1 Introduction
Algorithmic fractal dimensions, which quantify the density of algorithmic information in individual points [17, 1, 21], have recently been used to prove new theorems [25, 23, 26, 24, 20]
about their classical forerunners, the Hausdorff and packing dimensions of sets. Since algorithmic fractal dimensions are products of the theory of computing, and since the aforementioned new theorems are entirely classical (not involving logic or the theory of computing), these developments call for a more thorough investigation of the relationships between algorithmic and classical fractal dimensions. One significant facet of this investigation, initiated by Orponen
[31], is to look for purely classical proofs of these new classical theorems.In this paper, taking a different approach, we establish direct connections between algorithmic and classical fractal dimensions. Aside from the presence versus absence of algorithms, the most striking difference between algorithmic fractal dimensions and classical fractal dimensions is that the algorithmic dimensions are usefully defined for individual points in Euclidean space, while the classical Hausdorff and packing dimensions vanish on individual points. To bridge this gap, we examine the classical local dimensions (also called pointwise dimensions) of outer measures at individual points in Euclidean spaces [10]. These local fractal dimensions have been studied at least since the 1930s and are essential tools in multifractal analysis [11, 9]. Outer measures and the algorithmic and local dimensions are defined precisely in Section 2 below.
Outer measures, introduced by Carathéodory [4] in the “prehistory” of Hausdorff dimension [12] (defining what later became known as the 1dimensional Hausdorff measure), are now best known for their role in Carathéodory’s program [5] to generalize Lebesgue measure to a wide variety of settings [35]. However, it is the role of outer measures in local fractal dimensions that are of interest here.
The second author observed [22] that a particular, very nonclassical outer measure , defined in terms of Kolmogorov complexity, has the property that the classical local fractal dimensions of coincide exactly with the algorithmic fractal dimensions at every point in . This property of is analogous to Levin’s coding theorem [14, 15], which pertains to a particular, very nonclassical subprobability measure on strings. Levin’s theorem says that if we substitute
for the probability measure
in the classical Shannon selfinformation [33] , then the resulting quantity is essentially the prefix Kolmogorov complexity (i.e., the algorithmic information content) of the string .Levin defined as an optimal lower semicomputable subprobability measure, so the above analogy leads us to investigate here the algorithmic optimality properties of and other outer measures on Euclidean spaces.
We first investigate outer measures that are globally optimal, a property that is closely analogous to the optimality property of Levin’s . In Section 3 we prove that globally optimal outer measures on exist.
As it turns out, the outer measure is not globally optimal. In Section 4 we prove this fact, and we introduce and investigate the more general and more subtly defined class of locally optimal outer measures on . Our main theorem establishes that every locally optimal outer measure on a Euclidean space has the property that the classical local fractal dimensions of coincide exactly with the algorithmic dimensions at every point in .
In Section 5 we discuss implications of our results, especially for the pointtoset principles that have enabled the new classical results mentioned in the first paragraph of this introduction.
2 Algorithmic and Local Fractal Dimensions
This section reviews the algorithmic fractal dimensions and the classical local fractal dimensions.
Following standard practice [30, 8, 16]
, we fix a universal prefix Turing machine
and define the (prefix) Kolmogorov complexity of a string to bei.e., the minimum number of bits required to cause to output . By standard binary encodings, we extend this from to other countable domains. In particular, the Kolmogorov complexity of a rational point is well defined.
The Kolmogorov complexity of a point at a precision is
where is the Euclidean distance from to .
We now define the algorithmic fractal dimensions of points in .
Definition ([17, 28, 1]).
Let .

The algorithmic dimension of is
(2.1) 
The strong algorithmic dimension of is
(2.2)
The classical local fractal dimensions are local properties of outer measures. An outer measure on a set [35] is a function (where is the power set of ) with the following three properties.

[(i)]

(vanishes on empty set) .

(monotonicity) For all ,

(countable subadditivity) For all ,
An outer measure is finite if .
Definition ([10]).
If is a finite outer measure on , then the lower and upper local (or pointwise) dimensions of at a point are
(2.3) 
and
(2.4) 
respectively. (The logarithms here are base2, and is the open ball of radius about in .)
3 Global Algorithmic Optimality
The optimality notions that we discuss in this paper concern outer measures with three special properties that we now define.
Definition.
An outer measure on is finitely supported on if, for every , there is a finite set such that .
Note that an outer measure on that is finitely supported on is supported on in the usual sense that . The following example shows that the converse does not hold.
Example 3.1.
The function defined by
is an outer measure on that is supported, but not finitely supported, on .
Definition.
An outer measure on is strongly finite if is supported on and
It is clear that every strongly finite outer measure is finite. The outer measure of Example 3.1 shows that the converse does not hold.
Definition.
An outer measure on is lower semicomputable if it is finitely supported on and there is a computable function
(where is the finite power set of , i.e, the set of all finite subsets of ) with the following two properties.

[(i)]

For all and ,

For all ,
Lemma 3.2.
Let be a lower semicomputable outer measure on . If is a function testifying to the lower semicomputability of , then, for all ,
meaning that, for all , there exist and such that, for all and ,
Definition.
An outer measure on is globally optimal if the following conditions hold.

[(i)]

is strongly finite and lower semicomputable.

