2. Relation with percolation thresholds

We start by recalling some known results concerning the percolation thresholds $p_c=p_c(M)$ of fractal percolation in the plane. Chayes, Chayes and Durrett [Reference Chayes, Chayes and Durrett6] have already established that, for any $M\in {\mathbb N}_{\geq 2}$,

(2.1) \begin{align} \sqrt{1/M}\leq p_c(M)\leq 0.9999.\end{align}

In [Reference Chayes and Chayes5] it is shown that the percolation threshold $p_{c,\text{NN}}$ of site percolation on the nearest neighbor (NN) graph on ${\mathbb Z}^2$ is a lower bound, i.e. $p_{c,\text{NN}}\leq p_c(M)$ for any $M\in{\mathbb N}_{\geq 2}$. Since $ 0.556\leq p_{c,\text{NN}}$ (cf. [Reference Van den Berg and Ermakov27]), this improves the above lower bound for any $M\geq 4$. Moreover, $\lim_{M\to\infty} p_c(M)=p_{c,\text{NN}}$; see [Reference Chayes and Chayes5]. It is believed that $p_c(M')\leq p_c(M)$ for $M'\geq M$, but this monotonicity has been established only in special cases, e.g. if $M'=M^2$. These bounds have been improved in [Reference Don8, Reference White28] for some small values of M. The best known bounds for $M=2$ and $M=3$ are

(2.2) \begin{align} 0.881\leq p_c(2)\,\leq\, 0.993 \quad \text{ and }\quad 0.784\leq p_c(3)\leq 0.940,\end{align}

respectively, and $p_c(4)\leq 0.972$; cf. [Reference Don8].

In view of the aforementioned observations in [Reference Klatt, Schröder-Turk and Mecke12,Reference Mecke and Seyfried19–Reference Neher, Mecke and Wagner21], that the zero of the expected Euler characteristic per site is close to the percolation thresholds in many percolation models, let us now discuss the connections between the limit functionals for F introduced above and the connectivity properties in fractal percolation.

$p_0$ is a lower bound for $p_c$. Our first observation is that, for any $M\in{\mathbb N}_{\geq 2}$, the function $p\mapsto \overline{\mathcal V}_0(F(p))$ has a unique zero $p_0=p_0(M)$ in the open interval $(1/M^2,1)$, as suggested by Figure 4 (left). Moreover, $\overline{\mathcal V}_0(F(p))>0$ for $p<p_0$ and $\overline{\mathcal V}_0(F(p))<0$ for $p>p_0$. So far, this is in accordance with the observations in classical percolation models. Since $p_c>\frac12$, one would by analogy expect $p_0$ to be an upper bound for $p_c$. This is also what the naive heuristics suggests for our model: below $p_c$ the limit set F is totally disconnected and therefore, if $\overline{\mathcal V}_0(F)$ is naively interpreted as the ‘Euler characteristic’ of F (i.e., as #components – #holes), then it should be positive for all $p<p_c$. By comparing $p_0$ with the known lower bounds for $p_c$, we find in contrast that $p_0(M)$ is not an upper bound but a lower bound for $p_c(M)$, i.e.

\begin{align*} p_0(M)\leq p_c(M) \quad \text{ for all } M\in{\mathbb N}_{\geq 2}.\end{align*}

Indeed, for $M=2$ and $M=3$, $p_0(M)$ is below the lower bounds for $p_c(M)$ given by Don [Reference Don8] (cf. (2.2)), while for $M\geq 4$, $p_0(M)\leq 0.556$, which is the lower bound for the site percolation threshold $p_{c,\text{NN}}$ due to van den Berg and Ermakov [Reference Van den Berg and Ermakov27]. Although $p_0$ is a lower bound for $p_c$, unfortunately it is not very tight. In particular, it does not improve the known bounds.

The large-M limit of $p_0(M)$. In analogy with $p_c$, for which $p_c(M)\to p_{c,\text{NN}}$ as $M\to\infty$, we observe that the zeros $p_0(M)$ also converge to a limit as $M\to\infty$. The (pointwise) limit of the functions $p\mapsto\overline{\mathcal V}_0(F(p))$, as $M\to \infty$, is the function v given by

\begin{equation*} v(p)\,:\!=1-4p+4p^2-p^3, \qquad p\in(0,1], \end{equation*}

which is the red curve depicted in Figure 5 (left). It turns out that v(p) coincides (up to a factor p) with the mean Euler characteristic per site $\overline{V}_0({\mathbb Z}^{2,\text{NNN}};\,p)$ of site percolation on the next-to-nearest neighbor (NNN) graph on ${\mathbb Z}^2$; cf. [Reference Neher, Mecke and Wagner21]. In particular, this implies for the zeros that

\begin{align*} \lim_{M\to\infty} p_0(M)=p_{0,\text{NNN}}, \end{align*}

where $p_{0,\text{NNN}}=(3-\sqrt{5})/2$ is the unique zero of v in (0,1), i.e. of $p\mapsto \overline{V}_0({\mathbb Z}^{2,\text{NNN}};\,p)$. At first glance it might be surprising that a different site percolation model appears in the limit (NNN instead of NN, which showed up for the percolation thresholds). But this is consistent with the discussion before Theorem 1.3—there is too much connectivity in the approximation sets $F_n$. We will get back to this in a moment.

Figure 5: For increasing values of the number of subdivisions M (color-coded), the rescaled expected Euler characteristic of fractal percolation (left) and its complement (right) are plotted as functions of p. The limiting curve for $M\to\infty$ corresponds to the mean Euler characteristic per site (rescaled by the intensity) of site percolation on ${\mathbb Z}^2$ with eight or four neighbors, respectively.

$p_{\min}$—a bound for $p_c$? For any $M\in{\mathbb N}_{\geq 2}$, the function $p\mapsto \overline{\mathcal V}_0(F_p)$ has a unique minimum $p_{\min}=p_{\min}(M)$ in the open interval $(1/M^2,1)$, which lies always to the right of $p_0$ (i.e., potentially closer to $p_c$). This is another natural candidate to bound the percolation threshold. At $p_{\min}$ the geometry is extremal in the sense that, intuitively speaking, the ‘growth speed’ of the number of holes equals exactly the ‘growth speed’ of the number of clusters. For $M=2$, $p_{\min}$ is clearly a lower bound for $p_c(2)$, but as $M\to\infty$, $p_{\min}(M)\to 2/3$, which is above $p_{c,\text{NN}}$. So, for large M, $p_{\min}(M)$ is clearly not a lower bound for $p_c(M)$. This implies that $p_{\min}(M)$ can neither be a general lower nor a general upper bound for the percolation thresholds. Interesting open questions are at which values of M the quantities $p_c(M)$ and $p_{\min}(M)$ change their order, and whether $p_{\min}(M)$ (which can be interpreted as the parameter for which the difference between number of holes and the number of connected components is maximal) is related in some way to the percolation transition.

As the discussion before Theorem 1.3 suggests, there might be approximation sequences for F which better capture the percolative behavior of F, and one candidate is the sequence of the modified sets $F_n$ with all diagonal connections between cubes removed, which we studied by looking at the closed complements $C_n\,:\!=\overline{J\setminus F_n}$. Let us now discuss possible connections with percolation thresholds of the corresponding limit functionals $\overline{\mathcal V}^c_0(F)$.

