A fractal dimension for measures via persistent homology

The paper [70] studies the total length of a minimal spanning tree for random subsets of Euclidean space. Let $X_{n}$ be a random sample of points from a compact subset of $\mathbb{R}^{d}$ according to some probability distribution. Let $M_{n}$ be the sum of all the edge lengths of a minimal spanning tree on vertex set $X_{n}$. Then for $d\geq 2$, Theorem 1 of [70] says that

where the relation $\sim$ denotes asymptotic convergence, with the ratio of the terms approaching one in the specified limit. Here, $C$ is a constant depending on $d$ and on the integral $\int f^{(d-1)/d}$, where $f$ is the density of the absolutely continuous part of the probability distribution¹¹1If the compact subset has Hausdorff dimension less than $d$, then [70] implies $C=0$.. There has been a wide variety of related work, including for example [5, 6, 7, 42, 71, 72, 73, 74]. See [45] for a version of the central limit theorem in this context. The papers [58, 59] study the length of the longest edge in the minimal spanning tree for points sampled uniformly at random from the unit square, or from a torus of dimension at least two, and [47] extends this to any Ahlfors regular measure with connected support (i.e., to any connected semi-uniform metric measure space). By contrast, [46] studies Euclidean minimal spanning trees built on extremal finite subsets, as opposed to random subsets.

As Yukich explains in his book [79], there are a wide variety of Euclidean functionals, such as the length of the minimal spanning tree, the length of the traveling salesperson tour, and the length of the minimal matching, which all have scaling asymptotics analogous to (1). To prove such results, one needs to show that the Euclidean functional of interest satisfies translation invariance, subadditivity, superadditivity, and continuity, as in [22, Page 4]. Superadditivity does not always hold, for example it does not hold for the minimal spanning tree length functional, but there is a related “boundary minimal spanning tree functional” that does satisfy superadditivity. Furthermore, the boundary functional has the same asymptotics as the original functional, which is enough to prove scaling results. It is intriguing to ask if these techniques will work for functionals defined using higher-dimensional homology.

In this paper we consider simplicial complexes (say Vietoris–Rips or Čech) with randomly sampled points as the vertex set. The 1-skeleta of these simplicial complexes are random geometric graphs. We recommend the book [57] by Penrose as an introduction to random geometric graphs; related families of random graphs are also considered in [60]. Random geometric graphs are often studied when the scale parameter $r(n)$ is a function of the number of vertices $n$, with $r(n)$ tending to zero as $n$ goes to infinity. Instead, in this paper we are more interested in the behavior over all scale parameters simultaneously. From a slightly different perspective, the paper [44] studies the expected Euler characteristic of the union of randomly sampled balls (potentially of varying radii) in the plane.

Vanessa Robins’ thesis [65] contains many related ideas; we describe one such example here. Given a set $X\subseteq\mathbb{R}^{m}$ and a scale parameter $\varepsilon\geq 0$, let

denote the $\varepsilon$-offset of $X$. The $\varepsilon$-offset of $X$ is equivalently the union of all closed $\varepsilon$ balls centered at points in $X$. Furthermore, let $C(X_{\varepsilon})\in\mathbb{N}$ denote the number of connected components of $X_{\varepsilon}$. In Chapter 5, Robins shows that for a generalized Cantor set $X$ in $\mathbb{R}$ with Lebesgue measure 0, the box-counting dimension of $X$ is equal to the limit

Here Robins considers the entire Cantor set, whereas we study random subsets thereof.

The paper [51], which heavily influenced our work, introduces a fractal dimension defined using persistent homology. This fractal dimension depends on thickenings of the entire metric space $X$, as opposed to random or extremal subsets thereof. As a consequence, the computed dimension of some fractal shapes (such as the Cantor set cross the interval) disagrees significantly with the Hausdorff or box-counting dimension.

Schweinhart’s paper [67] takes a slightly different approach from ours, considering extremal (as opposed to random) subsets. After fixing a homological dimension $i$, Schweinhart assigns a fractal dimension to each metric space $X$ equal to the infimum over all powers $d$ such that for any finite subset $X^{\prime}\subseteq X$, the sum of the $i$-dimensional persistent homology bar lengths for $X^{\prime}$, each raised to the power $d$, is bounded. For low-dimensional metric spaces Schweinhart relates this dimension to the box counting dimension.

