Box Counting Dimensions of Generalised Fractal Nests

Home , Box counting, Minkowski sausage

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x π 2 γx = x . 2 Generalised fractal nests Γ( 2 + 1) 2.1 Box counting dimension For further discussion of this coefficient and its use in the Minkowski-Bouligand dimension, see [Res13] We start with the definition of the box counting di- and [KP99, Chapter 3.3]. mension as stated in [Fal14, pg. 28] as the alterna- tive definition. By an ǫ-mesh in Rn we understand Definition 2. Let S Rn. For ǫ > 0, we define ⊆ the partition of Rn into disjoint (except possibly at the ǫ-Minkowski sausage of S as the set the border) n-cubes of side ǫ. n (S)ǫ = x R d(x, S) ǫ . Definition 1. Let S Rn be a bounded Borel set. { ∈ | ≤ } ⊆ Rn By Nǫ(S) we denote the number of such n-cubes of Let λ be the Lebesgue measure on and denote n n,d the ǫ-mesh of R that intersect S. We define the by Aǫ (S) the ratio upper (lower) box counting dimension of S by n,δ λ(S)ǫ Aǫ (S) := n−δ . γn−δǫ dim S = inf δ ǫδN (S) 0 . B ǫ → We say that d is the upper (lower) Min- n o δ kowski-Bouligand dimension of S, dimMB S dimBS = sup δ ǫ Nǫ(S) + . → ∞ (dimMB S) if n o

m n,δ Example 1. The unit m-cube Km = [0, 1] d = inf δ lim Aǫ (S)=+ 0 n−m in Rn with m n has × ǫ→0 ∞ n o { } ≤ n,δ d = sup δ lim Aǫ (S)=0 . ǫ→0 dimBKm = dimBKm = m n o m because it intersects ǫ−1 n-cubes of side ǫ. A classical result (eg. [Pes97, Ch. I.2], [Fal14, Hence, for various δ we have, as ǫ 0, Prop. 2.4]) is that both upper and lower Minkow- → ski-Bouligand dimensions are exactly equal to the 0, for δ > m, corresponding upper and lower box counting di- − m ǫδ ǫ 1 1, for δ = m, mensions, so we will only use the box-dimension →  notation for the discussed dimension. + , for δ < m. ∞ Rn  Definition 3. For S , if dimBS = dimBS = The box counting dimension of a set S can be de- d, we say that S is of⊆ box counting dimension d, fined in terms of other counting functions, such as denoting the maximum number of disjoint ǫ-balls centred on dimB S = d. points of S, the minimal number of ǫ-balls needed to cover S, similar constructions in equivalent met- For S of box-dimension d, we define the nor- rics, etc. [Fal14, pg. 30]. malised upper (lower) Minkowski content of One important reformulation of the box counting S as

dimension is the Minkowski-Bouligand dimension. ∗ n,d d(S) = lim sup Aǫ (S), It is formulated in terms of the Lebesgue measure M ǫ→0

2 Figure 1: The Minkowski sausage of the unit square in R2.

n,d ∗d(S) = lim inf Aǫ (S)) . M ǫ→0 We say that the set S Rn of box-dimension d is Minkowski-non-degenerate⊆ if ∗(S) < + Md ∞ and ∗ (S) > 0 and denote M d Figure 2: Sets Eα of unit normalised Minkowski dimB S d. ≡ contents for α 0.2, 0.4,..., 5.0 . ∈{ } Example 2. As per the earlier discussion in Ex- ample 1, it is easy to show that K Rn is of nor- m ⊆ 2.2 Fractal analysis of α-regular gen- malised Minkowski content (both upper and lower) eralised fractal nests equal to 1 because Intuitively, an inner α-regular fractal nest is the −1 −α m m−k subset of the union of circles of radii n where λ(Km)ǫ ǫ (k) = γ − + γ − E , ǫn−m n m n k 2 m subsets per individual circle are homothetic to each kX=0 other. The outer α-regular fractal nest has radii of form 1 (k + 1)−α. (k) − where Em is the number of k-edges of the m-cube. For a set S Rn and x 0 we denote by (x)S (0) (1) ⊆ ≥ In Figure 1, we have E2 =4 and E2 =4 and the scaling of the set S by x. γ0 = 1, γ1 = 2 and γ2 = π, so we have λ(K2)ǫ = n−1 2 Deﬁnition 4. Let S S be a Borel subset of ǫ (1) ǫ (0) 2 ⊆ γ0 + 2 γ1E2 + 4 γ2E2 =1+4ǫ + πǫ . the unit sphere in Rn. We deﬁne the α-regular