$p_1$—a lower bound for $p_c$? Figure 4 (right) shows plots of the functions $p\mapsto-\overline{\mathcal V}_0^c(F(p))$ for different M (the black curves labeled ‘$n=\infty$’), again accompanied by some finite approximations for different n. In Figure 5 (right) there are plots of the functions $p\mapsto-\overline{\mathcal V}_0^c(F(p))$ for all M together with the limit curve as $M\to\infty$. Each of these curves possesses again a unique zero $p_1=p_1(M)$ in $(1/M^2,1)$. It is apparent from the plots in Figure 4 that $p_1(M)$ is larger than $p_0(M)$ and thus potentially closer to the percolation threshold $p_c(M)$. At least for $M=2, 3$, $p_1(M)$ is a better lower bound for $p_c(M)$. But is this true in general? Unfortunately not, as will become clear from looking at large M.

Large-M limit of $p_1(M)$. The (pointwise) limit of the functions $p\mapsto-\overline{\mathcal V}_0^c(F(p))$, as $M\to \infty$, is

\begin{equation*} v^c(p)\,:\!=-(1-p)(p^2+p-1)=p^3-2p+1, \quad p\in(0,1], \end{equation*}

which is the red curve depicted in Figure 5 (right). It turns out to coincide (up to a factor p) with the mean Euler characteristic per site $\overline{V}_0({\mathbb Z}^{2,\text{NN}},p)$ of site percolation on the nearest neighbor graph on ${\mathbb Z}^2$ as a function of $p\in[0,1]$; see e.g. [Reference Neher, Mecke and Wagner21, Eq.(5), p. 4]. In particular, one gets for the zeros that

\begin{align*} \lim_{M\to\infty} p_1(M)=p_{0,\text{NN}}, \end{align*}

where $p_{0,\text{NN}}=(\sqrt{5}-1)/2\approx 0.618$ is the unique zero of $v^c$ in (0,1). Note that $p_{0,\text{NN}}$ is strictly larger than $p_{c,\text{NN}}\approx 0.59$. Thus for large M, $p_c(M)<p_1(M)$, while for $M=2,3$, one has $p_c(M)>p_1(M)$. So $p_1$ can neither be a general lower bound nor a general upper bound for $p_c$. This observation also rules out the minimum of $p\mapsto \overline{\mathcal V}^c_0(F(p))$ as a good general bound in any way.

These findings show that there is not such a close connection between the Euler characteristics and percolation thresholds in this fractal model as there are in other percolation models. An explanation of why the phase transition leaves no signature in the studied functionals might be that percolation happens in fact on lower-dimensional subsets. Recently it has been shown (see [Reference Broman, Camia, Joosten and Meester4]) that for $p\geq p_c$ (and conditioned on F being nonempty), the union Z of all connected components of F larger than one point forms almost surely a set of strictly smaller Hausdorff dimension than the remaining set $F\setminus Z$ (the dust), which has dimension $\dim_H F\setminus Z=\dim_H F=D$ almost surely. The rescaling with $r^{Dn}$ of the geometric functionals essentially means that they do not see the lower-dimensional set Z on which percolation occurs. So from the point of view of the Hausdorff dimension, our result is consistent with the findings in [Reference Broman, Camia, Joosten and Meester4]. But in [Reference Broman, Camia, Joosten and Meester4], it is also shown that in contrast the Minkowski dimensions of Z and $F\setminus Z$ coincide almost surely for $p\geq p_c$. Since our approximation of F by unions of boxes $F_n$ is related to the Minkowski (or box) dimension rather than to the Hausdorff dimension, our results support the hypothesis that, also in the Minkowski setting, the effect of the dust dominates that of the larger components, though not on the level of dimension but on the refined level of associated measures or contents as provided by our functionals. Long before percolation occurs (i.e. for $p<p_c$), the expected Euler characteristic ${\mathbb E} V_0(F_n)$ becomes negative, i.e. it detects more holes than components in the approximations $F_n$, which indicates that the nth approximation of the dust must have a lot of structure which only disappears in the limit. More refined methods are necessary to separate the dust from the larger clusters. It might for instance be worthwhile to look at the Euler characteristic of the percolation cluster in finite approximations.

3. Outlook and outline

We emphasize that, although our work is motivated by questions regarding the percolation properties, our focus here is on establishing the existence of the geometric limit functionals $\overline{\mathcal V}_k(F)$ and $\overline{\mathcal V}_k^c(F)$ (Theorems 1.1 and 6.1), and on computing them explicitly in dimension 1 (Corollaries 5.1 and 6.2) and 2 (Theorems 1.2 and 1.3 and Proposition 6.4). The methods developed here can be transferred to other random (self-similar) models. The functionals may have other applications. Just like fractal curvatures, they clearly carry geometric information beyond the fractal dimension, but unlike them, they can be computed explicitly for random sets (at least in some cases). Even more importantly, they can be estimated well from the finite approximations; see Remarks 5.2 and 6.2 for a discussion of the speed of convergence of $r^{Dn}{\mathbb E} V_0(F_n)$ and $r^{Dn}{\mathbb E} V_0(C_n)$ as $n\to\infty$, and see Remark 5.3 for a practical demonstration. Hence the functionals may serve as robust and efficient geometric descriptors in applications and may e.g. help to distinguish different geometric structures of the same fractal dimension. It is an aim of future research to further develop this ‘box counting’ approach to geometric functionals to work for general (random) fractals. In view of the fast convergence of these functionals it is another intriguing question whether a similar speed of convergence can be expected for the percolation probabilities of $F_n$. The functionals $\overline{\mathcal V}_k(F)$ describe first-order properties of the random set F in a sense that can be made precise: for $p>M^{-d}$, the almost sure convergence, as $n\to\infty$, of the random variables $r^{n(D-k)} V_k(F_{n})$ to some limit random variable $Z^k_\infty$ can be shown, and the limit of expectations $\overline{\mathcal V}_k(F)$ appears as the expectation of this limit variable $Z^k_\infty$. Moreover, the functionals $\overline{\mathcal V}_k(F)$ turn out to determine essentially the covariance structure and also higher moments of the limit variables $Z^k_\infty$. We discuss these results further in [Reference Klatt and Winter13].

Outline. The remainder of this article is organized as follows. In Section 4, we describe fractal percolation as a random self-similar set and introduce some notation and basic concepts. In Section 5 we study in detail the approximation of F by the sets $F_n$, and in Section 6 the approximation by the sets $C_n$. In both cases we first prove a general formula for arbitrary dimensions (Theorems 1.1 and 6.1), which we then use to compute the limit functionals in ${\mathbb R}$ and ${\mathbb R}^2$. A careful analysis of the model in ${\mathbb R}$ is essential for the computations in ${\mathbb R}^2$. Additionally, it is necessary to understand the intersection of two independent copies of F in ${\mathbb R}$, the analysis of which also provides a new point of view on the lower bound for $p_c$ in (2.1) obtained in [Reference Chayes, Chayes and Durrett6]; see Remark 5.4. In the course of the proofs we derive not only explicit expressions for the limit functionals but also exact formulas for the nth approximations; see in particular Remarks 5.2 and 6.2. Following another idea in the physics literature [Reference Knüfing, Schollmeyer, Riegler and Mecke15, Reference Schönhöfer and Mecke26], these formulas are also used to derive in Remark 6.3 the fractal subdimensions and the exact associated amplitudes for this model, which is an alternative approach to refined information about fractal sets beyond fractal dimension.