More recently, Divol and Polonik [27] independently obtain generalizations of [70, 79] to higher homological dimensions. In particular, they prove our Conjecture 4.3 in the case when $X$ is a cube, and remark that a similar construction holds when the cube is replaced by any convex body. Related results are obtained in two papers by Schweinhart, which are in part inspired by our work: in [69] when $X$ is a ball or sphere, and afterwards in [68] when points are sampled from a fractal according to an Ahlfors regular measure.

There is a growing literature on the topology of random geometric simplicial complexes, including in particular the homology of Vietoris–Rips and Čech complexes built on top of random points in Euclidean space [13, 43, 3]. The paper [14] shows that for $n$ points sampled from the unit cube $[0,1]^{d}$ with $d\geq 2$, the maximally persistent cycle in dimension $1\leq k\leq d-1$ has persistence of order $\Theta((\frac{\log n}{\log\log n})^{1/k})$, where the asymptotic notation big Theta means both big O and big Omega. The homology of Gaussian random fields is studied in [4], which gives the expected $k$-dimensional Betti numbers in the limit as the number of points increases to infinity, and also in [12]. The paper [30] studies the number of simplices and critical simplices in the alpha and Delaunay complexes of Euclidean point sets sampled according to a Poisson process. An open problem about the birth and death times of the points in a persistence diagram coming from sublevelsets of a Gaussian random field is stated in Problem 1 of [29]. The paper [19] shows that the expected persistence diagram, from a wide class of random point clouds, has a density with respect to the Lebesgue measure. We refer the reader also to [41, 55], which are related to our Conjecture 6.2 in the setting of point processes.

The paper [16] explores what attributes of an algebraic variety can be estimated from a random sample, such as the variety’s dimension, degree, number of irreducible components, and defining polynomials; one of their estimates of dimension is inspired by our work.

In an experiment in [1], persistence diagrams are produced from random subsets of a variety of synthetic metric space classes. Machine learning tools, with these persistence diagrams as input, are then used to classify the metric spaces corresponding to each random subset. The authors obtain high classification rates between the different metric spaces. It is likely that the discriminating power is based not only on the underlying homotopy types of the shape classes, but also on the shapes’ dimensions as detected by persistent homology.

The concept of fractal dimension was introduced by Hausdorff and others [40, 15, 31] to describe spaces like the Cantor set. It was later popularized by Mandelbrot [52], and found extensive application in the study of dynamical systems. The attracting sets of a simple dynamical system is often a submanifold, with an obvious dimension, but in non-linear and chaotic dynamical systems the attracting set may not be a manifold. The Cantor set, defined by removing the middle third from the interval $[0,1]$, and then recursing on the remaining pieces, is a typical example. It has the same cardinality as $\mathbb{R}$, but it is nowhere-dense, meaning it at no point resembles a line. The typical fractal dimension of the Cantor set is $\log_{3}(2)$. Intuitively, the Cantor set has “too many” points to have dimension zero, but also should not have dimension one.

We speak of fractal dimensions in the plural because there are many different definitions. In particular, fractal dimensions can be divided into two classes, which have been called “metric” and “probabilistic” [33]. The former describe only the geometry of a metric space. Two widely-known definitions of this type, which often agree on well-behaved fractals, but are not in general equal, are the box-counting and Hausdorff dimensions. For an inviting introduction to fractal dimensions see [32]. Dimensions of the latter type take into account both the geometry of a given set and a probability distribution supported on that set—originally the “natural measure” of the attractor given by the associated dynamical system, but in principle any probability distribution can be used. The information dimension is the best known example of this type. For detailed comparisons, see [34]. Our persistent homology fractal dimension, Definition 4.1, is of the latter type.

For completeness, we exhibit some of the common definitions of fractal dimension. The primary definition for sets is given by the Hausdorff dimension [35].

The Hausdorff dimension of the Cantor set, for example, is $\log_{3}(2)$.

