:1)o . Let A be an oracle encoding h . We will show that for any x ∈ E ki , Dim ϕ,A ( x ) ≤ s ( k ). For x ∈ E ki and δ ∈ Q + , there is a d ∈ F ( i, k, δ ) such that x ∈ B δ ( d ). ThereforeK Aδ ( x | δ ) ≤ log N ( E ki , δ ) + K( i, k ) . Thus for all sufficiently small δ ∈ Q + ,K Aδ ( x | δ ) ≤ log 1 ϕ s ( k ) ( δ ) + O (1) , so Dim ϕ,A ( x ) ≤ s ( k ).2. Let s, s ′ be such that sup x ∈ E Dim ϕ,A ( x ) < s ′ < s . Then for each x ∈ E and all sufficientlysmall δ ∈ Q + , K Aδ ( x | δ ) < log 1 ϕ s ′ ( δ ) . For all δ ∈ Q + , let U δ = (cid:26) B δ ( f ( w )) (cid:12)(cid:12)(cid:12)(cid:12) K A ( w | δ ) ≤ log 1 ϕ s ′ ( δ ) (cid:27) , and for each i ∈ N , let E i = (cid:26) x (cid:12)(cid:12)(cid:12)(cid:12) ∀ δ < i , x ∈ U δ (cid:27) . Then E ⊆ S i ∈ N E i . For each δ < i , N ( E i , δ ) < ϕ s ′ ( δ ) , so N ( E i , δ ) ϕ s ( δ ) = o (1), and therefore dim ϕ M ( E i ) ≤ s . Assuming that there is a precisionfamily for ϕ , the result follows whenever by Lemma 2.3.15 Proofs from Section 5
Construction D.1.
Given a sequence R ∈ { , } ω , define a sequence A , A , . . . of 2 ℓ -element sets A ℓ ⊆ { , } ℓ by the following recursion.(i) A = { λ } , where λ is the empty string.(ii) Assume that A ℓ = n u i (cid:12)(cid:12)(cid:12) ≤ i < ℓ o ⊆ { , } ℓ has been defined, where the u i are in lexicographical order. Let b u , b u , b u , b u , . . . , b u ℓ , b u ℓ be the first 2 ℓ +1 bits of R that have not been used in earlier stages of this construction. Then A ℓ +1 = { uab ua | u ∈ A ℓ and a ∈ { , } } . For each string w ∈ { , } ∗ of even length, define the closed interval I w ⊆ [0 ,
1] of length 7 −| w | / bythe following recursion.(i) I λ = [0 , I w has been defined, and divide I w into seven equal-length closed intervals, callingthese K , J , J , K , J , J , and K , from left to right. Then, for each a, b ∈ { , } , wehave I wab = J ab .For each ℓ ∈ N , let E ℓ = [ w ∈ A ℓ I w , and let E = ∞ \ ℓ =0 E ℓ . This completes the construction.Intuitively, each E ℓ in Construction D.1 consists of 2 ℓ closed intervals of length 7 − ℓ , with gapsbetween them of length at least 7 − ℓ . For each of these, one bit of R decides which of the subintervals J and J is included in E ℓ +1 , and the next bit of R decides which of the subintervals J and J is included in E ℓ +1 . The set E is a Cantor-like set chosen in this fashion. It is clear that E iscompact. Observation D.2.
For all R ∈ { , } ω , the set E of Construction D.1 has Hausdorff and packingdimensions dim H ( E ) = dim P ( E ) = log 2log 7 ≈ . . Proof.
