Box Counting Dimensions of Generalised Fractal Nests

Box counting dimensions of generalised fractal nests Siniˇsa Miliˇcić∗ Juraj Dobrila University of Pula Faculty of Informatics 52 100 Pula, Croatia September 15, 2018 Abstract spect to the appropriate exponents. There are two general types of behaviour of fractal nests. The Fractal nests are sets defined as unions of unit n- well understood type of behaviour of fractal nests − spheres scaled by a sequence of k α for some α> 0. concerns sets that locally resemble Cartesian prod- In this article we generalise the concept to sub- ucts of fractals [LRZ17ˇ , pg. 227], [ZˇZ05ˇ , Remark sets of such spheres and find the formulas for their 6.]. In such sets, the dimension behaves naturally box counting dimensions. We introduce some novel as the sum of the dimensions of “base” sets. What classes of parameterised fractal nests and apply remains less understood are dimensions of fractal these results to compute the dimensions with re- nests of centre type. In this case, the dimension is spect to these parameters. We also show that these product-like in terms of dimensions of underlying dimensions can be seen numerically. These results elements, lacking an intuitive geometric interpreta- motivate further research that may explain the un- tion. intuitive behaviour of box counting dimensions for This article focuses on a more classical approach nest-type fractals, and in general the class of sets to the box-counting dimension, giving examples where the box-counting dimension differs from the that may further the understanding of how the box Hausdorff dimension. counting dimension behaves with respect to countable unions of similar sets. Whereas in [LRZ17ˇ ] and [ZˇZ05ˇ ] the fractal nests studied are based on (n 1)- − 1 Introduction spheres (hyper-spheres), we study fractal nests Sα based on fractal subsets of box counting dimension Research into rectifiability (eg. [Tri93]) has given δ of such spheres under similar scaling. Our results some unexpected results in the differences between are compatible with the cited ones, having them as the Hausdorff dimension [Fed69, pg. 171] and the limit cases of full dimension, δ = (n 1); notably, box counting dimension of some countable unions the dimension are − of sets and their smooth generalisations [Tri93, pg. δ +1 arXiv:1802.00870v1 [math.MG] 2 Feb 2018 121-122], such as unrectifiable spirals. Recently, dimB Sα = progress has been made in the application of the α +1 box counting dimension in the analysis of complex for the centre type and zeta functions, [LRZ17ˇ ]. We note especially the dis- covery of various interesting properties, including 1 dim S = δ + a relationship of Lapidus-style zeta functions with B α α +1 the Riemann zeta function and the generalisation of the concept of the box dimension to complex di- for the outer type. The proofs of these results hint mensions. Fractal nests, as presented and analysed at some more general geometric and topological in [LRZ17ˇ ], behave in an unexpected way with re- properties. This paper is divided into five sections. After ∗[email protected] this introduction we present the main concepts and 1 results of this paper – a dimensional analysis of in the ambient space and constructs a natural “con- generalised fractal nests, followed by applications tents” function at every dimension, in that regard of these results to novel fractal sets with numerical similar to the Hausdorff dimension and measure. examples. After that, we provide the proofs of our As ǫ-balls used here and in other literature cor- main results and in the final section we remark on respond to the Euclidean metric, we need to com- some issues and problems that naturally arise from pensate for the coefficient for volume of the ball these investigations. characteristic to this metric and dimension, 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.

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