−α fractal nest of centre (outer) type as Example 3. The set of points Eα = k k N for α > 0 is of box counting dimension{ 1| ,∈ with} ∞ 1+α S = (k−α)S normalised Minkowski contents (both upper and Fα k=1 lower) equal to [∞ −α α αS = (1 k )S . 2 α+1 α O − S = (α +1)Γ +1 . k=1 Md α√π 2(α + 1) [ Note that α 1 = Eα Figure 2 shows sets Eα of unit normalised Min- F { } kowski contents in dimensions ranging from 5/6 at and the bottom to 1/6 in the top row. 1 =1 E , Oα{ } − α

3 where for a set S the expression 1 S is deﬁned as sin x sin x sin x − ) − 1 2 ··· n 1 1 x x S , with Eα deﬁned in Example 3. { − | ∈ } is bi-Lipschitz. Theorem 1. Let S Sn−1 be a Borel subset of Thus, any subset of the unit hyper-cube can be ⊆ the unit hyper-sphere in Rn such that mapped to a corresponding set of equal dimension on the hyper-sphere. If we identify Km with its dimB S δ 0. Sn−1 ≡ ≥ embedding into , applying Theorem 1 we have that For every α> 0 the α-regular fractal nest of centre m +1 type generated by S has dimB αKm = . F α +1 δ+1 , for αδ < 1, We note that this formula also applies to m = 0. ≡ α+1 Also, for the outer nests, using Theorem 2 we have dim (S) = δ, for αδ =1 B Fα   1  δ, for αδ > 1. dim K = m + , ≡ B Oα m α +1  Sn−1 Theorem 2. Let S be a Borel subset of again, compatible with m = 0. the unit hyper-sphere⊆ in Rn such that

dim S δ 0. 3.2 (α, β)-bi-fractals B ≡ ≥ For every α > 0 that the α-regular fractal nest of Let α,β > 0. We can identify Eβ with its image on outer type generated by S has the unit circle of Φ1. Let Dβ be the union

π π 1 1 Dβ =Φ1 (1 Eβ) Φ1 (1 + Eβ) S . dimB αS δ + . 4 − ∪ 4 ⊆ O ≡ α +1 We have that The proofs of both theorems rely on the well- known technique of separating the “core” and the 1 dimB Dβ = . “tail” of the set, the “tail” part consisting of the 1+ β well-isolated components, and the “core” of the re- We call the set αDβ an (α, β)-bi-fractal. Its di- maining parts. mension is givenF by Theorem 1 as We take a novel approach to analysing the “core”, where we use the covering lemma (Lemma β +2 dimB αDβ = . 5) to replace the components of the core with F (β + 1)(α + 1) well-spaced sets without losing the asymptotic and hence dimensional properties, including Minkowski 3.3 Uniform Cantor nests (non)-degeneracy. C Intuitively, the uniform Cantor set N are Cantor sets “preserving” N copies of themselvesC totalling 3 Application of the gener- C in relative length in each iteration. alised nest formulas In [Fal14, pg. 71] uniform Cantor sets are defined in terms of the number of preserved copies m and In this section, we show some applications of our the relative gap r (see 5). Modifying that definition main results to some known and some novel fractal to our notation, uniform Cantor sets are defined as sets. follows. Definition 5 (Uniform Cantor set, [Fal14]). Let 3.1 Mapping m-cubes to m-spheres 1 N 2 be an integer and 0

4 (a) α ≈ 0.35 (b) α ≈ 0.67 (a) dimB FαDβ ≈ 0.25 (b) dimB FαDβ ≈ 0.5

Figure 3: (α, β)-bi-fractals of dimension 3/4 Figure 4: (α, β)-bi-fractals of various dimensions.

