The Fractal Dimension of Product Sets
aa r X i v : . [ m a t h . GN ] F e b T HE F R AC TAL D IMENSION OF P RODUCT S ETS
A P
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Clayton Moore Williams
Brigham Young University [email protected]
Machiel van Frankenhuijsen
Utah Valley University [email protected]
February 26, 2021 A BSTRACT
Using methods from nonstandard analysis, we define a nonstandard Minkowski dimension whichexists for all bounded sets and which has the property that dim( A × B ) = dim( A ) + dim( B ) . Thatis, our new dimension is “product-summable”. To illustrate our theorem we generalize an exampleof Falconer’s to show that the standard upper Minkowski dimension, as well as the Hausdorff di-mension, are not product-summable. We also include a method for creating sets of arbitrary rationaldimension. Introduction
There are several notions of dimension used in fractal geometry, which coincide for many sets but have important,distinct properties. Indeed, determining the most proper notion of dimension has been a major problem in geometricmeasure theory from its inception, and debate over what it means for a set to be fractal has often reduced to debate overthe proper notion of dimension. A classical example is the “Devil’s Staircase” set, which is intuitively fractal but whichhas integer Hausdorff dimension . As Falconer notes in The Geometry of Fractal Sets , the Hausdorff dimension is“undoubtedly, the most widely investigated and most widely used” notion of dimension. It can, however, be difficultto compute or even bound (from below) for many sets. For this reason one might want to work with the Minkowskidimension, for which one can often obtain explicit formulas. Moreover, the Minkowski dimension is computed usingfinite covers and hence has a series of useful identities which can be used in analysis. In particular, the Minkowskidimension of a product set is the sum of the dimensions of the factoring sets. We define this notion below for clarity. Definition 0.1.
Let f be a function defined for all bounded subsets of R k for all k. Then f is product-summable if f ( A ) + f ( B ) = f ( A × B ) , where A × B ⊂ R m + n is the Cartesian product of A and B, for all such bounded subsets A ⊂ R m and B ⊂ R n .Product-summability is a desirable property for any notion of dimension, by analogy with boxes in R n and their prod-ucts and because it agrees so strongly with human intuition. The Hausdorff dimension is not product-summable , noris the upper Minkowski dimension. The Minkowski dimension is product-summable (as is shown), but is unfortunatelydefined in terms of a limit which does not exist for all sets. In this paper we introduce a nonstandard Minkowski dimen-sion which exists for all bounded sets, agreeing with the standard Minkowski dimension whenever it exists. Moreover,this nonstandard Minkowski dimension is product summable. We make the argument that the nonstandard Minkowskidimension is a more appropriate tool of analysis for many applications in geometric measure theory. This nonstandard [1, page 73] Discussion in [2, page 82], [3, page 337]. [1, pages ix-x]. [4]. PREPRINT - F
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Minkowski dimension is introduced at the cost of ultrafilters and nonstandard analysis, however, we introduce theseideas in an intuitive and conceptually transparent way based in large part on Terence Tao’s pedagogical discussion in
Ultrafilters, Nonstandard Analysis, and Epsilon Management . In Fractal Geometry, Complex Dimensions, and Zeta Functions [3], Lapidus and van Frankenhuijsen give convincingevidence for the case that fractality ought to be defined in terms of the Minkowski dimension. We add to theirargument, which is grounded in surprising connections between zeta functions and the complex dimensions of fractalsets , by introducing and summarizing useful properties of the standard and nonstandard Minkowski dimensions fromthe perspective of geometric measure theory. Because the Minkowski dimension, when it exists, is equal to thenonstandard Minkowski dimension, we make the argument that the nonstandard Minkowski dimension is a naturalnotion of dimension.We begin by reviewing the standard Minkowski dimension and its identities, and proving the product-summability ofthe standard Minkowski dimension. Afterwards we briefly review the Hausdorff dimension. We then introduce ultrafil-ters and nonstandard limits in section 3, developing only the tools we will use, and define the nonstandard Minkowskidimension. Our main result, theorem 3.10, establishes the product-summability of the nonstandard Minkowski dimen-sion follows from its definition and a covering lemma, lemma 2.3. In section 4 we provide an interesting generalizationof an example of Falconer’s for which the Hausdorff dimension of the product is not the sum of the dimensions of thefactoring sets. The Minkowski dimension does not exist for these sets, which also provide an example for when theupper Minkowski dimension is not product-summable, however, the nonstandard Minkowski dimension does exist forthese sets. As a corollary to one of the identities of the Minkowski dimension we also introduce a means of generatingsets of arbitrary rational dimension. We conclude by elaborating on product-summability and providing a list of usefulproperties one would desire of any notion of dimension. To proceed we define the upper Minkowski dimension, which is computed using sets of fixed size. The best way tointroduce the upper Minkowski dimension is via the covering number N ( ǫ ) . Definition 1.1.
Let A be a bounded subset of R n and ǫ > . Then the covering number N A ( ǫ ) is the minimum numberof balls of radius ǫ in R n needed to cover A. We can now define the Minkowski dimension, as well as the covering dimension.
Definition 1.2.
Let A ⊂ R n be bounded, ǫ > , and N ( ǫ ) be its covering number. The upper Minkowski dimensionof A, dim M ∗ ( A ) , is then dim M ∗ ( A ) = lim sup ǫ → + log N ( ǫ )log ǫ . (1)If the limit exists then the standard Minkowski dimension is dim M ( A ) = lim ǫ → + log N ( ǫ )log ǫ , and equals the upperMinkowski dimension. The limit supremum is chosen to define the standard Minkowski dimension here becauseof its identities, as will be derived.In practice it is sometimes convenient, once one has found N A ( ǫ ) , to recognize the growth of N A ( ǫ ) is of particularorder in /ǫ. As will be shown, this observation allows one to compute the upper Minkowski dimension. We makethis notion precise below.
Definition 1.3.