Finally, in Section 7 some estimates are proved which ensure the convergence of the series occurring in the main formulas in Theorems 1.1 and 6.1. They are not needed for the further results in ${\mathbb R}$ and ${\mathbb R}^2$, as the convergence can be checked directly in these cases, but ensure their validity in higher dimensions.

4. Fractal percolation as a random self-similar set

Fractal percolation F in ${\mathbb R}^d$ with parameters $p\in[0,1]$ and $M\in{\mathbb N}_{\geq 2}$ is a random self-similar set generated by the following random iterated function system (RIFS) $\mathcal{S}$ constructed on the basic set $J=[0,1]^d$. Denote by $J_1,\ldots,J_{M^d}$ the $M^d$ subcubes of side length $r=1/M$ into which J is divided in the first step of the construction of F described above. $\mathcal{S}$ is a random subset of the set $\Phi\,:\!=\{\phi_1,\ldots, \phi_{M^d}\}$, where $\phi_j$, $j=1,\ldots, M^d$, is the similarity which maps J to $J_j$ (rotation- and reflection-free, for simplicity and uniqueness). Each map $\phi_j$ is included in $\mathcal{S}$ with probability p independent of all the other maps. It is obvious that $\mathcal{S}$ satisfies the open set condition (OSC) with respect to the interior $\textsf{int}(J)$ of J, since $\mathcal{S}$ is a random subset of $\Phi$ and even the full set $\Phi$ satisfies OSC with respect to $\textsf{int}(J)$.

For obtaining F as an invariant set of the RIFS $\mathcal{S}$, we employ a Galton–Watson tree on the set of all finite words $\Sigma_*\,:\!=\bigcup_{n=0}^\infty \Sigma_n$, where $\Sigma_n\,:\!=\{1,\ldots,M^d\}^n$, $n\in{\mathbb N}_0$. In particular, $\Sigma_0=\{{\varepsilon}\}$, where ${\varepsilon}$ is the empty word of length $|{\varepsilon}|=0$. For each $\sigma\in\Sigma_*$, let $\mathcal{S}_\sigma$ be an independent copy of the RIFS $\mathcal{S}$. $\mathcal{S}_\sigma$ contains a random number $\nu_\sigma$ of maps (with $\nu_\sigma$ being binomially distributed with p and $M^d$). Let $I_\sigma\subseteq \{1,\ldots,M^d\}$ be the set of indices of the maps in $\mathcal{S}_\sigma$. It is convenient to denote these maps by $\phi_{\sigma i}$, $i\in I_\sigma$. Note that $|I_\sigma|=\nu_\sigma$. In particular, $I_\sigma$ may be empty. We build a random tree $\mathcal{T}$ in $\Sigma_*$ as follows: set $\mathcal{T}_0\,:\!=\{{\varepsilon}\}$ and for $n\in{\mathbb N}_0$, define $\mathcal{T}_{n+1}\,:\!=\emptyset$ if $\mathcal{T}_n=\emptyset$, and

\begin{equation*}\mathcal{T}_{n+1}\,:\!=\{\sigma i\,:\, \sigma\in\mathcal{T}_n, i\in I_\sigma\!\}\end{equation*}

if $\mathcal{T}_n\neq\emptyset$. Finally, we set

\begin{equation*}\mathcal{T}\,:\!=\bigcup_{n=0}^\infty \mathcal{T}_n.\end{equation*}

Here $\mathcal{T}$ can be interpreted as the population tree of a Galton–Watson process in which $\mathcal{T}_n$ represents the nth generation and $\sigma i\in\mathcal{T}_{n+1}$, $i\in I_\sigma$ are the descendants of an individuum $\sigma\in\mathcal{T}_n$. The self-similar random set associated with the RIFS $\mathcal{S}$ is the set

\begin{equation*}F\,:\!=\bigcap_{n=1}^\infty \bigcup_{\sigma\in\mathcal{T}_n} J_\sigma,\end{equation*}

where, for any $\sigma\in\Sigma_*$ of length $|\sigma|=n\in{\mathbb N}$ and any set $K\subset{\mathbb R}^d$,

\begin{equation*}K_\sigma\,:\!=\phi_{\sigma|1}\circ \phi_{\sigma|2}\circ \ldots \circ \phi_{\sigma|n}(K).\end{equation*}

Here $\sigma|k$, $k\in\{1,\ldots,n\}$, denotes the word formed by the first k letters of $\sigma$. F is called self-similar because of the following stochastic self-similarity property (which characterizes F uniquely): if $F^{(i)}$, $i\in\{0,1,\ldots,M^d\}$, are independent and identically distributed copies of F and $\mathcal{S}$ is the corresponding RIFS as above, independent of the $F^{(i)}$, then

\begin{equation*}F^{(0)}=\bigcup_{\phi_i\in\mathcal{S}} \phi_i(F^{(i)}).\end{equation*}

In the language of the tree and the associated sets considered above, the construction steps $F_{n}$, $n\in{\mathbb N}$, of the fractal percolation process are given by

\begin{equation*}F_{n}=\bigcup_{\sigma\in\mathcal{T}_n} J_\sigma.\end{equation*}

Here the sets $J_\sigma$ with $|\sigma|=n$ encode the subcubes of level n of the construction, and the above union extends over those subcubes $J_\sigma$ which have survived all the previous steps, i.e. over all $\sigma$ for which all the cubes $J_{\sigma|i}$, $i\in\{1,\ldots, n\}$, have been kept in the ith step of the construction. We also introduce, for each $j\in\{1,\ldots,M^d\}$ and each $n\in{\mathbb N}$, the set

(4.1) \begin{align} F_n^j\,:\!=\bigcup_{{\sigma\in\mathcal{T}_n},{\sigma|1=j}} J_\sigma,\end{align}

which is the union of those cubes of level n which are subcubes of $J_j=\phi_j(J)$. We will not make much use of the limit objects and their self-similarity in the sequel; we will mainly use the following basic properties of the construction steps $F_n$ and their parts $F_n^j$. For any $j\in\{1,\ldots,M^d\}$ and any $n\in{\mathbb N}$, we have

(4.2) \begin{align} F_n^j=\phi_j(\widetilde F_{n-1})\end{align}

in distribution, where $\widetilde F_{n-1}$ is the random set which equals $F_{n-1}$ with probability p and is empty otherwise (i.e., $\widetilde F_{n-1}^j= F_{n-1}^j\cap \widetilde J_j$, where $\widetilde J_j$ is a random set independent of $F_{n-1}^j$, which equals $J_j$ with probability p and is empty otherwise). The homogeneity and motion-invariance of the intrinsic volumes implies now in particular that

(4.3) \begin{align} {\mathbb E} V_k\big(F_n^j\big)=p {\mathbb E} V_k(\phi_j(F_{n-1}))=p r^k {\mathbb E} V_k(F_{n-1}), \end{align}

for any $k\in\{0,\ldots,d\}$ and any $j\in\{1,\ldots,M^d\}$, where $r=1/M$ is the scaling ratio of $\phi_j$.