1 For the standard Cantor set, 3 we have, as is and the corresponding lower Minkowski content is C2 shown in [LP93] and [FC07], d r 2 1 d −1 1 log 2 ∗d N = − γ1−d. dim 3 , M C 1 d 2d B C2 ≡ log 3 − r Again, we can use Φ1 to identify N with its im- with different upper and lower Minkowski contents, age on S1, and so we have C ∗ 1 − − 3 1 log3 2 = γ − 4 2 2.27, d 2 1 log3 2 r 1 logr N M C · ≈ dimB α N = − and F C 1+ α log 2 and 1 3 3 3 −1 r 1 ∗d 2 =2γ − log 3 3 log4 2.19. dim = log N. M C 1 log3 2 2 2 ≈ B α N r O C 1+ α − An older proof of the following proposition can be found in [FC07], where the authors use the sim- 3.4 Numerical verification of the re- pler formula for (both upper and lower) Minkowski sults contents omitting the normalising factor. Since the main results of this article are asymptotic Proposition 1. The set r R is Minkowski CN ⊆ in nature, there is always a risk that we will not be non-degenerate with box counting dimension able to reproduce such results in numerical compu- log N tations. Luckily, as can be seen in figures 8 and dim r = d. B CN ≡− log r 9 we can observe the dimensions to a reasonable accuracy. The upper Minkowski content at the dimension d is We used the same algorithm for producing figures d ∗ s 1 r − 3, 4, 6 and 7 and also for the explicit computation r =2N − γ 1 MdCN 2 1 Nr 1−d of the dimensions. − 5 (a) α ≈ 0.33 (b) α ≈ 0.67 Figure 5: First four iterations of the uniform Can- tor set 1/4. C3

We use the same technique used in the proofs of the main results, in particular we use of Lemma 6 for producing numbers m1(ǫ) (the number of isolated components, the “tail” of the fractal) and m (ǫ), the number of 2ǫ-separated elements that 2 α ≈ . α ≈ . cover the “core” of the fractal nest. (c) 1 00 (d) 1 33 For figures 3, 4, 6 and 7 we used a program writ- r Figure 6: α nests of dimension 3/4. ten in the Python programming language (version F C3 3.6) that outputs an encapsulated PostScript (eps) description of the fractal. PostScript is well-suited In Figures 3 and 6 we show (α, β)-bi-fractals r as a page description language since it allows for αDβ and uniform Cantor nests α 3 of fixed di- global scaling and setting of line width using the mensionF d =3/4 and various α withF Cβ and r com- setlinewidth command that is defined in the stan- puted from dard as “up to two pixels” [Ado99, pg. 674] best − 1 approximation of the Minkowski sausage of half ra- r = N δ (3.1) dius of the given parameter when rendered. 1 β = 1 (3.2) The figures themselves have the half of the line δ − width parameter ǫ set at 1/300 of the height and δ = dα + d 1. (3.3) width of the picture. − For (α, β)-bi-fractals, we first obtain the ra- The constraint of the main result that α plays a (1) −α role in the total dimension only if αδ < 1 limits α dius of the element of the nest using rk = k (2) to for k 1,...,m1(ǫ) and rk = 2kǫ for k ∈ { } (i) ∈ 1 1 1,...,m2(ǫ) . Then, at each radius rk we draw α 1, (3.4) { } ∈ d − d the the set Dβ by the same construction, using 4ǫ 4ǫ m1 (i) and m2 (i) , both with respect to the πrk πrk In Figures 4 and 7 we vary the total dimension 1 exponent β instead of α. d , 1 and we set − − ∈ h 4 i For uniform Cantor nests, we repeat the same (i) 1 1 initial part to obtain rk , we use a standard recur- α = (3.5) sive algorithm for describing segments of the Can- d − 2 tor set, where the number of iterations depends on so that α is always in the centre of the interval given the radius of the nest element, since we are inter- in (3.4). Then we compute δ using (3.3), and finally ested only in segments that have gaps larger than we produce the parameters β and r from (3.2) and (i) 2ǫ/rk . (3.1).