Let A ⊂ R n be bounded and N ( ǫ ) be its covering number. Define dim C ( A ) to bedim C ( A ) = inf (cid:26) α | N ( ǫ ) = O (cid:18) ǫ α (cid:19) as ǫ → (cid:27) . (2)Then dim C ( A ) is the covering dimension . [5]. Their argument is that subsets L of R with a sequence of lengths { l i } should be defined as fractal if the geometric zeta functionof L , ζ L ( s ) = P i l si , has a nonreal pole with positive real part. [1, page 73]. PREPRINT - F
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We now prove that, for A ⊂ R n a bounded set, dim C ( A ) = dim M ∗ ( A ) . Theorem 1.4.
Let A ⊂ R be bounded. Let β = dim C ( A ) be the covering Minkowski dimension and dim M ∗ ( A ) = β ∗ be the upper Minkowski dimension. Then β = β ∗ . Proof.
This is most clearly demonstrated by use of the logical forms of the definitions. We will write definitions 1.2and 1.3 with regards to the relevant sets to make this clear.1. Note that inf (cid:8) α | N ( ǫ ) = O (cid:0) ǫ α (cid:1) as ǫ → (cid:9) = β if and only if, for all α > β, there exist a k > and a δ > such that for all < ǫ < δ we have N ( ǫ ) < k (1 /ǫ ) α ; and, for all α < β , for all k, δ > there existsan ǫ < δ such that N ( ǫ ) > k (1 /ǫ ) α .
2. Note also that lim sup ǫ → + log /ǫ N ( ǫ ) = β ∗ if and only if, for all α > β ∗ , there exists a δ > such that N ( ǫ ) < (1 /ǫ ) α for all ǫ < δ ; and, for all α < β ∗ , for all δ, there exists an ǫ < δ such that N ( ǫ ) > (1 /ǫ ) α . It is important to recognize that if 2 is true for β ∗ , then 1 is also true for β ∗ with k = 1 . All that remains in proving β = β ∗ is to show that 1 implies 2.We first show that, given a β as in 1, then for all α > β there exists a δ > such that N A ( ǫ ) < (1 /ǫ ) α . Suppose thatfor all α > β there exist k, δ > and an ǫ < δ such that N A ( ǫ ) < k (1 /ǫ ) α . Let α > β and consider α ′ = α − α − β . Let k, δ be such that N ( ǫ ) < k (1 /ǫ ) α ′ for all ǫ < δ. Then N A ( ǫ ) < ǫ α ǫ α − β . Let δ ′ = min n δ, (cid:0) k (cid:1) α − β o . Then for ǫ < δ ′ we have N A ( ǫ ) < ǫ α , and we are done.It remains to show that β is the greatest value such that for all α < β and for all δ > there exists an ǫ < δ such that N ( ǫ ) > (1 /ǫ ) α . This establishes β as the limit superior in equation 1 and concludes our demonstration of the equalityof β and β ∗ . But this property follows naturally from point 1 above when k = 1 . Then dim C ( A ) = dim M ∗ ( A ) . s -dimensional Outer Content It would be desirable to define an outer measure allowing us to directly relate the upper Minkowski dimension tothe Hausdorff dimension through properties of the covering. This is because the Hausdorff dimension, as stated indefinition 2.6, is defined in terms of an outer measure. While the analogous notion for the Minkowski dimensionis not a measure, it does still allows us to compare the Hausdorff and Minkowski dimensions. We will define theMinkowski outer content and relate the upper Minkowski dimension to it via equation 3. In what follows, let | A | =max {| x − y | | x, y ∈ A } , that is, let | A | be the diameter of a set A . Definition 1.5.
Let A ⊂ R n be bounded and { U i } be a minimal covering of A by balls of radius ǫ , that is, by N ( ǫ ) balls. Let s > and M sǫ ( A ) be M sǫ ( A ) = X i | U i | s = N A ( ǫ )(2 ǫ ) s , (3)where | U i | = 2 ǫ for all i . Then the Minkowski s -dimensional outer content of A is M s ( A ) = lim sup ǫ → M sǫ ( A ) . (4) We use the terminology content rather than measure for M s because it is not an outer measure. To see this, note if m ∗ ,J ( A ) = inf B ⊃ A,B = ·∪ ni =1 B i ,B i boxes m ( B ) , m ( B ) the elementary measure,is the outer Jordan content then M ( A ) = m ∗ ,J ( A ) . Since m ∗ ,J is not countably additive but only finitely additive M s is not anouter measure. See for example [6, page 149]. PREPRINT - F
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We can now relate M s to dim M ∗ . Theorem 1.6.
Let A ⊂ R n be bounded. Then dim M ∗ ( A ) = inf { s | M s ( A ) = 0 } . Proof.
Let { U i } be a minimal cover of A by ǫ -balls, and let β = dim M ∗ ( A ) be its upper Minkowski dimension. Then M sǫ ( A ) = N A ( ǫ )(2 ǫ ) s = O (cid:0) ǫ s − β (cid:1) . Hence there exists a k > such that M sǫ ( A ) ≤ kǫ s − β , for all ǫ within anappropriately small neighborhood of the origin. Then for s > β M s ( A ) = lim sup ǫ → + M sǫ ( A ) < lim sup ǫ → + kǫ s − β = 0 . Then dim M ∗ ( A ) ≤ inf { s | M ( A ) = 0 } . We now show that dim M ∗ ( A ) ≥ inf { s | M ( A ) = 0 } , establishing the desired equality. Let s < β , and s < α < β. Refer to point 2 of theorem 1.4 above. From this we see that for all k > , for all δ > , there exists an ǫ < δ suchthat N ( ǫ ) > kǫ α . Then k ǫ α ǫ s = kǫ s − α < N A ( ǫ ) ǫ s = M sǫ ( A ) . Since s − α < , we have M s ( A ) = lim sup ǫ → + kǫ s − α = ∞ 6 = 0 , and we are done. In conclusion, we have identified the upper Minkowski dimension and equated it to the covering dimension. Indeed,we have shown that dim M ∗ ( A ) = dim C ( A ) = inf { s | M s ( A ) = 0 } . (5)By doing so we have generated useful identities which can be used to compute the Minkowski dimension of a product,as well as relate the upper Minkowski dimension to other notions of dimension. The key feature of the Minkowskidimension is that it is computed using functions dependent on sets of fixed size. This is in contrast to the Hausdorffdimension, defined in terms of an infimum over all covers by sets with size bounded above (see definition 2.6). We now have the tools necessary to prove the product-summability of the Minkowski dimension. In this section weshow the standard Minkowski dimension of a Cartesian product is equal to the sum of the Minkowski dimensions ofthe factors, when the dimensions exist. Hence the Minkowski dimension is product-summable. In this section we alsouse the product-summability of the Minkowski dimension to generate subsets of R n of dimension within δ > of anyvalue less than or equal to k. While coverings by balls are conceptually simple, coverings by regular n -cubes are often more convenient computa-tionally (particularly when considering products, since the product of cubes is a higher dimensional cube). Fortunatelyit does not matter whether one defines the Minkowski dimension using squares or balls, as shown. Lemma 2.1.