5. Approximation of F by the sequence $\textbf{(\textit{F}}_{\textbf{\textit{n}}}\textbf{)}_{\textbf{\textit{n}}}$

Our first aim in this section is to prove Theorem 1.1. Let $M\in{\mathbb N}_{\geq 2}$ and $p\in(0,1]$ be arbitrary and let D be as defined in (1.2). (D is the Minkowski dimension of F in case $M^dp\geq 1$ and negative otherwise.) Set

(5.1) \begin{align}\overline{v}_k(n)\,:\!=r^{n(D-k)} {\mathbb E} V_k(F_{n}), \quad n\in{\mathbb N}_0,\end{align}

where $F_{0}\,:\!=J=[0,1]^d$. Since the latter is a deterministic set, we have $\overline{v}_k(0)= V_k(F_{0})=q_{d,k}$. We are going to show that the limit $\overline{\mathcal V}_k(F)=\lim_{n\to\infty} \overline{v}_k(n)$ exists for any k and, moreover, that it coincides with the expression stated in (1.4). The first step is to derive a kind of renewal equation for the $\overline{v}_k$. (The approach is similar to the methods in [Reference Winter29, Reference Zähle31] which are based on renewal theory. However, here we do not need the renewal theorem as it is possible to argue directly.)

Setting

\begin{equation*}w_k(n)\,:\!= \overline{v}_k(n)-\overline{v}_k(n-1), \quad n\in{\mathbb N},\end{equation*}

it is easy to see that

(5.2) \begin{align} \lim_{n\to\infty} \overline{v}_k(n)=\overline{v}_k(0)+\sum_{j=1}^\infty w_k(j);\end{align}

i.e., the limit on the left exists if and only if the sum on the right converges. (Indeed, by definition of $w_k$, we have

\begin{equation*}\overline{v}_k(n)=\overline{v}_k(n-1)+w_k(n)=\ldots =\overline{v}_k(0)+\sum_{j=1}^n w_k(j)\end{equation*}

for any $n\in{\mathbb N}$, and so in (5.2) the limit on the left exists if and only if the partial sums on the right converge.) Therefore, it is enough to compute the functions $w_k$, which turns out to be easier than computing the $\overline{v}_k$ directly. The relation

\begin{equation*}\overline{v}_k(n)=\overline{v}_k(n-1)+w_k(n),\qquad n\in{\mathbb N},\end{equation*}

can be viewed as a (discrete) renewal equation, with $w_k$ being the error term. By definition of $w_k$, we have

\begin{align*} \notag w_k(n)&= \overline{v}_k(n)-\overline{v}_k(n-1)= r^{n(D-k)} {\mathbb E} V_k(F_{n})- r^{(n-1)(D-k)} {\mathbb E} V_k(F_{n-1})\\ \notag &= r^{n(D-k)}\big( {\mathbb E} V_k(F_{n})- r^{k-D} {\mathbb E} V_k(F_{n-1})\big)\\ &= r^{n(D-k)}\big( {\mathbb E} V_k(F_{n})- M^dp \, r^{k} {\mathbb E} V_k(F_{n-1})\big), \end{align*}

where we employed the relation $M^dp=r^{-D}$ in the last step. Now the similarity relation (4.3) implies

\begin{equation*}\sum_{j=1}^{M^d} {\mathbb E} V_k\big(F_n^j\big)=M^d p r^k {\mathbb E} V_k(F_{n-1}),\end{equation*}

which we can insert in the above expression to obtain

(5.3) \begin{align} w_k(n)&= r^{n(D-k)}\Bigg( {\mathbb E} V_k(F_{n})- \sum_{j=1}^{M^d} {\mathbb E} V_k\big(F_n^j\big)\Bigg).\end{align}

Using the inclusion–exclusion principle, this can be expressed in a more convenient form. Since $F_{n}=\bigcup_{j=1}^{N} F^j_{n},$ we get

\begin{equation*}V_k(F_{n} )-\sum_{j=1}^{N} V_k\big(F^j_{n}\big)=\sum_{T\subset\{1,\ldots,M^d\},|T|\geq 2}({-}1)^{|T|-1} V_k\bigg(\bigcap_{j\in T} F^j_{n} \bigg).\end{equation*}

Taking expectations and plugging the resulting equation into (5.3), we obtain for each $n\in{\mathbb N}$ and each $k\in\{0,\ldots,d\}$ the representation

(5.4) \begin{align} w_k(n)&= \sum_{T\subset\{1,\ldots,M^d\},|T|\geq 2}({-}1)^{|T|-1} r^{n(D-k)} {\mathbb E} V_k\bigg(\bigcap_{j\in T} F_n^j\bigg).\end{align}

Note that this is a finite sum with a fixed number of terms (independent of n). Combined with (5.2), it yields

(5.5) \begin{align} \overline{\mathcal V}_k(F) &= q_{d,k}+\sum_{n=1}^\infty \sum_{T\subset\{1,\ldots,M^d\},|T|\geq 2}({-}1)^{|T|-1} r^{n(D-k)} {\mathbb E} V_k\bigg(\bigcap_{j\in T} F_n^j\bigg).\end{align}

This is almost the formula stated in Theorem 1.1 except for the different order of summation. The summations can be interchanged (and thus the formula (1.4) is verified) provided that the summations over n in (1.4) converge for each set T. This convergence is ensured by Proposition 5.1 below for any $p\in(0,1]$. Recall that the kth intrinsic volume of a polyconvex set K can be localized to a signed measure on K, the kth curvature measure $C_k(K,\cdot)$. Denote by $C_k^{\textsf{var}}(K)$ the total mass of the total variation measure of $C_k(K,\cdot)$.

Proposition 5.1. Let F be a fractal percolation in $[0,1]^d$ with parameters $M\geq 2$ and $p\in(0,1]$. For each $k\in\{0,\ldots,d\}$ and each $T\subset\{1,\ldots, M^d\}$ with $|T|\geq 2$,

\begin{equation*} \sum_{n=1}^\infty r^{n(D-k)} {\mathbb E} C_k^\textsf{var}\bigg(\bigcap_{j\in T} F_n^j\bigg)<\infty. \end{equation*}

In particular, the sums

\begin{equation*} \sum_{n=1}^\infty r^{n(D-k)} {\mathbb E} V_k\bigg(\bigcap_{j\in T} F_n^j\bigg) \end{equation*}

converge absolutely.

We postpone the proof of Proposition 5.1 to the last section, where we will discuss it together with the proof of a similar assertion needed in Section 6. With this statement in hand we can now complete the proof of the main theorem.

Proof of Theorem 1.1. To obtain the formula (1.4), all we have to do is to interchange the order of the summations in the formula (5.5). This is justified, since, by Proposition 5.1, all the series occurring in (1.4) converge for any $p\in(0,1]$.

Now we are going to apply formula (1.4) to derive explicit expressions for the functionals $\overline{\mathcal V}_k(F)$ for fractal percolation F in ${\mathbb R}^d$ for dimensions $d=1,2$. In particular, we will prove Theorem 1.2. The computations in dimension $d=2$ require explicit formulas for the nth construction steps of the one-dimensional case. The same method can, in principle, provide explicit expressions in dimension $d=3$ and higher. The derivation in dimension d relies on formulas for all dimensions up to $d-1$ for the nth construction steps $F_n$ of F (and for intersections of their independent copies). While the computations quickly become technically involved, including separate analyses of many cases of intersections and lengthy expressions, the method remains the same. In this sense the case $d=2$ is prototypical.