In practice it is difficult to compute the Hausdorff dimension of an arbitrary set, which has led to a number of alternative fractal dimension definitions in the literature. These dimensions tend to agree on well-behaved fractals, such as the Cantor set, but they need not coincide in general. Two worth mentioning are the box-counting dimension, which is relatively simple to define, and the correlation dimension.

The box-counting definition is unchanged if $N_{\epsilon}$ is instead defined by taking the number of open balls of radius $\varepsilon$, or the number of sets of diameter at most $\varepsilon$, or (for $S$ a subset of $\mathbb{R}^{n}$) the number of cubes of side-length $\varepsilon$ [77, Definition 7.8], [32, Equivalent Definitions 2.1]. It can be shown that $\dim_{B}(S)\geq\dim_{H}(S)$. This inequality can be strict; for example if $S=\mathbb{Q}\cap[0,1]$ is the set of all rational numbers between zero and one, then $\dim_{H}(S)=0<1=\dim_{B}(S)$ [32, Chapter 3]. If $S$ is a self-similar shape that is nice enough, i.e. satisfies an “open set” condition, then [32, Theorem 9.3] (for example) shows that the box-counting and Hausdorff dimensions agree: $\dim_{B}(S)=\dim_{H}(S)$.

In Section 4 we introduce a fractal dimension based on persistent homology which shares key similarities with the Hausdorff and box-counting dimensions. It can also be easily estimated via log-log plots, and it is defined for arbitrary metric spaces (though our examples will tend to be subsets of Euclidean space). A key difference, however, will be that ours is a fractal dimension for measures, rather than for subsets.

There are a variety of classical notions of a fractal dimension for a measure, including the Hausdorff, packing, and correlation dimensions of a measure [32, 61, 25]. We give the definitions of two of these.

We have $\dim_{H}(\mu)\leq\dim_{H}(\mathrm{supp}(\mu))$, and it is possible for this inequality to be strict [32, Exercise 3.10]²²2See also [33] for an example of a measure whose information dimension is less than the Hausdorff dimension of its support.. We also give the definition of the correlation dimension of a measure.

In [37, 38] it is shown that the correlation dimension gives a lower bound on the Hausdorff dimension of a measure. The correlation dimension can be easily estimated from a log-log plot, similar to the methods we use in Section 5. A different definition of the correlation dimension is given and studied in [24, 53]. The correlation dimension is a particular example of the family of Rènyi dimensions, which also includes the information dimension as a particular case [63, 64]. A collection of possible axioms that one might like to have such a fractal dimension satisfy is given in [53].

The field of applied and computational topology has grown rapidly in recent years, with the topic of persistent homology gaining particular prominence. Persistent homology has enjoyed a wealth of meaningful applications to areas such as image analyis, chemistry, natural language processing, and neuroscience, to name just a few examples [2, 10, 21, 26, 49, 50, 78, 80]. The strength of persistent homology lies in its ability to characterize important features in data across multiple scales. Roughly speaking, homology provides the ability to count the number of independent $k$-dimensional holes in a space, and persistent homology provides a means of tracking such features as the scale increases. We provide a brief introduction to persistent homology in this preliminaries section, but we point the interested reader to [8, 28, 39] for thorough introductions to homology, and to [17, 23, 36] for excellent expository articles on persistent homology.

Geometric complexes, which are at the heart of the work in this paper, associate to a set of data points a simplicial complex—a combinatorial space that serves as a model for an underlying topological space from which the data has been sampled. The building blocks of simplicial complexes are called simplices, which include vertices as 0-simplices, edges as 1-simplices, triangles as 2-simplices, tetrahedra as 3-simplices, and their higher-dimensional analogues as $k$-simplices for larger values of $k$. An important example of a simplicial complex is the Vietoris–Rips complex.

In constructing the Vietoris–Rips simplicial complex we translate our collection of points in $X$ into a higher-dimensional complex that models topological features of the data. See Figure 1 for an example of a Vietoris–Rips complex constructed from a set of data points, and see [28] for an extended discussion.