Let A ⊂ R n be bounded. Then one can compute the Minkowski s -dimensional outer measure usingeither cubes or balls.Proof. Let L A ( ǫ ) be the minimal number of cubes in R n of side length ǫ necessary to cover A. Define L sǫ ( A ) = L A ( ǫ ) ǫ s and L s ( A ) = lim sup ǫ → + L sǫ ( A ) . Note each cube in R n of side length ǫ is contained in a ball of radius √ n ǫ and contains a ball of radius ǫ . Hence N A (cid:0) ǫ (cid:1) ≤ L A ( ǫ ) ≤ N A (cid:16) √ n ǫ (cid:17) . Then M s ǫ ( A ) ≤ L sǫ ( A ) ≤ M s √ n ǫ ( A ) , PREPRINT - F
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26, 2021 and taking the limit supremum yields M s ( A ) = L s ( A ) . We can go further and restrict our attention to dyadic cubes of fixed side length and position. Doing so gives us greatcontrol over the cover without sacrificing anything in terms of the behavior of the fractal sets we are able to observe,introducing only a constant into our computations. Lemma 2.2.
Let A ⊂ R k be bounded, ǫ > , and n ∈ Z be such that − ( n +1) < ǫ ≤ − n . Let N ( ǫ ) be the set ofdyadic cubes of side length − n in R k , that is, N ( ǫ ) = { [ m , ( m + 1)2 − n ] × ... × [ m k , ( m k + 1)2 − n ] | m i ∈ Z } . Let S A ( ǫ ) be the minimal number of sets in N ( ǫ ) needed to cover A. Then N A ( ǫ ) ≤ S A ( ǫ ) ≤ n N A ( ǫ ) , N A ( ǫ ) is theminimal number of cubes of side length ǫ needed to cover A, analogous to definition 1.1.Proof. First note if ǫ < ǫ ′ then N A ( ǫ ) ≥ N A ( ǫ ′ ) , as shown by placing the center of an ǫ -cube at the center of eachcube in a minimal ǫ -cover of A. This is a cover of A using N A ( ǫ ) cubes of side length ǫ ′ . Now S A ( ǫ ) ≥ N A (2 − n ) , since it is the result of a more restrictive type of cover than N A (2 − n ) is. Let n, ǫ be as in the hypothesis. Then since ǫ ≤ − n , and N A ( ǫ ) ≥ N A (2 − n ) ≥ S A ( ǫ ) . It requires only 2 intervals of length − n +1 to cover an interval of length − n , hence it takes no more than k cubesof side length ǫ to cover a cube of side length − n . Therefore S A ( ǫ ) ≤ n N A ( ǫ ) . Hence the Minkowski dimension can be computed using covers by dyadic cubes. Dyadic cubes have the particularlyuseful net property, in which 2 dyadic cubes are either disjoint or one is a subset of the other. Because we’re restrictingour attention to cubes of constant size the dyadic cubes in N ( ǫ ) are either disjoint or equal, forming a regular tiling of R k . Other approaches to geometric measure theory, such as the Hausdorff measure, take the infimum over all coverswith diameter bounded above by ǫ . The Hausdorff s -dimensional outer measure can also be approximated by dyadiccubes, the so-called net measure, meaning the Hausdorff dimension can be computed using covers by dyadic cubes. From this we again see a distinction between what may be called the Minkowski approach to geometric measure theoryand the Hausdorff approach. We will see further distinctions between these approaches later.The net property of dyadic cubes allows us to prove that when the standard Minkowski dimension exists it is productsummable. To do so we will need a lemma on covers of a product set.
Lemma 2.3.
Let S U ( ǫ ) be the covering number of a bounded subset U as in lemma 2.2. Then for bounded sets A ⊂ R m and B ⊂ R n S A × B ( ǫ ) = S A ( ǫ ) × S B ( ǫ ) . Proof.
Let { U i } a cover of A by S A ( ǫ ) dyadic cubes of side length − n , − ( n +1) < ǫ ≤ − n . Let f a be the fiber in A × B ⊂ R m + n of a in A . For each fiber f a a minimum of S B ( ǫ ) dyadic cubes of side length − n , − ( n +1) < ǫ ≤ − n is necessary to cover f a . For each U i pick an a i ∈ U i ∩ A , note a i U j if i = j. Hence there are S A ( ǫ ) such points a i . Covering all f a i by S B ( ǫ ) dyadic cubes requires then S A ( ǫ ) S B ( ǫ ) dyadic cubes. Since these cubes are disjoint nomore efficient cover is possible at this size, hence S A × B ( ǫ ) = S A ( ǫ ) S B ( ǫ ) . Theorem 2.4.
Let A ⊂ R m , B ⊂ R n be bounded with Minkowski dimensions dim M ( A ) , dim M ( B ) respectively. Thendim M ( A × B ) = dim M ( A ) + dim M ( B ) . Proof.