The case $\textbf{\textit{d}}=\textbf{1}$. Fractal percolation in one dimension (for which we use throughout the letter K instead of F) is not very interesting as a percolation model. However, the limiting behavior of the studied geometric functionals is of independent interest. Moreover, as indicated above, the case $d=1$ is essential for the computations in the two-dimensional case. First, we derive explicit expressions for the expected intrinsic volumes of the approximation steps $K_{n}$, $n\in{\mathbb N}_0$, of a fractal percolation K in [0, 1], from which it is easy to determine the rescaled limits $\overline{\mathcal V}_k(K)$. Then we study the intersection of two such random sets (cf. Proposition 5.3), which is needed too for the discussion of the case $d=2$.

Proposition 5.2. Let K be a fractal percolation on the interval [0, 1] with parameters $M\in{\mathbb N}_{\geq 2}$ and $p\in(0,1]$. Denote by $K_{n}$ the nth step of the construction of K. Then, for any $n\in{\mathbb N}_0$,

\begin{align*} {\mathbb E} V_1(K_{n})=p^{n} \end{align*}

and

\begin{align*} {\mathbb E} &V_0(K_{n})= (Mp)^n\left(1-\frac{(M-1)p}{M-p}\left[1-\left(\frac {p}M\right)^n\right]\right)\!. \end{align*}

Proof. For $j=1,\ldots,M$, let $K_{n}^j$ be the union of the surviving intervals of level n contained in $J_j=\phi_j([0,1])$; cf. (4.1). Then $K_{n}=\bigcup_{j=1}^M K_{n}^j$, and since in this union only sets $K_{n}^j$ with consecutive indices can have a nonempty intersection, by the inclusion–exclusion formula we get

(5.6) \begin{align} {\mathbb E} V_k(K_{n})=\sum_{j=1}^M {\mathbb E} V_k\big(K_{n}^j\big)-\sum_{j=1}^{M-1} {\mathbb E} V_k\big(K_{n}^j\cap K_{n}^{j+1}\big). \end{align}

For $k=1$, the second sum vanishes, since these intersections consist of at most one point. Moreover, by (4.3), the terms in the first sum satisfy

(5.7) \begin{align} {\mathbb E} V_k\big(K_{n}^j\big)=pr^k{\mathbb E} V_k(K_{n-1}),\quad n\in{\mathbb N}. \end{align}

Since $V_1(K_0)=V_1([0,1])=1$, this yields

\begin{align*} {\mathbb E} V_1(K_{n})=\sum_{j=1}^M (p/M) {\mathbb E} V_1(K_{n-1})=p {\mathbb E} V_1(K_{n-1})=\ldots = p^n \end{align*}

as claimed. For $k=0$, the terms in second sum in (5.6) contribute. The Euler characteristic $V_0\big(K_{n}^j\cap K_{n}^{j+1}\big)$ equals 1 with probability $p^{2n}$ (and is 0 otherwise), since for a nonempty intersection at each level from 1 to n the two intervals containing the possible intersection point need to survive (which has probability p for each of these intervals). Using this and (5.7), we conclude from (5.6) that

\begin{align*} {\mathbb E} V_0(K_{n})=\sum_{j=1}^M p {\mathbb E} V_0(K_{n-1})-\sum_{j=1}^{M-1} p^{2n}=Mp {\mathbb E} V_0(K_{n-1})-(M-1)p^{2n}. \end{align*}

This is a recursive relation for the sequence $({\mathbb E} V_0(K_{n}))_{n\in{\mathbb N}_0}$ where ${\mathbb E} V_0(K_0)=1$. By an induction argument, it is easy to obtain the explicit representation

\begin{align*} {\mathbb E} V_0(K_{n})=(Mp)^n-(M-1)\sum_{i=1}^n (Mp)^{n-i} p^{2i} =(Mp)^n\bigg(1-(M-1)\sum_{i=1}^n \left(\frac pM\right)^i\bigg), \end{align*}

which yields the asserted formula.

Corollary 5.1. Let K be a fractal percolation on the interval [0, 1] with parameters $M\in{\mathbb N}_{\geq 2}$ and $p\in(0,1]$. Then

\begin{align*} \overline{\mathcal V}_1(K)&=1 \qquad \textit{ and }\qquad \overline{\mathcal V}_0(K)=\frac{M(1-p)}{M-p}. \end{align*}

Proof. Since $D=\log (Mp)/\log M$, we have $Mp=M^D=r^{-D}$, and so, by Proposition 5.2,

\begin{align*} \overline{\mathcal V}_0(K)=\lim_{n\to\infty} r^{Dn} {\mathbb E} V_0(K_{n})=\lim_{n\to\infty} 1-\frac{(M-1)p}{M-p}\left[1-\left(\frac {p}M\right)^n\right]=1-\frac{(M-1)p}{M-p} \end{align*}

and

\begin{equation*}\overline{\mathcal V}_1(K)=\lim_{n\to\infty} r^{(D-1)n} {\mathbb E} V_1(K_{n})=\lim_{n\to\infty} p^{-n} p^n=1,\end{equation*}

as claimed.

Figure 6 (left) shows plots of $\overline{\mathcal V}_0(K)$ as a function of p for different values of the parameter M. It is apparent that these are positive and monotone decreasing functions in p for any M and that the limit as $M\to\infty$ is given by $f(p)=1-p$.

Figure 6: The rescaled limits $\overline{\mathcal V}_0(K)$ (left) and $\overline{\mathcal V}_0(K^{(1)}\cap K^{(2)})$ (right) as functions of $p\in(0,1]$ for different values of M (color-coded) as given by Corollary 5.1 and Remark 5.1, respectively. The limit curves as $M\to\infty$ are shown in light blue.

Proposition 5.3. Let $K^{(1)}, K^{(2)}$ be independent fractal percolations on the interval [0, 1] with the same parameters $M\in{\mathbb N}_{\geq 2}$ and $p\in(0,1]$. Then, for any $n\in{\mathbb N}_0$,

\begin{align*} {\mathbb E} V_1\big(K_n^{(1)}\cap K_n^{(2)}\big)=p^{2n}\end{align*}

and

\begin{align*} {\mathbb E} V_0\big(K_n^{(1)}\cap K_n^{(2)}\big)&= (Mp^2)^n\\ & \quad\times \bigg(3-2 M^{-n}-4p \frac{M-1}{M-p}\left[1-\left(\frac pM\right)^n\right]+\frac{(M-1)p^2}{M-p^2}\bigg[1-\left(\frac {p^2}M\right)^n\bigg]\bigg). \end{align*}