Let R ∈ { , } ω , and let R = 0 ω . Let E be the set constructed from R as in Construc-tion D.1, and let E be the set constructed from R . Note that E is the set of all reals in [0 , g : { , } ω → { , , , } ω g ( S )[ n ] = S [ n ] + R (cid:2) n + S [ n ] (cid:3) for all S ∈ { , } ω and n ∈ N . Note that g transforms each 1 in S to a 1 or 2 in g ( S ), and g transforms each 4 in S to a 4 or 5 in g ( S ). If we identify sequences in { , , , } with the reals thatthey represent in base 7, then we now have a bijection g : E −−→ onto E. Moreover, if x, y ∈ E are distinct, and n is the first position at which x and y have different base-7digits, then | x − y | , | g ( x ) − g ( y ) | ∈ (cid:2) − n , − n (cid:3) , so g is bi-Lipschitz and hence preserves Hausdorff and packing dimensions [8]. We thus havedim H ( E ) = dim H ( E ) , dim P ( E ) = dim P ( E ) . (D.1)The set E is the self-similar fractal given by an iterated function system consisting of two contrac-tions, each with ratio . It follows by the fundamental theorem on self-similar fractals [30, 7, 8]that the Hausdorff and packing dimensions of E are both the unique solution s of the equation2 · − s = 1, i.e., that dim H ( E ) = dim P ( E ) = log 2log 7 . (D.2)The observation follows from (D.1) and (D.2). Observation D.3. If R ∈ { , } ω is Martin-L¨of random, then the set E of Construction D.1 hasalgorithmic dimensions dim( E ) = Dim( E ) = ∞ . Proof of Observation D.3 (sketch).
Let R and E be as given. By (3.1) it suffices to show that, forall sufficiently large r , K r ( E ) > r/ . For this is suffices to show that, for all sufficiently large r and all F ∈ D , ρ H ( F, E ) ≤ − r = ⇒ K( F ) > r/ . (D.3)Let r ∈ N be large, and assume that ρ H ( F, E ) ≤ − r . Let ℓ = ⌈ r/ ⌉ . Then 2 − r < / · − ℓ , sothe finite set F can be used to compute the set A l of Construction D.1. This implies that F canbe used to compute the (2 ℓ − w of R that was used to decide the set A ℓ . Since R israndom and r is large, this implies that K( F ) > r/ .In addition to illustrating the difference between classical and algorithmic dimensions, Obser-vation D.3 combines with our general point-to-set principle to give a very non-classical proof of thefollowing known classical fact. Corollary D.4. dim H ( K ([0 , P ( K ([0 , ∞ .Proof. Let A ⊆ N . By Theorem 4.1 applied to K ([0 , E ∈ K ([0 , A ( E ) = ∞ . If we choose R ∈ { , } ω to be Martin-L¨of random relative to A , thenObservation D.3, relativized to A , tells us that Construction D.1 gives us just such a point.17 roof of Theorem 5.1. Let E ⊆ X and ϕ be a gauge family. Let δ > F ⊆ X be such that | F | = N ( E, δ ) and E ⊆ [ x ∈ F B δ ( x ) . For every L ∈ K ( E ), we have ρ H ( L, { x ∈ F | B δ ( x ) ∩ L = ∅} ) < δ , so K ( E ) ⊆ [ T ⊆ F B δ ( T ) , and therefore N ( K ( E ) , δ ) ≤ | F | = 2 N ( E,δ ) .Now suppose that lim inf δ → + N ( E, δ ) ϕ s ( δ ) = 0. Then since ϕ s ( δ ) → + as δ → + , we havelim inf δ → + N ( K ( E ) , δ ) e ϕ s ( δ ) ≤ lim inf δ → + N ( E,δ ) − /ϕ s ( δ ) = lim inf δ → + N ( E,δ ) ϕs ( δ ) − ϕs ( δ ) = 0 , so dim e ϕ M ( K ( E )) ≤ dim ϕ M ( E ).We now show that dim e ϕ M ( K ( E )) ≥ dim ϕ M ( E ). Let δ >
0, let P be a set of 2 δ -separated pointsin E , and observe that | P | ≤ N ( E, δ ). Let
F ⊆ K ( E ) satisfy |F | = N ( K ( E ) , δ ) and K ( E ) ⊆ [ F ∈F B δ ( F ) . For every distinct pair
S, S ′ ⊆ P , we have ρ H ( S, S ′ ) ≥ δ , so, for each F ∈ F , the ball B δ ( F ) cancontain at most one subset of P . Hence, N ( K ( E ) , δ ) = |F | ≥ | P | ≥ N ( E,δ ) . Now let t > s > dim e ϕ M ( K ( E )). Thenlim inf δ → + N ( K ( E ) , δ ) e ϕ s ( δ ) = 0 = ⇒ lim inf δ → + N ( E,δ ) − /ϕ s ( δ ) = 0 ⇐⇒ lim inf δ → + (cid:18) N ( E, δ ) − ϕ s ( δ ) (cid:19) = −∞⇐⇒ lim inf δ → + N ( E, δ ) ϕ s ( δ ) − ϕ s ( δ ) = −∞ = ⇒ lim inf δ → + N ( E, δ ) ϕ s ( δ ) < ⇒ lim inf δ → + N ( E, δ ) ϕ t ( δ ) = 0= ⇒ dim ϕ M ( E ) ≤ t. The argument for upper Minkowski dimension is completely analogous.