By our dyadic cubes lemma 2.2, N A × B ( ǫ ) ≤ S A ( ǫ ) S B ( ǫ ) ≤ m + n N A × B ( ǫ ) . Then lim ǫ → + log( N A × B ( ǫ ))log(1 /ǫ ) ≤ lim ǫ → + log( S A ( ǫ ) S B ( ǫ ))log(1 /ǫ ) ≤ lim ǫ → + log(2 m + n N A × B ( ǫ ))log(1 /ǫ ) . The cover by dyadic cubes of fixed size used here, N ( ǫ ) , is a subset of the more general cover Falconer uses to prove theHausdorff dimension is not product-summable. Indeed, using general dyadic cubes is equivalent to using arbitrary covers for thepurpose of computing the Hausdorff dimension. See [1, chapter 5]. [1, pages 64-65]. PREPRINT - F
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Note lim ǫ → + log(2 m + n )log( ǫ ) = 0 , hence this becomes lim ǫ → + log( N A × B ( ǫ ))log(1 /ǫ ) ≤ lim ǫ → + (cid:18) log( S A ( ǫ ))log(1 /ǫ ) + log( S B ( ǫ ))log(1 /ǫ ) (cid:19) ≤ lim ǫ → + log( N A × B ( ǫ ))log(1 /ǫ ) . Then dim M ( A × B ) = lim ǫ → + log( N A × B ( ǫ ))log(1 /ǫ ) = dim M ( A ) + dim M ( B ) . Note while this establishes product-summability for the Minkowski dimension, the Minkowski dimension does notalways exist. A more general notion of dimension having the product-summability property would be desirable andwill be presented in section 3.
The Hausdorff dimension is perhaps the most commonly used notion of dimension in geometric measure theory.While in form it seems nearly identical to the Minkowski dimension as presented in theorem 1.6, in practice it hasimportant properties differing from the Minkowski dimension. This is because the Hausdorff dimension is not definedusing covers of a fixed size, say ǫ , but rather using covers consisting of sets of arbitrary diameter less than ǫ. Thisdistinction not only makes the Hausdorff dimension difficult to compute for some sets, but also prevents it frombeing product-summable. To define the Hausdorff dimension we need some intermediate concepts, namely the s -dimensional Hausdorff outer measure. Definition 2.5.
Let A ⊂ R n be bounded, ǫ > , and { V i } be a cover of A by sets of diameter | V i | , where | V i | ≤ ǫ. In this case we call { V i } an ǫ cover of A . Define H sǫ ( A ) to be H sǫ ( A ) = inf X i | V i | s . (6)Then the Hausdorff s-dimensional outer measure of A is [1] H s ( A ) = lim ǫ → + H sǫ ( A ) = sup ǫ> H sǫ ( A ) . (7)Note that the Minkowski s -dimensional outer measure was defined in terms of sets of fixed size and shape while theHausdorff s -dimensional outer measure is defined in terms of sets of arbitrary shape and bounded size.Because the Hausdorff s -dimensional outer measure is not defined in terms of sets of a fixed size we have the followinginequality. If { V i } is an ǫ cover of A and { W i } is an ǫ cover of A , with ǫ < ǫ , then { V i } is also an ǫ cover of A. Then H sǫ ( A ) ≤ H sǫ ( A ) by the approximation property of infima. Hence, since H sǫ ( A ) is bounded below by and is monotonic, the limit inequation 7 always exists.Now to define the Hausdorff dimension. Definition 2.6.
Let A be a bounded subset of R n . Then the
Hausdorff dimension of A , dim H ( A ) , isdim H ( A ) = inf { s | H s ( A ) = 0 } . (8)Using the s -dimension Minkowski outer content we can relate the Minkowski dimension to the Hausdorff dimension. Theorem 2.7.
Let A ⊂ R n be bounded. Then dim M ∗ ( A ) ≥ dim H ( A ) . (9) Proof.
First, note if { U i } is a cover of A by balls of radius ǫ then it is an ǫ cover of A. Hence H sǫ ( A ) ≤ X i | U i | s = M sǫ ( A ) . Taking the limit yields H s ( A ) ≤ M s ( A ) . Then inf { s | H s ( A ) = 0 } ≤ inf { s | M s ( A ) = 0 } , satisfying the inequality in equation 9. PREPRINT - F
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In this section we will show that it is possible to extend the standard Minkowski dimension so it exists for all boundedsubsets of R n while retaining the product-summability property. Before doing so it is appropriate to remark that theupper Minkowski dimension is not such an extension, as will be shown in section 4. In order to properly extend thestandard Minkowski dimension we will need some tools from nonstandard analysis. A non-principal ultrafilter Q on a set X is a subset of the power set P ( X ) with the following proper-ties:
1. Monotonicity: If A ∈ Q and A ⊂ B then B ∈ Q ,2. Closure under intersection: If A, B ∈ Q then A ∩ B ∈ Q .3. Dichotomy: For any subset A of X either A ∈ Q or X − A ∈ Q
4. Closure under subtraction of finite sets: If A ∈ Q and B is finite then A − B ∈ Q . A subset of P ( X ) obeying just 1 and 2 is a filter , and a subset obeying 1, 2, and 3 only is an ultrafilter and notnecessarily a non-principal one. There is one more desirable property we can require of our ultrafilter:5. Proper: ∅ is not in Q .We note a basic property of ultrafilters. Lemma 3.2. If A ∪ B ∈ Q , Q ⊂ P ( X ) an ultrafilter, then either A ∈ Q or B ∈ Q . Proof.
Suppose for contradiction neither A nor B is in Q . Then X − ( A ∪ B ) ⊂ X − A ∈ Q , and similarly X − B ∈ Q , by propery 3. Hence X − ( A ∪ B ) = ( X − A ) ∩ ( X − B ) ∈ Q by De’Morgan’s law, hence A ∪ B
6∈ Q . The following lemma will be useful in proving the existence of ultrafilters over any infinite set. Lemma 3.3.
The union of a chain of filters is again a filter.Proof.
Let {F i } be a chain of filters partially ordered by set inclusion, so that F i ⊂ F j if j > i. Let
A, B ∈ ∪ i F i .Then A ∈ F i , B ∈ F j for some i, j. Hence A ∪ B ∈ F max { i,j } , since the chain is partially ordered. Hence A ∪ B ∈ F i or F j . Now to show the union has property 2, let
A, B ∈ ∪ i F i . As before, A ∈ F i , B ∈ F j for some i, j, so A, B are bothin F max { i,j } . Since this is a filter A ∩ B ∈ F max { i,j } ⊂ ∪ i F i . And we are done.Note that the set of all cofinite sets is a non-principal, proper filter on X . This filter is called the Fréchet Filter, andany filter containing this filter is called a free filter. We can prove the existence of a non-principal free ultrafilter onany infinite set using Zorn’s Lemma. The proof of the existence of an ultrafilter over X , an infinite set, follows froma dichotomy property guaranteeing that a maximal proper filter on X is an ultrafilter, as shown below. Lemma 3.4.