For every strongly finite, lower semicomputable outer measure on , there is a constant such that, for all ,
In proving that globally optimal outer measures exist, we will use a computable enumeration of all strongly finite, lower semicomputable outer measures that take values in . Our main technical lemma, which we now state, shows that such an enumeration exists.
Lemma 3.3.
Let be the set of all strongly finite, lower semicomputable outer measures . There is a computable function
such that, if we write for all and , and if we define by
for all , then .
Proof.
Let be an enumeration of all prefix Turing machines that take inputs in , give outputs in , and satisfy for all . For each , we define the following functions.
is given by
is given by
is given by
Now fix . It is immediate from our construction that takes values in and is finitely supported on , and that is computable. To prove that , then, it remains to show that satisfies conditions (i) and (ii) from the definition of lower semicomputable outer measures, and that is countably subadditive.
For condition (i), let and with . Then , and for all , , so .
For condition (ii), let with , let , and let be such that . Then
so
To prove that is countably subadditive, we first show that, for all , is finitely subadditive on . Fix , let , and let be such that and
Suppose that, for some , we have . Then there are sets such that
which contradicts the minimality of our choice of . Hence, for all ,
so
Now let be such that . Let , and let and be such that
For each , let . Since is a finite set, there is some such that . By Lemma 3.2 and the finite subadditivity of ,
Letting , we have
and we conclude that , so .
For the converse, let , and let be a witness to the lower semicomputability of . Then is computable and for all , so there is some such that, for all , we have .
We now show that for all ,
Let and . Then
Now let , let be such that, for all ,
and let be such that . Then for each ,
It follows that
by the countable subadditivity of .
We have shown that for every , every , and every sufficiently large , and there exists some such that . This implies that for all ,
and therefore for all ,
We conclude that , so . ∎
Theorem 3.4.
Globally optimal outer measures exist.
Proof.
Define the strongly finite outer measure by
where is defined as in Lemma 3.3. This outer measure is supported on , and the function given by
is a witness to the lower semicomputability of .
Let be any lower semicomputable outer measure that is strongly finite and supported on , and let
Then the function given by
belongs to . By Lemma 3.3, there is some such that, for all ,
We have, for all ,
so is globally optimal. ∎
4 Local Algorithmic Optimality
This paper’s investigation of algorithmic optimality is primarily driven by a specific outer measure . To define , we first define the Kolmogorov complexity of a set to be
That is, is the minimum number of bits required to cause the universal prefix Turing machine to print some rational point in . (Shen and Vereschagin [34] introduced a similar notion for a different purpose.)
Definition ([22]).
Define the function by
for all .
Observation 4.1 ([22]).
is an outer measure on .
Our primary interest in is the following connection between classical local fractal dimensions and algorithmic fractal dimensions.
Observation 4.2 ([22]).
For all ,
and
We next investigate the algorithmic properties of the outer measure .
Observation 4.3.
is strongly finite and lower semicomputable.
Proof.
It suffices to show three things.

is finitely supported on . For this, let . Let
Then is a finite subset of , and
so .

. For this, just note that
by the Kraft inequality for prefix Kolmogorov complexity.

is lower semicomputable. This follows immediately from the well known upper semicomputability of the Kolmogorov complexity function.
∎
Lemma 4.4.
is not globally optimal.
Proof.
Define the function by
We now show that is a strongly finite, lower semicomputable outer measure on . It is clear that is an outer measure on . It thus suffices to prove that has the properties 1, 2, and 3 proven for in the proof of Observation 4.3. For properties 2 and 3, the proofs for are identical to those for . For property 1, that is finitely supported on , let . By the Kraft inequality for prefix Kolmogorov complexity,
so there is a finite set such that
Hence, to prove the lemma, it suffices to exhibit a set such that, for all ,
(4.1) 
Let . Let be a constant such that, for all ,
Let be some parameter, let , and define the set
Then , and
There are fewer than rational points with . For all such that ,
so there are at least rational points with , and we have . Thus,
Choosing such that
yields (4.1). ∎
Notwithstanding Lemma 4.4, does have an optimality property, which we next define. For each , let
be the unit cube at . For each such and each , let
be the dyadic cube with address . Note that each is “halfclosed, halfopen” in such a way that, for each , the family
is a partition of .
Definition.
Let and be outer measures on , and let be a sequence of families of subsets of . We say that dominates on if there is a function such that as and, for every and every set ,
We say that dominates on dyadic cubes if dominates on . We say that dominates on balls if dominates on , where is the set of all open balls of radius in .
Definition.
An outer measure on is locally optimal if the following two conditions hold.

[(i)]

is strongly finite and lower semicomputable.

For every strongly finite, lower semicomputable outer measure on , dominates on dyadic cubes.
Theorem 4.5.
The outer measure is locally optimal.
Proof.
We rely on machinery created for a different purpose by Case and the first author [6]. Just as we have “lifted” Kolmogorov complexity from to via routine encoding, we lift Levin’s optimal lower semicomputable subprobability measure [14, 15] from to so that, for all ,
We also set
for all . The LDS coding theorem of [6] is a mild generalization of Levin’s coding theorem [14, 15] that tells us that there is a constant such that, for all and ,
(4.2) 
To prove the present theorem, it suffices by Observation 4.3 to prove that satisfies condition 2 of the definition of local optimality. For this, let be a strongly finite, lower semicomputable outer measure on . Define by for all . Then is lower semicomputable and , so the optimality property of tells us that there is a constant such that, for all ,
(4.3) 
Corollary 4.6.
A strongly finite, lower semicomputable outer measure on is locally optimal if and only if it dominates on dyadic cubes.
Lemma 4.7.
There is a constant such that, for every , every dyadic cube , and every open ball of radius that intersects ,
Proof.
Lemma 3.5 of [6] gives us a constant such that, for all , , and as in the present lemma,
Hence it suffices to show that there is a constant such that, for all , , and as in the present lemma,
Let be a prefix Turing machine that, on input where , and , and , outputs the lexicographically ^{th} point in the product set
Let , , and be as in the present lemma. Let be such that <
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