6 In Figure 9, we display the relative error of the same method against a varying total box counting dimension, d 0.25, 1 as in figures 4 and 7, for α defined by (3.5∈). h i Since α here is defined to be quite distant from δ−1 with 1 d αδ = , 2 − 4 we expect the error to be relatively low, with Figure r r (a) dimB FαC3 ≈ 0.25 (b) dimB FαC3 ≈ 0.50 9 showing the error under 5% for both types of nests. To see the behaviour more clearly, in figures 10 and 11 we show the relative deviations from linear regression for more detailed samples, with 300 samples for ǫ between 2−5 and 2−35. The dimension of the set is fixed at 3/4. For α = 4/5 and α = 3 in both figures the approximation of the dimension is very good, the numerically obtained dimension is between 0.751 and 0.754, while at the critical point (where we lose F Cr ≈ . F Cr ≈ . (c) dimB α 3 0 75 (d) dimB α 3 1 00 Minkowski-non-degeneracy, α = 4/3), the approx- Figure 7: r nests. imate dimension is around 0.82. At the critical FαC3 point, the relative error shows the most bias in both figures. For Figure 8 we fix the dimension of the whole In figure 11 we can also observe the effects of uni- fractal nest to be 3/4 and we vary the parameter form Cantor sets having distinct upper and lower α. Minkowski content, with visible oscillations of con- We compute the parameters β for the (α, β)-bi- tent at different scales. r fractal and r for the α 3 Cantor nest. For ten F C −25 −10 different values of ǫ ranging from 2 to 2 we 4 Proofs of main results count the number Nǫ of points necessary to draw the fractals. Using Python’s scikit-learn library [PVG+11], we find the slope for the linear regres- In this section we prove the main results. After the sion of ln N against ln ǫ. introduction of the asymptotic notation we use and ǫ − As can be seen in Figure 8, the results of the lemmas necessary for the proof of theorems 1 these computations come mostly within 10% preci- and 2. sion, with relatively short execution time, finishing within minutes on a laptop computer. The falling 4.1 Asymptotic notation dotted line at the left side of Figure 8 represents In studying box counting dimensions, asymptotic 1 notation is often useful. Here we opt for -style dimB Eα = ∼ α +1 notation, like in [Tri93] for mutually bounded sequences and functions, corresponding to Θ in clas- and the rising dotted line on the right side is δ, the sical big-Oh notation [Knu76]. dimension of the set copied by the nest. The Can- tor nests are sensitive in the beginning, following Definition 6 (Sequence and function equivalence). −1 the (α + 1) curve. On the right-hand side, as Let an and bn be two positive sequences. We say we approach the Minkowski degenerate point when that ak and bk are equivalent and denote it by α = δ−1 we should expect the error to grow, as it does. a b k ∼k k

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Figure 8: Relative error for numerical computation of the dimensions of αDβ (grey dashed line) and r (black dashed line) for ﬁxed dimension 3/4 (bold horizontal line). F FαC3 if there exists a number M 1 such that for all Proof. Let x J and k N. Since f I g we have k N, ≥ ∈ ∈ ∼ −1 ∈ −1 M (g φ)(x, k) (f φ)(x, k) M(g φ)(x, k). M ak bk Mak. ◦ ≤ ◦ ≤ ◦ ≤ ≤ for some M 1. Taking the sum of parts of the Similarly, let I be a set and let f,g : I R+. We inequality to≥m(x), we get say that f and g are equivalent on I,→ denoting m(x) m(x) − f(x) x∈I g(x) M 1 (g φ)(x, k) (f φ)(x, k) ∼ ◦ ≤ ◦ k=1 k=1 if there exists a constant M 1 such that for every X X ≥ m(x) x I we have M (g φ)(x, k), ∈ ≤ ◦ k=1 M −1g(x) f(x) Mg(x). X ≤ ≤ proving the lemma. When the domain of equivalence is unambigu- The following Minkowski non-degeneracy condi- ous, we will use only the symbol to denote the tion is a useful intuition on what the box-counting equivalence. ∼ dimension represents, namely that the ambient area of the Minkowski sausage is asymptotically 4.2 The lemmas and the proofs equivalent to radius to the power of the “comple- mentary” dimension of the ambient space. Lemma 1. Let f,g : I R and let φ: J N I → × → Rn and m: J N. If Lemma 2. Let S be such that dimB S d → and let L> 0. Then⊆ that for ǫ 0,L] we have ≡ h f(x) ∈ g(x) ∼x I λ(S) ǫn−d. ǫ ∼ then n,d Proof. This is an direct consequence of Aǫ being bounded from both above and below as ǫ 0 and m(x) m(x) → the fact that λ(S)ǫ is continuous and monotonous (g φ)(x, k) x∈J (h φ)(x, k). ◦ ∼ ◦ as a function of ǫ. kX=1 Xk=1