There exists a non-principal ultrafilter on X , X an infinite set.Proof. Let F be the set of free filters on X . Note F is not empty as the Fréchet filter is in F. We can partially order F by set inclusion. Since the union of any chain of free filters is again a free filter, each chain is bounded above by theunion of all the filters in the chain. Hence by Zorn’s lemma there exists a maximal filter in F. Call it Q . We show Q is a non-principle ultrafilter. To do so we need show only property 3 above, as Q is in F and hence a filter.Note for all A ⊂ X, A ∪ ( X − A ) ∈ Q . Suppose for contradiction neither A nor X − A is in Q . If this is the case, weclaim T = { U ∈ P ( X ) | A ∪ U ∈ Q} is a free filter properly containing Q .• T is a filter. [5]. [7, §1.2]. [7]. PREPRINT - F
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26, 2021 – Monotonicity: Let α ∈ T, and α ⊂ β. Then A ∪ α ∈ Q . But A ∪ α ⊂ A ∪ β , so by the monotonicity of Q we have A ∪ β ∈ Q . Hence β ∈ T . – Closure under intersection: If α, β ∈ T then α = A ∪ α ∈ Q , and similarly for β. then A ∪ ( α ∩ β ) =( A ∪ α ) ∩ ( A ∪ β ) ∈ Q since Q is a filter. Hence α ∩ β ∈ T. • Q ⊆ T : Let α ∈ Q . Then by monontonicity A ∪ α ∈ Q , so α ∈ T. • T properly contains Q : This is clear from the fact that X = A ∪ ( X − A ) , hence X − A ∈ T but X − A
6∈ Q . Hence, supposing Q does not have the dichotomy property we come to a contradiction, namely, that Q is not a maximalfilter. Therefore Q has the dichotomy property, property 3 above.Therefore there exists a non-principal, proper ultrafilter, say Q , on any infinite set. It is worth recognizing at this pointa couple of useful properties of Q . The first is that ultrafilters are closed under unions, guaranteed by monotonicity.The second is that no finite set S is in Q a non-principal, proper ultrafilter. The proof follows from the dichotomyproperty of ultrafilters, together with the fact that Q is proper. Now that we have an ultrafilter we can define limits of sequences and functions. We first give an intuitive, algorithmicdefinition of the limit of a sequence, then show this is equivalent to a more powerful definition.
Definition 3.5. An Algorithm for Computing Non-standard Limits of Bounded Sequences
Let { a n } ⊂ R be a bounded sequence, a n ∈ ( x , z ) for all n. We can view { a n } as a function a : N → R . Let Q be a non-principal ultrafilter on N . Compute the Q -limit of { a n } by choosing y ∈ ( x , z ) and determining whether a − (( x , y )) = U or a − ([ y , z )) = V is in Q . Note that U ∩ V = ∅ , so only one of U or V is in Q . Moreover, U ∪ V = N . Say U ∈ Q . Repeat with a y i ∈ ( U i − ) , choosing a sequence of nested intervals { U i } with widthtending to for which a − ( U i ) ∈ Q . The limit of this sequence of intervals is a point, which is the non-standard limitor Q -limit denoted by lim Q ( a n ) . If this non-standard limit equals l , we may write { a n } → Q l. For this algorithm to be useful, it will have to yield a unique limit regardless of the choice of intervals used to computeit.
Lemma 3.6.
Let { a n } ⊂ R be a bounded sequence and l the Q -limit computed by a sequence of nested intervals { U i } → l, let l ′ be the Q -limit computed by a sequence of nested intervals { V i } → l ′ . Then l = l ′ . Proof.
Define W i = U i ∩ V i for all i. Since a − ( W i ) = U i − ∩ V i − , a − ( W i ) ∈ Q for all i. Hence ∩ ∞ i =1 W i ⊂∩ ∞ i =1 U i = { l } , and ∩ ∞ i =1 W i ⊂ ∩ ∞ i =1 V i = { l ′ } . Hence l = l ′ and the limit is unique.It is important to note that the limit of the sequence is not shift-invariant, as shifting the index may affect whichinverse image is in Q . Therefore the non-standard limit is not, in general, characterized sequentially for functions,and we will have to introduce more powerful tools to define the limits of functions. The above lemma motivates theintroduction of a more convenient characterization of the sequential Q -limit of a sequence. Theorem 3.7.
Let { a n } be a bounded sequence, say a n ∈ ( x, y ) for all n , and Q be a nonprincipal ultrafilter on N . Then lim Q ( a n ) = l if and only if for all intervals U containing l a − ( U ) ∈ Q . Proof.
Suppose first all intervals U containing l are such that a − ( U ) ∈ Q . Let { U i } → l be any sequence of nestedintervals converging to l. Then lim Q ( a n ) = l by the algorithm above.Now suppose lim Q ( a n ) = l. Let U be any interval containing l. Then either a − ( U ∩ ( x, y )) or a − (( x, y ) − U ) is in Q . If the latter then for no subinterval of U is the inverse image under a in Q . Therefore the Q -limit cannot be anypoint of U ∩ ( x, y ) . To define the nonstandard limit of a function it is necessary to specify not only an ultrafilter Q on N but also a sequence,say { ǫ n } , tending to . This inspires the following definition for the nonstandard limit of a function. [5]. PREPRINT - F
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Definition 3.8.
Let { a n } be a bounded sequence in the domain of f : R m → R n . Let f n = f ( a n ) . Then lim Q ,a n f ( a ) = lim Q f n . Note that the dichotomy property of an ultrafilter guarantees this limit always exists and is unique, moreover it obeysthe algebra homomorphism laws for limits (in particular lim Q ( a n + b n ) = lim Q a n + lim Q b n . ). Applying the nonstandardlimit to the Minkowski dimension is then particularly nice, as we then have a notion of Minkowski dimension whichalways exists, and which is equal to the standard Minkowski dimension when it exists. Definition 3.9.