It is readily observed that for various data sets, there is not necessarily an ideal choice of the scale parameter so that the associated Vietoris–Rips complex captures the desired features in the data. The perspective behind persistence is to instead allow the scale parameter to increase and to observe the corresponding appearance and disappearance of topological features. To be more precise, each hole appears at a certain scale and disappears at a larger scale. Those holes that persist across a wide range of scales often reflect topological features in the shape underlying the data, whereas the holes that do not persist for long are often considered to be noise. However, in the context of this paper (estimating fractal dimensions), the holes that do not persist are perhaps better described as measuring the local geometry present in a random finite sample.

For a fixed set of points, we note that as scale increases, simplices can only be added and cannot be removed. Thus, for $r_{0}<r_{1}<r_{2}<\cdots$, we obtain a filtration of Vietoris–Rips complexes

The associated inclusion maps induce linear maps between the corresponding homology groups $H_{k}(\mathrm{VR}(X;r_{i}))$, which are algebraic structures whose ranks (roughly speaking) count the number of independent $k$-dimensional holes in the Vietoris–Rips complex. A technical remark is that homology depends on the choice of a group of coefficients; it is simplest to use field coefficients (for example $\mathbb{R}$, $\mathbb{Q}$, or $\mathbb{Z}/p\mathbb{Z}$ for $p$ prime), in which case the homology groups are furthermore vector spaces. The corresponding collection of vector spaces and linear maps is called a persistent homology module.

A useful tool for visualizing and extracting meaning from persistent homology is a barcode. The basic idea is that each generator of persistent homology can be represented by an interval, whose start and end times are the birth and death scales of a homological feature in the data. These intervals can be arranged as a barcode graph in which the $x$-axis corresponds to the scale parameter. See Figure 2 for an example. If $Y$ is a finite metric space, then we let $\mathrm{PH}^{i}(Y)$ denote the corresponding collection of $i$-dimensional persistent homology intervals. Indeed, any persistent homology module decomposes uniquely as a direct sum of interval summands.

Zero-dimensional barcodes always produce one infinite interval, as in Figure 2, which are problematic for our purposes. Therefore, in the remainder of this paper we will always use reduced homology, which has the effect of simply eliminating the infinite interval from the 0-dimensional barcode while leaving everything else unchanged. As a consequence, there will never be any infinite intervals in the persistent homology of a Vietoris–Rips simplicial complex, even in homological dimension zero.

We display several experimental results, for shapes of both integral and non-integral fractal dimension. In Figure 3, we show the log-log plots of $L^{i}(X_{n})$ as a function of $n$, where $X_{n}$ is sampled uniformly at random from a disk, a square, and an equilateral triangle, each of unit area in the plane $\mathbb{R}^{2}$. Each of these spaces constitutes a manifold of dimension two, and we thus expect these shapes to have persistent homology fractal dimension $d=2$ as well. Experimentally, this appears to be the case, both for homological dimensions $i=0$ and $i=1$. Indeed, our asymptotically estimated slopes lie in the range $0.49$ to $0.54$, which is fairly close to the expected slope of $\frac{d-1}{d}=\frac{1}{2}$.

In Figure 4 we perform a similar experiment for the cube in $\mathbb{R}^{3}$ of unit volume. We expect the cube to have persistent homology fractal dimension $d=3$, corresponding to a slope in the log-log plot of $\frac{d-1}{d}=\frac{2}{3}$. This appears to be the case for homological dimension $i=0$, where the slope is approximately $0.65$. However, for $i=1$ and $i=2,$ our estimated slope is far from $\frac{2}{3}$, perhaps because our computational limits do not allow us to take $n$, the number of randomly chosen points, to be sufficiently large.