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r Figure 9: Numerical computation of the dimensions of αDβ (grey dashed line) and α 3 (black dashed line) for varying dimensions. F F C

n −1 Lemma 3. Let S R be a Borel set and let x,ǫ > Proof. Since M ak M for some M 1, we 0. Then, ⊆ have that ≤ ≤ ≥ ǫ ((x)S)ǫ = (x)(S) x λ(S) λ(S) M nλ(S) , and, consequently, ǫak ≤ ǫM ≤ ǫ

n ǫ and likewise for the lower bound, we have λ((x)S)ǫ = x λ(S) x . λ(S) λ(S) . Proof. Suppose a ((x)S)ǫ. This is true if and ǫak ∼k ǫ only if ∈ inf a xb ǫ. Taking the sum of both sides from 1 to m, we prove b∈S k − k≤ the lemma. Since x> 0, we get Lemma 5 (Dense covering lemma). Let S Rn 1 ǫ be a Borel set. Suppose that for every ǫ > 0⊆there inf a b , + b∈S x − ≤ x is a set Iǫ R and a (possibly inﬁnite) sequence ⊆ aǫ , such that −1 k so a ((x)S) if and only if x a (S) ǫ , which is ǫ x ∈ ∈ ǫ equivalent to a (x)(S) ǫ , proving the lemma. I (A ) (I ) . ∈ x ǫ ⊆ ǫ ⊆ ǫ ǫ Corollary 1. Under the same assumptions, with (ǫ) where m(ǫ) is the (possibly inﬁnite) length of ak x 1 (0 < x 1) we have ǫ ≥ ≤ and A = ak k m(ǫ) . Then, n { | ≤ } λ(S)xǫ x λ(S)ǫ ≤ m(ǫ) (ǫ) n λ (x)S λ (a )S , λ(S)xǫ x λ(S)ǫ . ∼  k  ≥ x∈I ! k=0 [ǫ ǫ [ ǫ Lemma 4. Let S Rn be a Borel set and let ǫ> 0.   with respect to ǫ. For a sequence a ⊆ 1, k ∼k m Proof. The inclusions

λ(S)ǫak m mλ(S)ǫ. ǫ ∼ Iǫ (A )ǫ (Iǫ)ǫ Xk=1 ⊆ ⊆

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Figure 10: Relative deviations of ln Nǫ from linear regression with respect to ln ǫ for (α, β)-bi-fractals of dimension 3/4 with α =4/5 (dashed grey line), α =4/3 (dashed black line)− and α = 3 (dotted grey line). imply, by triangle inequality, (m + 1)−α 2m ǫ m−α 1 ≤ 2 ≤ 1 ǫ ǫ (A )ǫ (Iǫ)ǫ (A )2ǫ. such that ⊆ ⊆ −1 m1(ǫ) m2(ǫ) ǫ 1+α . Hence, ∼ ∼

Proof. The existence of m1 and the required m(ǫ) (ǫ) asymptotic behaviour of m1 follows from the fact (x)S (ak )S , that ∈ ! ⊆   x Iǫ ǫ k=0 −α −α −(α+1) [ [ 2ǫ n (n + 1) n n .   − ∼ and by monotonicity of the Lebesgue measure, and −α For m2 we observe that 2ǫ ﬁts between m1 and applying Corollary 1 we have −α (m1 + 1) , so we can construct

m(ǫ) −α −α n (ǫ) (m1(ǫ)+1) (m1 + 1) λ (x)S 2 λ (a )S m2(ǫ)= +1. ≤  k  2ǫ ≤ 2ǫ x∈Iǫ ! k=0 [ ǫ [ ǫ   −α By the same argument applied to (Aǫ) (I ) , Suppose 2ǫm2 >m1 . Then ǫ ⊆ ǫ ǫ we have − (m + 1) α m−α < 2ǫ 1 +1 m(ǫ) 1 2ǫ (ǫ) λ (a )S 2nλ (x)S , −α  k  ≤ = (m1 + 1) +2ǫ k=0 x∈Iǫ ! [ ǫ [ ǫ m−α   ≤ 1 proving the lemma. so we have arrived at a contradiction. Lemma 6. Let α> 0. For every ǫ> 0 there exist numbers m1(ǫ) and m2(ǫ) (denoted further by m1 Now we turn to the proofs of our main theorems. and m2) satisfying: Proof of Theorem 1. Let ǫ > 0. We apply Lemma (m + 1)−α (m + 2)−α <2ǫ m−α (m + 1)−α 6 to obtain the functions m (ǫ) and m (ǫ). 1 − 1 ≤ 1 − 1 1 2