Proof. For $i\in\{1,2\}$ and $j\in\{1,\ldots,M\}$, let $K_n^{(i),j}$ be the union of those level-n intervals in the union $K_{n}^{(i)}$ which are contained in $J_j$ (similarly as in (4.1)). Then $K_n^{(i)}=\bigcup_{j=1}^M K_n^{(i),j}$. Since $K_i^j\subset J_j$ and $J_j\cap J_l\neq\emptyset$ if and only if $|j-l|\le 1$, we can write the intersection $K_n^{(1)}\cap K_n^{(2)}$ as

\begin{equation*}K_n^{(1)}\cap K_n^{(2)}=\bigcup_{j=1}^M K_n^{(1),j} \cap \bigcup_{l=1}^M K_n^{(1),l}=\bigcup_{j=1}^M \Bigg( K_n^{(1),j}\cap \bigcup_{l=j-1}^{j+1} K_n^{(2),l}\Bigg)=\!:\,\bigcup_{j=1}^M L_j,\end{equation*}

where we have set $K_n^{(2),0}=K_n^{(2),M+1}\,:\!=\emptyset$ for convenience. The random sets $L_j$ (whose dependence on n we suppress in the notation) satisfy $L_j\subset J_j$ almost surely, and therefore in the union $\bigcup_j L_j$ only sets with consecutive indices can have a nonempty intersection. Thus, by the inclusion–exclusion principle, we conclude for the expected intrinsic volumes that

(5.8) \begin{align} {\mathbb E} V_k\big(K_n^{(1)}\cap K_n^{(2)}\big)=\sum_{j=1}^M {\mathbb E} V_k(L_j)- \sum_{j=1}^{M-1} {\mathbb E} V_k(L_j\cap L_{j+1}).\end{align}

Now observe that, for $j=1,\ldots,M-1$,

\begin{equation*}L_j\cap L_{j+1}= K_n^{(1),j}\cap K_n^{(1),j+1}\cap \big(K_n^{(2),j}\cup K_n^{(2),j+1}\big),\end{equation*}

and this random set either is empty or consists of exactly one point $z_j$ (namely, the unique point in the intersection $J_j\cap J_{j+1}$). The latter event occurs if and only if (1) for each of the two sets $K_n^{(1),j}, K_n^{(1),j+1}$, at each level $k=1,\ldots,n$ the subinterval of level k that contains $z_j$ survives (each of these events has probability $p^n$), and (2) a similar survival of all subintervals containing $z_j$ also occurs for at least one of the sets $K_n^{(2),j}, K_n^{(2),j+1}$. The probability of this latter event is $2p^{n}-p^{2n}$. Hence ${\mathbb E} V_k(L_j\cap L_{j+1})=(2p^{3n}-p^{4n}) V_k(\{z_j\})$, and therefore

(5.9) \begin{align} {\mathbb E} V_0(L_j\cap L_{j+1})=2p^{3n}-p^{4n} \quad \text{ and } \quad {\mathbb E} V_k(L_j\cap L_{j+1})=0 \text{ for } k\ge 1.\end{align}

It remains to determine ${\mathbb E} V_k(L_j)$. By definition of $L_j$, we have

\begin{align*} L_j= \bigcup_{l=j-1}^{j+1} K_n^{(1),j}\cap K_n^{(2),l},\end{align*}

and therefore the inclusion–exclusion formula gives

\begin{align*} {\mathbb E} V_k( L_j)= \sum_{l=j-1}^{j+1} {\mathbb E} V_k \big(K_n^{(1),j}\cap K_n^{(2),l}\big)-\sum_{l=j-1}^j {\mathbb E} V_k\big(K_n^{(1),j}\cap K_n^{(2),l}\cap K_n^{(2),l+1}\big).\end{align*}

Now again $K_n^{(1),j}\cap K_n^{(2),l}$ is a singleton with probability $p^{2n}$ and empty otherwise, provided $l=j-1$ or $l=j+1$ (and $l\notin\{0,M+1\}$). Similarly, $K_n^{(1),j}\cap K_n^{(2),l}\cap K_n^{(2),l+1}$ is a singleton with probability $p^{3n}$ and empty otherwise, provided $l\notin\{0,M\}$. (For the exceptional l, these intersections are empty almost surely.) This implies

\begin{align*} {\mathbb E} V_0( L_j)= {\mathbb E} V_0\big(K_n^{(1),j}\cap K_n^{(2),j}\big)+ \left\{\begin{array}{l@{\quad}l} 2(p^{2n}-p^{3n}), & j\in\{2,\ldots,M-1\},\\[4pt] p^{2n}-p^{3n}, & j\in\{1,M\}, \end{array} \right.\end{align*}

and ${\mathbb E} V_k(L_j)= {\mathbb E} V_k\big(K_n^{(1),j}\cap K_n^{(2),j}\big)$ for any $k\geq 1$. Plugging this and (5.9) into (5.8), we conclude that, for $k=0,1$ and any $n\in{\mathbb N}$,

(5.10) \begin{align} {\mathbb E} V_k\big(K_n^{(1)}\cap K_n^{(2)}\big)=p_k(n)+\sum_{j=1}^M {\mathbb E} V_k\big(K_n^{(1),j}\cap K_n^{(2),j}\big),\end{align}

where $p_0(n)\,:\!=(M-1)(2p^{2n}-4p^{3n}+p^{4n})$ and $p_1(n)\,:\!=0$, $n\in{\mathbb N}.$ Now observe that, by (4.2), we have

\begin{equation*}K_n^{(1),j}\cap K_n^{(2),j}=\phi_j\big(\widetilde K_{n-1}^{(1)}\cap \widetilde K_{n-1}^{(2)}\big)\end{equation*}

in distribution, where $\widetilde K_{n-1}^{(i)}$ is (similarly as in (4.2)) the random set which equals $K_{n-1}^{(i)}$ with probability p and is empty otherwise. This implies

\begin{align*} {\mathbb E} V_k\big(K_n^{(1),j}\cap K_n^{(2),j}\big)= p^2 r^k {\mathbb E} V_k\big(K_{n-1}^{(1)}\cap K_{n-1}^{(2)}\big),\end{align*}

for any $j=1,\ldots,M$ and any $n\in{\mathbb N}$, where $K_{0}^{(i)}=[0,1]$, and thus ${\mathbb E} V_k\big(K_{0}^{(1)}\cap K_{0}^{(2)}\big)=V_k([0,1])=1$ for $k=0,1$. Setting $\alpha_n\,:\!={\mathbb E} V_1\big(K_n^{(1)}\cap K_n^{(2)}\big)$, $n\in{\mathbb N}_0$, we have $\alpha_0 =1$, and we infer from (5.10) that

\begin{equation*}\alpha_n=\sum_{j=1}^M {\mathbb E} V_1\big(K_n^{(1),j}\cap K_n^{(2),j}\big)=M p^2 r \alpha_{n-1}=p^2 \alpha_{n-1}, \quad n\in{\mathbb N}.\end{equation*}

It is easy to see now that $\alpha_n=p^{2n}$, proving the first formula in Proposition 5.3.