Let Q be an ultrafilter on N , { ǫ n } a sequence of positive reals tending to . Let X ⊂ R n . Let S ( ǫ ) beas in lemma 2.2. Then the nonstandard Minkowski dimension of X relative to Q , { ǫ n } is dim Q ,ǫ n ( X ) = lim Q ,ǫ n log( S ( ǫ ))log( ǫ ) . (10)From the algebraic homomorphism properties of the nonstandard limits we can establish the main result of this paper,namely the product summability of the nonstandard Minkowski dimension. Theorem 3.10.
Let X ⊂ R m , Y ⊂ R n , Q , { ǫ n } as in definition 3.9. Then dim Q ,ǫ n ( X × Y ) = dim Q ,ǫ n ( X ) + dim Q ,ǫ n ( Y ) . Proof.
Note log( S X × Y ( ǫ ))log( ǫ ) = log( S X ( ǫ ))log( ǫ ) + log( S Y ( ǫ ))log( ǫ ) by lemma 2.3. The result follows from the algebraic homomor-phism property of the nonstandard limit.We now have a notion of dimension, dim Q ,ǫ n , which is product summable and exists for every set, since the non-standard limit always exists. Moreover, the nonstandard Minkowski dimension agrees with the Minkowski dimensionwhen it exists, and so has all the desirable properties of the Minkowski dimension. In The Geometry of Fractal Sets
Falconer discusses an example of two subsets
A, B of [0 , , each of which haveHausdorff dimension dim H ( A ) = dim H ( B ) = 0 , but which have a Cartesian product of dimension greater than . Falconer constructs sets which have dimension by using a sequence of subsets for which the Hausdorff dimensiontends to . This works because the Hausdorff dimension is defined as an infimum, that is, dim H ( X ) ≡ inf { s : H s ( X ) = 0 } . Here we present a generalization which, aside from bringing clarity to Falconer’s example, is also aninteresting case of when the Hausdorff and upper Minkowski dimensions fundamentally differ, and feature the utilityof the nonstandard Minkowski dimension.Let A , B , A , B , ... be a partition of N , so A S B S A S B S ... = { , , , ... } , with sets A i , B j connected(in the sense that if a < b < c and a, c ∈ A i then b ∈ A i , similarly for B j ) for all i, j. Thus the A i and B j areintersections of N with connected sets in R . We consider expansions in base d . Define the subsets A, B of [0 , by A = { x ∈ [0 ,
1] : x = ∞ X i =1 a i d i , a i = 0 if i ∈ B k for some k } (11) B = { x ∈ [0 ,
1] : x = ∞ X i =1 b i d i , b i = 0 if i ∈ A k for some k } . (12)We use closed intervals of length d − ( m +1) < ǫ ≤ d − m to cover A. Note that x ∈ A has the form x =0 .a a a ...a m − a m ... . Note also that N A ( ǫ ) is the number of choices of digits up to a m , because intervals of length ǫ [5]. [1, page 73]. PREPRINT - F
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26, 2021 are insensitive to variations in size less than or equal to d − ( m +1) . Denote the number of degrees of freedom in A and B by f A , f B respectively. Then f A = (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) ∞ [ i =1 A i ! \ { , ..., m } (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) , (13) f B = (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) ∞ [ i =1 B i ! \ { , ..., m } (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) . (14)Then N A ( ǫ ) and N B ( ǫ ) are, for d − ( m +1) < ǫ ≤ d − m , N A ( ǫ ) = d f A , (15) N B ( ǫ ) = d f B . (16)Computing log( N A ( ǫ ))log(1 /ǫ ) is straightforward. If m ∈ A k for some k , we have log N A ( ǫ )log( ǫ ) = f A m . Similarly log N B ( ǫ )log( ǫ ) = f B m . We can construct the sets A and B so the function log N ( ǫ )log(1 /ǫ ) = fm oscillates between and as ǫ tends to , bycontrolling the size of each set A i or B i . When f A ≈ m the upper Minkowski dimension of A will be observed to benear while the dimension of B will be observed to be near zero. Hence the limit infimum is and the limit supremumis for both, by construction. Then the limit does not exist for these sets.Because we defined the upper Minkowski dimension in terms of the limit supremum, however, we have dim M ∗ ( A ) = dim M ∗ ( B ) = 1 . Similarly, the Hausdorff dimension is an infimum, so dim H ( A ) = dim H ( B ) = 0 . As an example, consider the case when | A n +1 | = n |∪ ni =1 ( A i ∪ B i ) | and | B n +1 | = n |∪ ni =1 ( A i ∪ B i ) ∪ A n +1 | . Let m = max A n +1 . Then f A ≥ m − mn + 1 = nn + 1 mf B ≤ mn + 1 . Then dim M ∗ ( A ) is observed to be near when measured just as A n +1 is complete, while dim M ∗ ( B ) when measuredat this scale would be near . To discuss this notion with more clarity, we may say A is thick when measured at ǫ if log /ǫ ( N A ( ǫ )) > log /ǫ N B ( ǫ ) , indeed, since these two sum to m when d − ( m +1) < ǫ ≤ d − m , we may say A is thick at ǫ if log /ǫ ( N A ( ǫ )) =0 . m, at this scale B will be thick (or thin, if you prefer). We can refer to the dimension in a similar manner,saying dim M ( A ) = x when measured at ǫ if log /ǫ N A ( ǫ ) = x. This is, of course, an abuse of terminology, butfacilitates the use of naturalistic language during computation. A × B Here we demonstrate by counterexample that the Hausdorff and upper Minkowski dimensions are not product-summable. Before doing so, note the following. If we define A + B = { x + y | x ∈ A, y ∈ B } . Then dim M ( A + B ) = 1 PREPRINT - F