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Figure 11: Relative deviations of ln Nǫ from linear regression with respect to ln ǫ for uniform Cantor nests of dimension 3/4 with α =4/5 (dashed grey line), α =4/3 (dashed black− line) and α = 3 (dotted grey line).

m2 We define the ǫ-tail, Tǫ(S) as the part of the nest n = (2kǫ) λ(S) 1 (4.2) consisting of ǫ-isolated components, 2k kX=0 m1 m2 m2 −α n n δ−n n δ TǫS = (k )S, ǫ k k ǫ k (4.3) ∼ · ∼ k=1 [ kX=0 kX=0 n 1+δ n− δ+1 and we define the ǫ-core, C (S) as the remaining ǫ m ǫ α+1 , ǫ ∼ 2 ∼ components applying Lemma 5 twice at step (4.1), first time to −α −α CǫS = (k )S. the infinite sequence of k for k>m1 and then

k>m1 to the finite sequence 2ǫk. At step (4.2) we apply [ Lemma 3 and to obtain (4.3) we apply Proposition Now, we have that 2 and Lemma 1. λ ( S) = λ (T S) + λ (C S) . For the tail, we have Fα ǫ ǫ ǫ ǫ ǫ m By construction, 2ǫm2(ǫ) is an upper bound on 1 −α the Hausdorff distance of the limit set 0 of the λ (TǫS)ǫ = λ (k )S ǫ { } whole nest and the ǫ-core. Xk=1 m1 First, we find the asymptotic behaviour of the −nα = k λ(S) α (4.4) core by computing k ǫ Xk=1 m1 −α −nα α n−δ λ (CǫS)ǫ = λ (k )S k (k ǫ) ! ∼ k>m[ 1 ǫ Xk=1 m1 m2 ǫn−δ k−αδ, (4.5) λ (2kǫ)S (4.1) ∼ ∼ ! k=1 k[=0 ǫ X m2 = λ ((2kǫ)S) where we apply Lemmas 1 and 3 at step (4.4) and ǫ Proposition 2 at step (4.5). kX=0

11 Now, if αδ < 1, we have For the tail, we compute m − − − δ+1 1 n δ 1 αδ n α+1 −α λ (TǫS)ǫ ǫ m1 ǫ , λ (T S) = λ (1 k )S (4.6) ∼ ∼ ǫ ǫ − ǫ kX=1 in which case we have m1 −α n = 1 k λ(S) − −α −1 (4.7) δ +1 − ǫ(1 k ) dim F (S) . k=1 B α ≡ α +1 X n−δ− 1 m1λ(S)ǫ ǫ α+1 , (4.8) If αδ > 1, we set β = αδ 1, and therefore the ∼ ∼ ∞ −αδ − applying Lemma 3 at step (4.6) and Lemmas 1 and series k=1 k converges, so we have 4 at step (4.7). P λ (T S) ǫn−δ, For the core, we have ǫ ǫ ∼ and so the total area is − λ (C S) = λ (1 k α)S (4.9) ǫ ǫ − − − δ+1 − β k>m ! n δ n α+1 n δ α+1 1 ǫ λ ( αS)ǫ ǫ + ǫ = ǫ (1 + ǫ ), [ F ∼ m2 λ (1 2kǫ)S (4.10) proving ∼ − k=0 ! dimB Fα(S) δ. [ ǫ ≡ m2 n In the case αδ = 1 we have = (1 2kǫ) λ(S) ǫ (4.11) − 1−2kǫ k=0 δ +1 X n−δ− 1 = δ m λ(S) ǫ α+1 , (4.12) α +1 ∼ 2 ǫ ∼ so for the total area we have by applying Lemma 5 at step (4.9), Lemma 3 at step (4.10). To apply Lemma 4 at step (4.11) we λ ( S) ǫn−δ(1 ln ǫ). note that Fα ǫ ∼ − 1 m−α < 1 2kǫ 1 For any ζ > 0 we have − 1 − ≤ from the defining condition on m2 (Lemma 6) on λ ( αS)ǫ ζ m2. At step (4.12) we also apply Lemma 1. n−F(δ+ζ) ǫ (1 ln ǫ) 0 ǫ ∼ − → Thus, we have shown that for the upper dimension and 1 dim S δ + . B Oα ≡ α +1 λ ( S) Fα ǫ ǫ−ζ (1 ln ǫ) + , ǫn−(δ−ζ) ∼ − → ∞ for the lower dimension, proving 5 Closing remarks and open dimB F 1 S = δ, δ problems a Minkowski-degenerate case. In this article we have shown that an α-regular frac- Proof of Theorem 2. Let ǫ > 0. Again, we intro- tal nests based on a set S of non-degenerate box duce m1 and m2 using Lemma 6. Also, as in the counting dimension δ, has dimensions previous proof, we define the ǫ-tail and ǫ-core; the δ +1 dim ( S)= or dim ( S)= δ ǫ-tail, TǫS as the part of the nest consisting of ǫ- B Fα α +1 B Fα isolated components, and the ǫ-core, Cǫ(S) as the remaining components. for nests of centre type, with αδ < 1 for the first case and αδ 1 for the second, and we have shown Now, we have that ≥ 1 λ ( S) = λ (T S) + λ (C S) . dimB( αS)= δ + Oα ǫ ǫ ǫ ǫ ǫ O α +1