Setting $\beta_n\,:\!={\mathbb E} V_0\big(K_n^{(1)}\cap K_n^{(2)}\big)$, $n\in{\mathbb N}_0$, we infer in a similar way that $\beta_0=1$ and

\begin{equation*}\beta_n=\sum_{j=1}^M {\mathbb E} V_0\big(K_n^{(1),j}\cap K_n^{(2),j}\big)+ p_0(n)=M p^2 \beta_{n-1}+ p_0(n), \quad n\in{\mathbb N},\end{equation*}

which provides a recursive relation for the sequence $(\beta_n)_n$. By an induction argument, we obtain

\begin{equation*}\beta_n=\big(Mp^2\big)^n+ \sum_{j=1}^n \big(Mp^2\big)^{n-j} p_0(j), \quad n\in{\mathbb N}_0.\end{equation*}

Plugging in the $p_0(j)$ and computing the sum, we conclude that, for any $n\in{\mathbb N}_0$,

\begin{equation*}\beta_n=(Mp^2)^n\bigg(3-\frac 2{M^n}-4p \frac{M-1}{M-p}\left[1-\left(\frac pM\right)^n\right]+\frac{(M-1)p^2}{M-p^2}\Big[1-\Big(\frac {p^2}M\Big)^n\Big]\bigg),\end{equation*}

which shows the second formula in Proposition 5.3 and completes the proof.

Remark 5.1. It is easy to see from Proposition 5.3 that for $D'\,:\!=\log\!(Mp^2)/\log M$ the rescaled expressions $r^{n(D'-k)}{\mathbb E} V_k\big(K_n^{(1)}\cap K_n^{(2)}\big)$ converge as $n\to\infty$. Indeed, since $r^{D'-1}=p^{-2}$ and $r^{D'}=(1/M)^{D'}=(M p^2)^{-1}$, we obtain

\begin{align*} \overline{\mathcal V}_1(K^{(1)}\cap K^{(2)})&\,:\!=\lim_{n\to\infty} r^{n(D'-1)}{\mathbb E} V_1\big(K_n^{(1)}\cap K_n^{(2)}\big)=1 \qquad \text{ and }\\ \overline{\mathcal V}_0(K^{(1)}\cap K^{(2)})&\,:\!=\lim_{n\to\infty} r^{nD'}{\mathbb E} V_0\big(K_n^{(1)}\cap K_n^{(2)}\big) =3-4p \frac{M-1}{M-p}+p^2\frac{M-1}{M-p^2}. \end{align*}

Again the rescaled length $\overline{\mathcal V}_1(K^{(1)}\cap K^{(2)})$ is constant, while the rescaled Euler characteristic of $K^{(1)}\cap K^{(2)}$ depends on p and M. Figure 6 (right) shows plots of $\overline{\mathcal V}_0(K^{(1)}\cap K^{(2)})$ as a function of p for different values of the parameter M. It is apparent that these are positive and monotone decreasing functions in p for any M and that the limit as $M\to\infty$ is given by $f(p)=\break 3-4p+p^2$. From the existence of the limits $\overline{\mathcal V}_k(K^{(1)}\cap K^{(2)})$ it is clear that Dʹ as chosen above is the correct scaling exponent. The notation for the limit is justified by the fact that Dʹ is almost surely the Hausdorff dimension of $K^{(1)}\cap K^{(2)}$, as the following statement clarifies.

Proof. For any $p\in(0,1]$, the set $K^{(1)}\cap K^{(2)}$ can be coupled with a fractal percolation F on [0, 1] with parameter $p^2$ (and the same M) by retaining an interval $I_\sigma$ of level n if and only if it is contained in both sets $K_n^{(1)}$ and $K_n^{(2)}$. Then $K^{(1)}\cap K^{(2)}$ dominates F. Hence, almost surely, $\dim_H\left(K^{(1)}\cap K^{(2)}\right)\geq \dim_H F$. Now observe that conditioning on the event $\{K^{(1)}\cap K^{(2)}\neq\emptyset\}$ is the same as conditioning on $\{F\neq\emptyset\}$. Indeed, on the one hand the first event is obviously satisfied whenever the latter is. On the other hand, if $\{F=\emptyset\}$ holds, then there is some $n\in{\mathbb N}$ such that $F_n=\emptyset$. This implies that, for any $m\geq n$, $K_m^{(1)}\cap K_m^{(2)}$ consists of finitely many isolated points contained in the set $\{kM^{-n}\,:\,k\in\{1,\ldots,M^n-1\}\}$ (cf. the proof of Proposition 5.3). In particular, there are no new points generated after the nth step. Each point at level $m\geq n$ is independently retained in the next step with probability $p^2$. This means in particular that ${K^{(1)}\cap K^{(2)}}$ is empty almost surely under the condition $F=\emptyset$. We conclude that, for $p\leq\sqrt{M}$, the set $K^{(1)}\cap K^{(2)}$ is empty almost surely, since F has this property. Moreover, since conditioned on $F\neq\emptyset$ we have $\dim_H F=D'$ almost surely for any $p\geq\sqrt{M}$, we infer from the above inequality that conditioned on $K^{(1)}\cap K^{(2)}\neq\emptyset$, Dʹ is almost surely a lower bound for $\dim_H\left(K^{(1)}\cap K^{(2)}\right)$. (The same is true for the Minkowski dimension.)

We show that Dʹ is also an upper bound for $\dim_H\left(K^{(1)}\cap K^{(2)}\right)$. For any realization of ${K^{(1)}\cap K^{(2)}}$ and any $\delta>0$, a $\delta$-cover of $K^{(1)}\cap K^{(2)}$ is obtained by taking the cubes of level n (for some n large enough that $M^{-n}<\delta$) contained in $F_n$ (which cover F) and adding the finitely many singletons $\left\{kM^{-n}\right\}$, $k=1,\ldots,M^{n}-1$, which clearly cover the additional isolated points in $K^{(1)}\cap K^{(2)}$ not already covered by the chosen intervals. Using these covers and noting that the singletons have diameter zero and the intervals diameter $M^{-n}$, we get for any $s>0$ that $\mathcal{H}^s_\delta(K^{(1)}\cap K^{(2)})\leq Z_n M^{-ns}$, where $Z_n$ is the number of cubes in $F_n$. Since $Z_n$ is the size of the nth generation of a Galton–Watson process in which the expected number of offspring of an individuum is $M^{D'}=M^2p$, it is well known that $Z_n M^{-nD'}\to 1$ almost surely as $n\to\infty$. This shows $\mathcal{H}^{D'}(K^{(1)}\cap K^{(2)})<\infty$ almost surely and thus $\dim_H(K^{(1)}\cap\break K^{(2)})\leq D'$.

The case $\textbf{\textit{d}}=\textbf{2}$. Now we provide proofs of the formulas stated in Theorem 1.2 for the three limit functionals $\overline{\mathcal V}_k(F)$, $k=0,1,2$, for fractal percolation F in ${\mathbb R}^2$. The starting point is again the general formula in Theorem 1.1, which can be simplified further by using on the one hand the various symmetries in the fractal percolation model and on the other hand the properties of the functionals.