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26, 2021 because, for all y ∈ [0 , , there exists an a ∈ A and a b ∈ B such that a + b = y ; therefore A + B = [0 , which is a1-dimensional set. And indeed, log N A ( ǫ )log(1 /ǫ ) + log N B ( ǫ )log(1 /ǫ ) = | A | + | A | + ... + | A k ∩ { , ..., m }| m + | B | + ... + | B k − | m = 1 . (17)We start with the Hausdorff dimension dim H ( A × B ) . Note that projections are distance-decreasing, and so do notincrease the Hausdorff outer measure. Let proj ( A × B ) be the orthogonal projection of A × B onto the line y = x . Noteproj ( A × B ) has the same measure as the interval [0 , , by the logic above. Hence dim H ( A × B ) ≥ dim H ( proj ( A × B )) > , so dim H ( A × B ) > dim H ( A ) + dim H ( B ) and the Hausdorff dimension is not product-summable.We can also show the upper Minkowski dimension is not product-summable. Let d − ( m +1) < ǫ ≤ d − m , then N A × B ( ǫ ) ≤ N A ( ǫ ) N B ( ǫ ) = d f A + f B . Since f A + f B = m , we have N A × B ( ǫ ) ≤ d − m = ǫ − . Hencedim M ( A × B ) ≤ , and the upper Minkowski dimension is not product-summable.We can do better, however, and show dim M ( A × B ) = 1 . Fix b ∈ B . Then N A ( ǫ ) = N A ×{ b } ( ǫ ) ≤ N A × B ( ǫ ) , but N A ( ǫ ) = ǫ − . Then ǫ − ≤ N ǫ ( A ) ≤ ǫ − , and dim M ( A × B ) = inf { α | N A × B ( ǫ ) = O ( ǫ − α } = 1 . Finally, it is here the nonstandard Minkowski dimension shows its utility. Let Q be an ultrafilter on N and ǫ n be asequence of positive numbers tending to 0. Here N A,B ( ǫ ) = S A,B ( ǫ ) and, since N A ( ǫ ) N B ( ǫ ) = d m , dim Q ,ǫ n ( A × B ) = lim Q ,ǫ n log( S A × B ( ǫ ))log(1 ǫ )= lim Q ,ǫ n (cid:18) log( N A ( ǫ ))log(1 /ǫ ) + log( N B ( ǫ ))log(1 /ǫ ) (cid:19) = lim Q ,ǫ n log( d m )log( d m )= 1 . Hence dim Q ,ǫ n ( A × B ) = dim Q ,ǫ n ( A ) + dim Q ,ǫ n ( B ) . Moreover, dim Q ,ǫ n ( B ) = 1 − dim Q ,ǫ n ( A ) , regardless of thesequence of ǫ n chosen. In this subsection we demonstrate a method for creating sets of arbitrary rational dimension. To do so we first prove yetanother useful identity of the Minkowski dimension, this time for subsets of 1-dimensional sets, and use that identityto create subsets of [0 , which have a Minkowski dimension within ǫ of any value in [0 , . The method used to createthese subsets also provides a means for calculating their dimension exactly.Our first lemma provides an efficient means of calculating the Minkowski dimension for subsets of R which havea particular structure. This lemma bears some resemblance to results used in the generalization of an example ofFalconer’s in section 4.1. Lemma 4.1.
Let X ⊂ [0 , be such that for all x ∈ X, if x is expressed in d -adic notation, so x = x d − + x d − + x d − + ... , then each x i can take on any of f < d values, where f is constant for all x i . We call f the number ofdegrees of freedom of x i . Then dim M ( X ) = log d ( f ) . (18) Proof.
Let d − j ≤ ǫ < d − j +1 . Then N ( ǫ ) = f j , since there are f possibilities for each of j x i values. Covering X with closed intervals of width ǫ = d − j yields, from definition 1.2,dim M ∗ ( X ) = lim sup ǫ → + log( N ( ǫ ))log(1 /ǫ )= lim j →∞ log( f j )log(1 /d − j )= log d ( f ) . PREPRINT - F
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Since the limit lim ǫ → N ( ǫ ))log(1 /ǫ ) exists, the limit supremum and limit infimum exist and are equal. Hence theMinkowski dimension exists and is equal to log d ( f ) , and we are done.Using equation 18 we can generate sets of dimension within ǫ of any value in [0 , , a corollary theorem 4.2. Anexample of such a set is the Cantor set expressed in triadic notation. In particular, we can generate sets of anydimension in Q ∩ [0 , , as shown below. Theorem 4.2.
Let r, s ∈ N with s > r. Then there exists a set X ⊂ [0 , such that dim M ( X ) = rs . Proof.
Expressing the elements of our set in d -adic notation, generate X by taking digits in blocks of length s and,out of d s possible strings, choose d r combinations. Now let d − ns ≤ ǫ < d − ( n − s . Then having d r combinations inthis block of s digits yields d ( n − r < N ( ǫ ) ≤ d nr , since there are n such blocks. Nowdim M ( X ) = lim ǫ → + log( N ( ǫ ))log(1 /ǫ )= lim n →∞ log( d nr )log(1 /d − ns )= rs , and we are done.A corollary of theorem 4.2 establishes a method for generating sets of dimension within ǫ > of any positive realnumber, using a product with a finite number of terms. Corollary 4.2.1.
Let a > , ǫ > . Then there exists a set X ⊂ R n , n > a, such that | dim M ( X ) − a | < ǫ. Proof.