12 for the outer type. Acknowledgements These results concur with examples given in [LRZ17ˇ ] for hyper-spheres Sn−1 Rn, where I would like to thank Vedran Caˇcić,ˇ Ida DelaˇcMar- ⊆ ion, and Irina Pucićfor their helpful input and com- n−1 n ments and my wife Marija GalićMiliˇcićfor her pa- dimB αS = max n 1, , F − α +1 tience. − α All of the code used to generate the dim Sn 1 = n . B Oα − α +1 figures in this article is available at https://github.com/sinisa-milicic/nests1. We have also shown that for sets S of dimen- Fα δ sion d based on Minkowski non-degenerate sets Sδ of dimension δ we have the following relations: References 1 1 α 1, , [Ado99] Adobe Systems Incorporated. ∈ d − d PostScript language reference, third δ = dα + d 1. edition. Addison-Wesley, 1999. − These relationships allow us to study the efficacy [Bou91] Nicolas Bourbaki. General topology. of simpler numerical techniques for fractal sets pre- Chapters 1–4. Springer-Verlag, Heidel- sented in this article. berg, 1991. It is well known [Fal14, Fed69, Tri93] that the box counting dimension is not continuous with re- [Bre93] Glen E Bredon. Topology and Geom- spect to countable unions. The exact behaviour in etry (Graduate Texts in Mathematics). examples given here points to a subtler structure Springer-Verlag, New York, 1993. that explains the formulas for fractal nests. [Fal14] Kenneth Falconer. Fractal Geometry: For the outer type of nests, this has already been Mathematical Foundations and Applica- discussed in [ZˇZ05ˇ , Remark 6], but the behaviour tions, 3rd Edition. Wiley, Feb 2014. of centre-type nests remains less well understood. The formula for the centre-type nests obtained here [FC07] Jiang Feng and Shirong Chen. The points to a multiplicative operation on the dimen- Minkowski content of uniform Can- sions. Such behaviour of dimensions is well known tor set. Acta Mathematica Scientia, mostly in vector spaces for tensor products. We 27(4):641–647, 2007. propose here that there exists an abstract tensor- like product on Borel sets such that [Fed69] Herbert Federer. Geometric Measure ⊗ Theory, volume 153 of Grundlehren αS (S I) α 1 der mathematischen Wissenschaften. F ≈ × ⊗F { } Springer-Verlag, Berlin, 1969. with guaranteeing the existence of a bi-Lipschitz ≈ map between the sets. We expect the following to [Knu76] Donald E Knuth. Big omicron and hold, big omega and big theta. ACM Sigact News, 8(2):18–24, 1976. dim (A B) = dim A dim B, B ⊗ B · B [KP99] Steven G. Krantz and Harold R. Parks. along with being distributive with respect to ⊗ The Geometry of domains in space. Cartesian products (which behave additively). Birkh’:auser, Boston, 1999. Perhaps such a product could be found through an analogy between Lipschitz (or bi-Lipschitz) [LP93] Michel L Lapidus and Carl Pomerance. maps that don’t increase or preserve the box di- The Riemann zeta-function and the mension, respectively, in uniform spaces [Bou91, one-dimensional Weyl-Berry conjecture Chapter 2] and linear maps in vector spaces, pos- for fractal drums. Proceedings of the sibly using a kind of a smash product [Bre93, pg. London Mathematical Society, 3(1):41– 435]. 69, 1993.

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