In Figure 5 we use log-log plots to estimate some persistent homology fractal dimensions of the Cantor set cross the interval (expected dimension $d=1+\log_{3}(2)$), of the Sierpiński triangle (expected dimension $d=\log_{2}(3)$), of Cantor dust in $\mathbb{R}^{2}$ (expected dimension $d=\log_{3}(4)$), and of Cantor dust in $\mathbb{R}^{3}$ (expected dimension $d=\log_{3}(8)$). As noted in Section 3, various notions of fractal dimension tend to agree for well-behaved fractals. Thus, in each case above, we provide the Hausdorff dimension $d$ in order to define an expected persistent homology fractal dimension. The Hausdorff dimension is well-known for the Sierpiński triangle, Cantor dust in $\mathbb{R}^{2}$, and Cantor dust in $\mathbb{R}^{3}.$ The Hausdorff dimension for the Cantor set cross the interval can be shown to be $1+\log_{3}(2),$ which follows from [32, Theorem 9.3] or [54, Theorem III]. In Section 5.2 we define these fractal shapes in detail, and we also explain our computational technique for sampling points from them at random.

Summarizing the experimental results for self-similar fractals, we find reasonably good estimates of fractal dimension for homological dimension $i=0.$ More specifically, for the Cantor set cross the interval, we expect $\frac{d-1}{d}\approx 0.3869$, and we find slope estimates from a linear fit of all data and an asymptotic fit to be $0.3799$ and $0.36488$, respectively. In the case of the Sierpiński triangle, the estimate is quite good: we expect $\frac{d-1}{d}\approx 0.3691$, and the slope estimates from both a linear fit and an asymptotic fit are approximately $0.37$. Similarly, the estimates for Cantor dust in $\mathbb{R}^{2}$ and $\mathbb{R}^{3}$ are close to the expected values: (1) For Cantor dust in $\mathbb{R}^{2}$, we expect $\frac{d-1}{d}\approx 0.2075$ and estimate $\frac{d-1}{d}\approx 0.25$. (2) For Cantor dust in $\mathbb{R}^{3}$, we expect $\frac{d-1}{d}\approx 0.4717$ and estimate $\frac{d-1}{d}\approx 0.49$. For $i>0$ many of these estimates of the persistent homology fractal dimension are not close to the expected (Hausdorff) dimensions, perhaps because the number of points $n$ is not large enough. The theory behind these experiments has now been verified in [68].

It is worth commenting on the Cantor set, which is a self-similar fractal in $\mathbb{R}$. Even though the Hausdorff dimension of the Cantor set is $\log_{3}(2)$, it is not hard to see that the $0$-dimensional persistent homology fractal dimension of the Cantor set is $1$. This is because as $n\to\infty$ a random sample of points from the Cantor set will contain points in $\mathbb{R}$ arbitrarily close to 0 and to 1, and hence $L_{0}(X_{n})\to 1$ as $n\to\infty$. This is not surprising—we do not necessarily expect to be able to detect a fractional dimension less than one by using minimal spanning trees (which are $1$-dimensional graphs). For this reason, if a measure $\mu$ is defined on a subset of $\mathbb{R}^{m}$, we sometimes restrict attention to the case $m\geq 2$. See Figure 6 for our experimental computations on the Cantor set.

Finally, we include one example with data drawn from a two-dimensional manifold in $\mathbb{R}^{3}$. We sample points from a torus with major radius 5 and minor radius 3. We expect the persistent homology fractal dimensions to be 2, and this is supported in the experimental evidence for 0-dimensional homology shown in Figure 7 with approximate slope $\frac{d-1}{d}=\frac{1}{2}$.

The Cantor set $C=\cap_{l=0}^{\infty}C_{l}$ is a countable intersection of nested sets $C_{0}\supseteq C_{1}\supseteq C_{2}\supseteq\cdots$, where the set $C_{l}$ at level $l$ is a union of $2^{l}$ closed intervals, each of length $\frac{1}{3^{l}}$. More precisely, $C_{0}=[0,1]$ is the closed unit interval, and $C_{l}$ is defined recursively via

In our experiment for the Cantor set (Figure 6), we do not sample from the Cantor distribution on the entire Cantor set $C$, but instead from the left endpoints of level $C_{l}$ of the Cantor set, where $l$ is chosen to be very large (we use $l=100{,}000$). More precisely, in order to sample points, we choose a binary sequence $\{a_{i}\}_{i=1}^{l}$ uniformly at random, meaning that each term $a_{i}$ is equal to either $0$ or $1$ with probability $\frac{1}{2}$, and furthermore the value $a_{i}$ is independent from the value of $a_{j}$ for $i\neq j$. The corresponding random point in the Cantor set is $\sum_{i=1}^{l}\frac{2a_{i}}{3^{i}}$. Note that this point is in $C$ and furthermore is the left endpoint of some interval in $C_{l}$. So we are selecting left endpoints of intervals in $C_{l}$ uniformly at random, but since $l$ is large this is a good approximation to sampling from the entire Cantor set according to the Cantor distribution.