Define A as in theorem 4.2, with dim M ( A ) ∈ Q ∩ ( a/n − ǫ/n, a/n + ǫ/n ) . Then the set X = Q ni =1 A i , with A i = A for all i, is such that dim M ( X ) = P ni =1 dim M ( A i ) = n dim M ( A ) is in ( a − ǫ, a + ǫ ) . And we are done.Note in the above corollary if a is rational then we can create a set of dimension a. The Hausdorff dimension, as stated in definition 2.6, is perhaps the most commonly used notion of dimension inmodern geometric measure theory, and is the main object of discussion in Falconer’s classic
The Geometry of FractalSets . Its effectiveness has been shown over many decades, however, it has some drawbacks. The first is that anycountable subset of R n has a Hausdorff dimension of zero. The proof of this claim is nearly identical to the proofthat an at most countable set has Lebesgue measure zero. This prevents the Hausdorff dimension from being ableto distinguish between sets within a large class. Another problem encountered when using the Hausdorff dimensionis its sensitivity to the arrangement of the so called “pieces” in the set. The Minkowski dimension is invariant underrearrangements, however, a key property which Lapidus and van Frankenhuijsen call “Independence of GeometricRealization”. Lastly, computation of the Hausdorff dimension of any set is difficult, and even when obtained doesno more than bound the dimension of a Cartesian product set in terms of its factors. Falconer’s remarks on theHausdorff measure and dimension bear mentioning here.“[The] Hausdorff dimension, defined in terms of the Hausdorff measure, has the overriding advantage from the mathe-matician’s point of view that the Hausdorff measure is a measure (i.e. is additive on countable collections of disjointsets). Unfortunately the Hausdorff measure and dimension of even relatively simple sets can be hard to calculate; inparticular it is often awkward to obtain lower bounds for these quantities. This has been found to be a considerabledrawback in physical applications and has resulted in a number of variations on the definition of Hausdorff dimensionbeing adopted, in some cases inadvertently.” [1]. See for example [8, page 332]. [3, remark 1.4]. [1, corollary 5.10 and theorem 5.11]. [1, pages ix-x]. PREPRINT - F
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While the choice of which dimension to use depends on the application, the Minkowski dimension has several prop-erties which make it a more natural choice over other notions. Among the most important is that the Minkowskidimension is product-summable, when it exists. This conforms with human intuition and is hence a desirable property.Indeed, a short list of properties the authors would desire from any notion of dimension would be the following:
1. Obtains expected values for well understood sets.2. Dimensions of subsets are less than or equal to the dimension of the set.3. Describes the behavior of sets when measured at different scales.4. Can distinguish between a wide variety of sets.5. Can be computed or estimated for most sets.6. Is product-summable, by analogy with linear algebra and products of boxes in R n .7. Is Lipschitz invariant and invariant under rigid motions.8. Is stable (so dim( A ∪ B ) = max { dim( A ) ∪ dim( B ) } ) .
9. Is independent of geometric realization.
10. Is defined in terms of a measure.Despite its nice mathematical properties, the Hausdorff dimension does not have properties 4, 5, 6, and 9 in general.The nonstandard Minkowski dimension, on the other hand, epitomizes all these properties. The Hausdorff dimensiondoes, however, relate to a measure (when restricted to an appropriate σ algebra), while the upper Minkowskidimension is related to the Minkowski content.Nonstandard analysis has been applied to geometric measure theory by previous authors, one of which extends theHausdorff measure to the nonstandard universe (in so doing introducing a nonstandard Hausdorff dimension). Thisresearcher, Potgieter, introduces the nonstandard Hausdorff dimension for reasons differing from our own - Potgieterintroduces nonstandard analysis because it allows for more intuitive proofs of some classical results, while in ourcase nonstandard analysis is introduced to extend the notion of Minkowski dimension to sets which do not have astandard Minkowski dimension. Because the Hausdorff dimension exists for every set our motivation does not applyto nonstandard extensions of the Hausdorff dimension.Additional motivation for the study of the Minkowski dimension can be found in the theory of the zeta functions offractal strings. Lapidus and van Frankenhuijsen define a fractal string as a bounded open subset of R with fractalboundary. In particular, a fractal string is a sequence of lengths L = { l i } , and the zeta function on such a string is ζ L ( s ) = P i l − si . It is an interesting fact that the Minkowski dimension of a fractal string determines the largest half-plane on which the zeta function is holomorphic. If one defines a fractal set as one whose zeta function has at leastone nonreal pole with positive real part, then the sets we would ordinarily consider as fractal, such as self-similar sets,are fractal. In particular, the Devil’s staircase (a rearrangement of the Cantor set) is fractal under this definition, thoughit is not under more classical definitions (because, for example, its Hausdorff dimension (which is integer) equals itstopological dimension). The theory can be extended beyond R , as done by Lapidus in [11]. From this example, aswell as because it satisfies most of the properties above (including product-summability), the nonstandard Minkowskidimension may well be a more natural notion of dimension than the Hausdorff dimension, and more useful in manyapplications. References [1] Kenneth J Falconer.
The geometry of fractal sets . Cambridge university press, 1986.[2] Benoit Mandelbrot.
The Fractal Geometry of Nature , volume revised from 1977 edition. W.H. Freeman, NewYork, 1983. Similar lists by various authors exist, for example Falconer has a similar list in [9, chapter 3]. Our list differs from Falconer’sin several elements, most notably 6 and 9 as well as in our interpretation of point 4. See [3, remark 1.4]. [10]. [3]. [3, theorem 1.10]. [2, page 82],[3, page 337]. PREPRINT - F
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26, 2021 [3] Michel L Lapidus and Machiel Van Frankenhuijsen.
Fractal geometry, complex dimensions and zeta functions:geometry and spectra of fractal strings . Springer Monographs in Mathematics, 2nd edition, 2013.[4] John Marstrand. The dimension of cartesian product sets.
Proceedings of the Cambridge Philosophical Society ,50:198–202, 1954.[5] Terence Tao. Ultrafilters, nonstandard analysis, and epsilon management, Jun 2007. https://terrytao.wordpress.com/2007/06/25/ultrafilters-nonstandard-analysis-and-epsilon-management/ .[6] Terence Tao.
An Introduction to Measure Theory . American Mathematical Society, 2010.[7] Max Garcia. Filters and ultrafilters in real analysis. arXiv , 2012. https://arxiv.org/abs/1212.5740 .[8] William R. Wade.
An Introduction to Analysis . Pearson Education, 4 edition, 2018.[9] Kenneth Falconer.
Fractal geometry: mathematical foundations and applications . John Wiley & Sons, 2004.[10] Paul Potgieter. Nonstandard analysis, fractal properties and brownian motion. arXiv , 2010. https://arxiv.org/abs/math/0701640 .[11] Michel L Lapidus, Goran Radunovic, and Žubrinic Darko. Fractal zeta functions and complex dimensions: Ageneral higher-dimensional theory. Technical report, 2015. https://arxiv.org/abs/1502.00878v3 ..