Observation D.5.
Let ϕ be any gauge family, X any metric space, E ⊆ X , and δ > .1. If dim ϕ M ( E ) < ∞ , then there exists a point x ∈ E such that dim ϕ M ( E ∩ B δ ( x )) = dim ϕ M ( E ) .2. If dim ϕ M ( E ) < ∞ , then there exists a point x ∈ E such that dim ϕ M ( E ∩ B δ ( x )) = dim ϕ M ( E ) . roof. It follows from the monotonicity of Minkowski dimensions thatdim ϕ M ( E ∩ B δ ( x )) ≤ dim ϕ M ( E )holds for every x ∈ X .If dim ϕ M ( E ) < ∞ , then N ( E, δ/
2) is finite; let the set { x , . . . , x N ( E,δ/ } ⊆ X testify to thisvalue. Then every E ∩ B δ/ ( x i ) is nonempty, and E ⊆ N ( E,δ/ [ i =1 ( E ∩ B δ/ ( x i )) , so by the finite stability of dim ϕ M , there is some 1 ≤ i ≤ N ( E, δ/
2) such thatdim ϕ M ( E ) ≤ dim ϕ M ( E ∩ B δ/ ( x i )) . Now let x ∈ E ∩ B δ/ ( x i ) and observe that E ∩ B δ/ ( x i ) ⊆ E ∩ B δ ( x ), so monotonicity givesdim ϕ M ( E ∩ B δ/ ( x i )) ≤ dim ϕ M ( E ∩ B δ ( x )) . This proves the first statement, and the proof of the second statement is completely analogous.
Lemma D.6.
Let ϕ be any gauge family, X any metric space, and E ⊆ X a compact set.1. There exists a point x ∈ E such that dim ϕ M ( E ∩ B δ ( x )) = dim ϕ M ( E ) holds for all δ > .2. There exists a point x ∈ E such that dim ϕ M ( E ∩ B δ ( x )) = dim ϕ M ( E ) holds for all δ > .Proof. By the compactness of E , N ( E, δ ) is finite for every δ >
0, and the Minkowski dimensionsof E are finite as well. Hence, Observation D.5 yields a sequence { x r } r ∈ N of points in X such thatdim ϕ M ( E ∩ B − r ( x r )) = dim ϕ M ( E )for all r ∈ N . Since E is compact, there is a subsequence { x r i } i ∈ N of { x r } r ∈ N that converges tosome point x ∈ E . Thus, for all δ >
0, there is an i ∈ N such that ρ ( x r i , x ) < − r i < δ/
2, so B − ri ( x r i ) ⊆ B − ri ( x ) ⊆ B δ ( x ) . By the monotonicity of Minkowski dimensions, then,dim ϕ M ( E ) = dim ϕ M ( E ∩ B − ri ( x r i )) ≤ dim ϕ M ( E ∩ B δ ( x )) ≤ dim ϕ M ( E ) . The proof of the second statement is completely analogous.
Proof of Theorem 5.2.
Lemma 2.3 immediately gives dim e ϕ P ( K ( E )) ≤ dim e ϕ M ( K ( E )).For the other direction, apply the general point-to-set principle for packing dimension (Theo-rem 4.2) to let A be an oracle such thatdim e ϕ P ( K ( E )) ≥ sup L ∈K ( E ) Dim e ϕ,A ( L ) , (D.4)and let t = dim ϕ M ( E ). Applying Lemma D.6, let x ∈ E be a point such that for all δ > ϕ M ( E ∩ B δ ( x )) = t.
