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The Fractal Dimension of Product Sets THE FRACTAL DIMENSION OF PRODUCT SETS A PREPRINT Clayton Moore Williams Brigham Young University [email protected] Machiel van Frankenhuijsen Utah Valley University [email protected] February 26, 2021 ABSTRACT Using methods from nonstandard analysis, we define a nonstandard Minkowski dimension which exists for all bounded sets and which has the property that dim(A × B) = dim(A) + dim(B). That is, our new dimension is “product-summable”. To illustrate our theorem we generalize an example of Falconer’s1 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 rational dimension. 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 geometric measure theory from its inception, and debate over what it means for a set to be fractal has often reduced to debate over the propernotion of dimension. A classical example is the “Devil’s Staircase” set, which is intuitively fractal but which has integer Hausdorff dimension2. As Falconer notes in The Geometry of Fractal Sets3, the Hausdorff dimension is “undoubtedly, the most widely investigated and most widely used” notion of dimension. It can, however, be difficult to compute or even bound (from below) for many sets. For this reason one might want to work with the Minkowski dimension, for which one can often obtain explicit formulas. Moreover, the Minkowski dimension is computed using finite covers and hence has a series of useful identities which can be used in analysis. In particular, the Minkowski dimension of a product set is the sum of the dimensions of the factoring sets. We define this notion below for clarity. arXiv:2102.13050v1 [math.GN] 25 Feb 2021 Definition 0.1. Let f be a function defined for all bounded subsets of Rk for all k. Then f is product-summable if f(A)+ f(B)= f(A × B), where A × B ⊂ Rm+n is the Cartesian product of A and B, for all such bounded subsets A ⊂ Rm and B ⊂ Rn. Product-summability is a desirable property for any notion of dimension, by analogy with boxes in Rn and their prod- ucts and because it agrees so strongly with human intuition. The Hausdorff dimension is not product-summable4, nor is the upper Minkowski dimension. The Minkowski dimension is product-summable(as is shown), but is unfortunately defined 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 Minkowski dimension is a more appropriate tool of analysis for many applications in geometric measure theory. This nonstandard 1[1, page 73] 2Discussion in [2, page 82], [3, page 337]. 3[1, pages ix-x]. 4[4]. A PREPRINT - FEBRUARY 26, 2021 Minkowski dimension is introduced at the cost of ultrafilters and nonstandard analysis, however, we introduce these ideas in an intuitive and conceptually transparent way based in large part on Terence Tao’s pedagogical discussion in Ultrafilters, Nonstandard Analysis, and Epsilon Management.5 In Fractal Geometry, Complex Dimensions, and Zeta Functions[3], Lapidus and van Frankenhuijsen give convincing evidence for the case that fractality ought to be defined in terms of the Minkowski dimension. We add to their argument, which is grounded in surprising connections between zeta functions and the complex dimensions of fractal sets6, by introducing and summarizing useful properties of the standard and nonstandard Minkowski dimensions from the perspective of geometric measure theory. Because the Minkowski dimension, when it exists, is equal to the nonstandard Minkowski dimension, we make the argument that the nonstandard Minkowski dimension is a natural notion of dimension. We begin by reviewing the standard Minkowski dimension and its identities, and proving the product-summability of the 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 Minkowski dimension. Our main result, theorem 3.10, establishes the product-summability of the nonstandard Minkowski dimen- sion follows from its definition and a coveringlemma, lemma 2.3. In section 4 we provide an interesting generalization of an example