We use a similar procedure to sample at random for our experiments on the Cantor set cross the interval, on Cantor dust in $\mathbb{R}^{2}$, on Cantor dust in $\mathbb{R}^{3}$, and on the Sierpiński triangle (Figure 5). The Cantor set cross the interval is $C\times[0,1]\subseteq\mathbb{R}^{2}$, equipped with the Euclidean metric. We computationally sample by choosing a point from $C_{l}$ as described in the paragraph above for $l=100{,}000$, and by also sampling a point from the unit interval $[0,1]$ uniformly at random. Cantor dust is the subset $C\times C$ of $\mathbb{R}^{2}$, which we sample by choosing two points from $C_{l}$ as described previously. The same procedure is done for the Cantor dust $C\times C\times C$ in $\mathbb{R}^{3}$. The Sierpiński triangle $S\subseteq\mathbb{R}^{2}$ is defined in a similar way to the Cantor set, with $S=\cap_{l=0}^{\infty}S_{l}$ a countable intersection of nested sets $S_{0}\supseteq S_{1}\supseteq S_{2}\supseteq\cdots$. Here each $S_{l}$ is a union of $3^{l}$ triangles. We choose $l=100{,}000$ to be large, and then sample points uniformly at random from the bottom left endpoints of the triangles in $S_{l}$. More precisely, we choose a ternary sequence $\{a_{i}\}_{i=1}^{l}$ uniformly at random, meaning that each term $a_{i}$ is equal to either $0$, $1$, or $2$ with probability $\frac{1}{3}$. The corresponding random point in the Sierpiński triangle is $\sum_{i=1}^{l}\frac{1}{2^{i}}\vec{v}_{i}\in\mathbb{R}^{2}$, where vector $\vec{v}_{i}$ is given by

Note this point is in $S$ and furthermore is the bottom left endpoint of some triangle in $S_{l}$.

In the case where $\mu$ is the uniform distribution on the unit interval $[0,1]$, then a weaker version of Conjecture 6.1 (convergence distribution-wise) is known to be true, and furthermore a formula for the limiting rescaled distribution is known. If $X_{n}$ is a subset of $[0,1]$ drawn uniformly at random, then (with probability one) the points in $X_{n}$ divide $[0,1]$ into $n+1$ pieces. The joint probability distribution function for the lengths of these pieces is given by the flat Dirichlet distribution, which can be thought of as the uniform distribution on the $n$-simplex (the set of all $(t_{0},\ldots,t_{n})$ with $t_{i}\geq 0$ for all $i$, such that $\sum_{i=0}^{n}t_{i}=1$). Note that the intervals in $\mathrm{PH}^{0}(X_{n})$ have lengths $t_{1},\ldots,t_{n-1}$, omitting $t_{0}$ and $t_{n}$ which correspond to the two subintervals on the boundary of the interval.

The probability distribution function of each $t_{i}$, and therefore of each interval length in $\mathrm{PH}^{0}(X_{n})$, is the marginal of the Dirichlet distribution, which is given by the Beta distribution $B(1,n)$ [11]. After simplifying, the true cumulative distribution function (which we denote by $F_{n}^{(0)}$ instead of the empirical cumulative distribution function $\hat{F}_{n}^{(0)}$) of $B(1,n)$ is given by  [66]

As $n$ goes to infinity, $F^{(0)}_{n}(t)$ converges pointwise to the constant function 1. However, after rescaling, $F^{(0)}_{n}(n^{-1}t)$ converges to a more interesting distribution independent of $n$. Indeed, we have $F^{(0)}_{n}\left(\tfrac{t}{n}\right)=1-(1-\tfrac{t}{n})^{n}$, and the limit as $n\to\infty$ is

This is the cumulative distribution function of the exponential distribution with rate parameter one. Therefore, the rescaled interval lengths in the limit as $n\to\infty$ are distributed according to the exponential distribution $\mathrm{Exp}(1)$.