19y the hyperspace Minkowski dimension theorem, then, we also havedim e ϕ M ( K ( E ∩ B δ ( x ))) = t (D.5)for all δ > s < t . We will recursively define a compact set L ∈ K ( E ) such thatDim e ϕ,A ( L ) > s. Let A be an oracle that encodes both A and x . By (D.5) and Observation 2.2,lim sup δ → + ˆ N ( K ( E ∩ B ( x )) , δ ) e ϕ s ( δ ) = ∞ . Thus there is a precision δ ∈ Q + such thatˆ N ( K ( E ∩ B ( x )) , δ ) e ϕ s ( δ ) > . That is, it requires at least 1 / e ϕ s ( δ ) open balls of radius δ (in the ρ H metric), with centers thatare finite subsets of D , to cover K ( E ∩ B ( x )). By the pigeonhole principle, the number of finitesets J ⊆ D satisfying K A ( J | δ ) ≤ log (cid:18) e ϕ s ( δ ) (cid:19) is at most 2 log (cid:16) e ϕs ( δ (cid:17) +1 − < e ϕ s ( δ ) . Hence there is some compact set L ∈ K ( B ( x ) ∩ E ) withK A δ ( L | δ ) ϕ s ( δ ) > log (cid:18) e ϕ s ( δ ) (cid:19) ϕ s ( δ )= 1 − ϕ s ( δ ) > / . Define the compact set L ′ = ( L \ B δ ( x )) ∪ { x } , and notice that ρ H ( L ∪ { x } , L ′ ) ≤ δ , soK A δ ( L | δ ) ≤ K A δ ( L , x | δ ) + O (1) ≤ K A δ ( L ′ | δ ) + O (1) , since A encodes x . Thus, as long as δ is sufficiently small, we haveK A δ ( L ′ | δ ) ϕ s ( δ ) ≥ / . Now, for each i ≥
1, let A i be an oracle encoding A i − and δ i − . Let δ i ∈ Q + and L ′ i ∈ K (cid:0) B δ i − / ( x ) ∩ E (cid:1) be such that L ′ i ∩ B δ i ( x ) = { x } and K A i δ i ( L ′ i | δ i ) ϕ s ( δ i ) ≥ / . (D.6)20his pair exists for exactly the same reason that δ and L ′ exist.Define the set L = [ i ∈ N L ′ i . Notice first that this set belongs to K ( E ). Consider any sequence { x r } r ∈ N of points in L . If thesequence is contained within { x } ∪ S ni =0 L ′ i for some finite n —i.e., within a finite union of compactsets, which is compact—then it has a convergent subsequence that converges to a point in thatunion. Otherwise, the sequence has points in infinitely many of the L ′ i , and there is a subsequence { x r j } j ∈ N such that, for every pair j ′ > j there exists a pair i ′ > i such that x r j ∈ L ′ i \ { x } and x r j ′ ∈ L ′ i ′ \ { x } ; such a subsequence converges to x . Thus L is sequentially compact and thereforecompact.Let D be a countable dense set in X , and recall that U is a fixed universal oracle prefix Turingmachine. Consider an oracle prefix Turing machine M , with access to an oracle for x and δ i . Oninput π such that U ( π ) = F ⊆ D , M outputs the set { y ∈ F | ρ ( x, y ) < δ i / } . Now let π testify to K A i δ i ( L ). Then M ( π ) ⊆ D is a set of points satisfying ρ H ( L ′ i , U ( π )) < δ i , so wehave K A i δ i ( L ′ i | δ i ) ≤ | π | + c M = K A i δ i ( L | δ i ) + c M (D.7)where c M is an optimality constant for the machine M . Furthermore,K A i δ i ( L | δ i ) ≤ K Aδ i ( L | δ i ) + O (1) . Combining this fact with (D.6) and (D.7) yieldsK Aδ i ( L | δ i ) ϕ s ( δ i ) ≥ / − ϕ s ( δ i ) · O (1) . The latter term vanishes as i → ∞ , solim sup δ → + K Aδ ( L ) ϕ s ( δ ) ≥ / . By Theorem 3.7, this implies that Dim e ϕ,A ( L ) > s . We conclude that Dim e ϕ,A ( L ) ≥ t , so by (D.4),the proof is complete. References [1] Krishna B. Athreya, John M. Hitchcock, Jack H. Lutz, and Elvira Mayordomo. Effectivestrong dimension in algorithmic information and computational complexity.
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