of Falconer’s7 for which the Hausdorff dimension of the product is not the sum of the dimensions of the factoring sets. The Minkowski dimension does not exist for these sets, which also provide an example for when the upper Minkowski dimension is not product-summable, however, the nonstandard Minkowski dimension does exist for these sets. As a corollary to one of the identities of the Minkowski dimension we also introduce a means of generating sets of arbitrary rational dimension. We conclude by elaborating on product-summability and providing a list of useful properties one would desire of any notion of dimension. 1 The Minkowski Dimension 1.1 Identities of the Minkowski Dimension To proceed we define the upper Minkowski dimension, which is computed using sets of fixed size. The best way to introduce the upper Minkowski dimension is via the covering number N(ǫ). n Definition 1.1. Let A be a boundedsubset of R and ǫ> 0. Then the covering number NA(ǫ) is the minimum number of balls of radius ǫ in Rn needed to cover A. We can now define the Minkowski dimension, as well as the covering dimension. Definition 1.2. Let A ⊂ Rn be bounded, ǫ > 0, and N(ǫ) be its covering number. The upper Minkowski dimension of A, dimM ∗ (A), is then log N(ǫ) dim (1) M ∗ (A) = lim sup 1 . ǫ 0+ log → ǫ log N(ǫ) If the limit exists then the standard Minkowski dimension is dimM (A) = lim 1 , and equals the upper ǫ 0+ log ǫ Minkowski dimension. The limit supremum is chosen to define the standard Minkowski→ dimension here because of its identities, as will be derived. In practice it is sometimes convenient, once one has found NA(ǫ), to recognize the growth of NA(ǫ) is of particular order in 1/ǫ. As will be shown, this observation allows one to compute the upper Minkowski dimension. We make this notion precise below. n Definition 1.3. Let A ⊂ R be bounded and N(ǫ) be its covering number. Define dimC (A) to be 1 dim (A) = inf α | N(ǫ)= O as ǫ → 0 . (2) C ǫα Then dimC (A) is the covering dimension. 5[5]. 6 Their argument is that subsets L of R with a sequence of lengths {li} should be defined as fractal if the geometric zeta function s of L, ζL(s)= Pi li , has a nonreal pole with positive real part. 7[1, page 73]. 2 A PREPRINT - FEBRUARY 26, 2021 n We now prove that, for A ⊂ R a bounded set, dimC (A)= dimM ∗ (A). Theorem 1.4. Let A ⊂ R be bounded. Let β = dimC (A) be the covering Minkowski dimension and dimM ∗ (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 1.2 and 1.3 with regards to the relevant sets to make this clear. 1 1. Note that inf α | N(ǫ)= O ǫα as ǫ → 0 = β if and only if, for all α > β, there exist a k > 0 and a δ > 0 such that for all 0 <ǫ<δ we have N(ǫ) < k(1/ǫ)α; and, for all α<β, for all k,δ > 0 there exists an ǫ<δ such that N(ǫ) > k(1/ǫ)α. 2. Note also that lim supǫ 0+ log1/ǫ N(ǫ) = β∗ if and only if, for all α>β∗, there exists a δ > 0 such that α → α N(ǫ) < (1/ǫ) for all ǫ<δ; and, for all α<β∗, for all δ, there exists an ǫ<δ such that N(ǫ) > (1/ǫ) . 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. α We first show that, given a β as in 1, then for all α>β there exists a δ > 0 such that NA(ǫ) < (1/ǫ) . Suppose that α α β for all α>β there exist k,δ > 0 and an ǫ<δ such that NA(ǫ) < k(1/ǫ) . Let α>β and consider α′ = α − −2 . Let k,δ be such that N(ǫ) < k(1/ǫ)α′ for all ǫ < δ. Then 1 α β N (ǫ) < ǫ −2 . A ǫα 2 1 α β Let δ′ = min δ, k − . Then for ǫ<δ′ we have n o 1 N (ǫ) < , A ǫα and we are done. It remains to show that β is the greatest value such that for all α<β and for all δ > 0 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 equality of β and β∗. But this property follows naturally from point 1 above when k =1. Then dimC (A)= dimM ∗ (A). 1.1.1 The Minkowski Dimension in Terms of the s-dimensional Outer Content It would be desirable to define an outer measure allowing us to directly relate the upper Minkowski dimension to the Hausdorff dimension through properties of the covering.
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