We now move to the case where $\mu$ is the uniform distribution on the unit square in $\mathbb{R}^{2}$. It is known that the sum of the edge lengths of the minimal spanning tree, given by $L^{0}(X_{n})$ where $X_{n}$ is a random sample of $n$ points from the unit square, converges as $n\to\infty$ to $Cn^{1/2}$, for a constant $C$ [70]. However, to our knowledge the limiting distribution of all (rescaled) edge lengths is not known. We instead analyze this example empirically. The experiments in Figure 9 suggest that as $n$ increases, it is plausible that both $F_{n}^{(0)}(n^{-1/2}t)$ and $F_{n}^{(1)}(n^{-1/2}t)$ converge point-wise to a limiting probability distribution. We have tried to fit these limiting probability distributions to standard distributions, without yet having found obvious candidates.

In this section we give experimental evidence that the assumption of being a rescaled Lebesgue measure in Conjecture 6.1 is necessary. Our example computation is done on a separated Sierpiński triangle.

For a given separation value $\delta\geq 0$, the separated Sierpiński triangle can be defined as the set of all points in $\mathbb{R}^{2}$ of the form $\sum_{i=1}^{\infty}\frac{1}{(2+\delta)^{i}}\vec{v}_{i}$, where each vector $\vec{v}_{i}\in\mathbb{R}^{2}$ is either $(0,0)$, $(1,0)$, or $(\frac{1}{2},\frac{\sqrt{3}}{2})$. The Hausdorff dimension of this self-similar fractal shape is $\log_{2+\delta}(3)$ ([32, Theorem 9.3] or [54, Theorem III]), and note that when $\delta=0$, we recover the standard (non-separated) Sierpiński triangle. See Figure 10 for a picture when $\delta=2$. Computationally, when we sample a point from the separated Sierpiński triangle, we sample a point of the form $\sum_{i=1}^{l}\frac{1}{(2+\delta)^{i}}\vec{v}_{i}$, where in our experiments we use $l=100{,}000$.

In the following experiment we sample random points from the separated Sierpiński triangle with $\delta=2$. As the number of random points $n$ goes to infinity, it appears that the rescaled³³3Since the separated Sierpiński triangle has Hausdorff dimension $\log_{2+\delta}(3)$, the rescaled distributions we plot are $F_{n}^{(0)}(n^{-1/m}t)$ with $m=\log_{2+\delta}(3)$. CDF of $H_{0}$ interval lengths are not converging to a fixed probability distribution, but instead to a periodic family of distributions, in the following sense. If you fix $k\in\mathbb{N}$ then the distributions on $n=k,3k,9k,27k,\ldots,3^{j}k,\ldots$ points appear to converge as $j\to\infty$ to a fixed distribution. Indeed, see Figure 11 for the limiting distribution on $3^{j}$ points, and for the limiting distribution on $3^{j}\cdot 2$ points. However, the limiting distribution for $3^{j}k$ points and the limiting distribution for $3^{j}k^{\prime}$ points appear to be the same if and only if $k$ and $k^{\prime}$ differ by a power of $3$. See Figure 12, which shows four snapshots from one full periodic orbit.

Here is an intuitively plausible explanation for why the rescaled CDFs for the separated Sierpiński triangle converge to a periodic family of distributions, rather than a fixed distribution: Imagine focusing a camera at the origin of the Sierpiński triangle and zooming in. Once you get to $(2+\delta)\times$ magnification, you see the same image again. This is one full period. However, for magnifications between $1\times$ and $(2+\delta)\times$ you see a different image. In our experiments sampling random points, zooming in by a factor of $(2+\delta)\times$ is the same thing as sampling three times as many points (indeed, the Hausdorff dimension is $\log_{2+\delta}(3)$). When zooming in you see the same image only when the magnification is at a multiple of $2+\delta$, and analogously when sampling random points perhaps we should expect to see the same probability distribution of interval lengths only when the number of points is multiplied by a power of 3.