Proof of Theorem 1.2. Let $M\in{\mathbb N}_{\geq 2}$, $p\in(0,1]$, and $k\in\{0,1,2\}$. By (1.4) in Theorem 1.1, we have

(5.11) \begin{align} \overline{\mathcal V}_k(F)= q_{2,k} +\sum_{T\subset\{1,\ldots,M^2\},|T|\geq 2}({-}1)^{|T|-1} \sum_{n=1}^\infty r^{n(D-k)} {\mathbb E} V_k\bigg(\bigcap_{j\in T} F^j_{n}\bigg). \end{align}

Observe that among the intersections $\bigcap_{j\in T} F_n^j$ occurring in (5.11) only those need to be considered for which the corresponding intersection $\bigcap_{j\in T} J_j$ of the subcubes $J_j=\phi_j(J)$ is nonempty. All other intersections are empty almost surely and hence their expected intrinsic volumes are zero. The nonempty intersections of subcubes can be reduced to four basic cases (see Figure 7). There are only two ways in which two subcubes can have a nonempty intersection: they can intersect in a common face (like $J_1$ and $J_4$ in Figure 7) or in a common corner (like $J_1$ and $J_2$). Three subcubes can only have a nonempty intersection at a common corner (like $J_1$, $J_2$, and $J_3$), and similarly four subcubes can only intersect in a common corner (like $J_1$, $J_2$, $ J_3$, and $J_4$). Only the number of intersections of each of these four types changes with M. These numbers are given by $2M(M-1)$, $2(M-1)^2$, $4(M-1)^2$ and $(M-1)^2$, respectively, independently of p and n. Hence the formula (5.11) reduces to

(5.12) \begin{align} \overline{\mathcal V}_k(F) \ =\ & q_{2,k} -2M(M-1)\sum_{n=1}^\infty r^{n(D-k)} {\mathbb E} V_k\big(F_n^1\cap F_n^4\big)\\ &-2(M-1)^2 \sum_{n=1}^\infty r^{n(D-k)} {\mathbb E} V_k\big(F_n^1\cap F_n^2\big) \notag \displaybreak\\ &+4(M-1)^2 \sum_{n=1}^\infty r^{n(D-k)} {\mathbb E} V_k\big(F_n^1\cap F_n^2\cap F_n^3\big) \notag \\ &-(M-1)^2 \sum_{n=1}^\infty r^{n(D-k)} {\mathbb E} V_k\bigg(\bigcap_{j=1}^4 F_n^j\bigg) .\nonumber \end{align}

Figure 7: Possible mutual positions of the basic cubes $J_j$ which produce nonempty intersections.

For $k=2$, i.e. for the area $V_2$ in ${\mathbb R}^2$, it is enough to observe that the area of all the intersections of the level sets $F_n^j$ in this formula are almost surely zero (they are all contained in a line segment), implying that ${\mathbb E} V_2\big(\!\bigcap_{j\in T} F_n^j\big)=0$ for all $n\in{\mathbb N}$ and all index sets T with $|T|\geq 2$. Therefore,

\begin{equation*}\overline{\mathcal V}_2(F)=q_{2,2}=V_2([0,1]^2)=1,\end{equation*}

independently of M and p, as asserted in Theorem 1.2.

For $k=1$, i.e. for the ‘boundary length’ $V_1$, only the intersections of the first type $F_n^1\cap F_n^4$ need to be considered; for the other three types the intersection is at most one point, implying that the expected boundary length vanishes, independently of n. This yields

(5.13) \begin{align} \overline{\mathcal V}_1(F) &=q_{2,1} -2M(M-1)\sum_{n=1}^\infty r^{n(D-1)} {\mathbb E} V_1\big(F_n^1\cap F_n^4\big). \end{align}

We claim that, for each $n\in{\mathbb N}$,

(5.14) \begin{align} {\mathbb E} V_1\big(F_n^1\cap F_n^4\big)= p^{2n}/M.\end{align}

We will show below that this follows from Proposition 5.3. Plugging (5.14) into (5.13) and recalling that $r^{D-1}=M^{-D+1}=(M\,p)^{-1}$, we conclude

\begin{align*} \overline{\mathcal V}_1(F) &=2 - 2(M-1)\sum_{n=1}^\infty (M\,p)^{-n} p^{2n} =2-2(M-1)\sum_{n=1}^\infty (p/M)^n\\ &=2-2(M-1)\frac{p}{M-p}=\frac{2M(1-p)}{M-p}. \end{align*}

For $k=0$, i.e. for the Euler characteristic $V_0$, all terms in the above formula (5.12) are relevant and contribute to the limit. It is rather easy to see that $V_0(F_n^1\cap F_n^2\cap F_n^3\cap F_n^4)=1$ with probability $p^{4n}$, since at all levels $m=1,\ldots,n$, in each of the four cubes $J_i$, $i=1,\ldots, 4$, the subcube of level m which intersects the common corner needs to survive (which happens with probability p, independently of all the other subcubes of any level). Otherwise the intersection of the four sets $F^j_n$ will be empty. Hence, for each $n\in{\mathbb N}$ (and each $M\geq 2$),

(5.15) \begin{align} {\mathbb E} V_0\big(F_n^1\cap F_n^2\cap F_n^3\cap F_n^4\big)=p^{4n}.\end{align}

Therefore, the sum in the last line of formula (5.12) is given by

(5.16) \begin{align} \sum_{n=1}^\infty r^{nD} {\mathbb E} V_0\bigg(\bigcap_{j=1}^4 F_n^j\bigg)= \sum_{n=1}^\infty (r^D\, p^4 )^{n}=\frac{r^D\,p^4}{1-r^D\,p^4}=\frac{p^3}{M^2-p^3},\end{align}

where the last equality is due to the relation $p\, r^D=r^2=M^{-2}$. (Note that the geometric series above converges, since $p^4 \,r^D=p^3\,M^{-2}<1$ for any $p\in(0,1]$ and any integer $M\geq2$.)

Similarly, one observes that $V_0\big(F_n^1\cap F_n^2\cap F_n^3\big)=1$ with probability $p^{3n}$ and $V_0\big(F_n^1\cap F_n^2\big)=1$ with probability $p^{2n}$ for $n\in{\mathbb N}$, which yields for the sums in the third and the second line in the formula (5.12) the expressions

(5.17) \begin{align} \sum_{n=1}^\infty r^{nD} {\mathbb E} V_0\big(F_n^1\cap F_n^2\cap F_n^3\big)= \sum_{n=1}^\infty (r^D\,p^3)^{n}=\frac{p^2}{M^2-p^2}\end{align}

and

(5.18) \begin{align} \sum_{n=1}^\infty r^{nD} {\mathbb E} V_0\big(F_n^1\cap F_n^2\big)= \sum_{n=1}^\infty (r^D p^2)^{n}=\frac{p}{M^2-p}.\end{align}

It remains to compute the expected Euler characteristic for the type $F_n^1\cap F_n^4$. We claim that, for any $n\in{\mathbb N}$,

(5.19) \begin{align} {\mathbb E} V_0\big(F_n^1\cap F_n^4\big)&= (Mp^2)^n\\[4pt] &\quad \times \bigg(\frac 3M-2 M^{-n}-4 \frac{M-1}{M-p}\Big[\frac pM-\left(\frac pM\right)^{n}\Big]+\frac{M-1}{M-p^2}\bigg[\frac{p^2}M-\left(\frac {p^2}M\right)^{n}\bigg]\bigg).\notag\end{align}

We will demonstrate below that this follows from Proposition 5.3. Plugging (5.16)–(5.19) into (5.12) and computing the remaining series yields the missing terms of $\overline{\mathcal V}_0(F)$. More precisely, we get for the last sum on the first line of (5.12) the expression

\begin{align*} E_1&\,:\!=\frac{2(M-1)^2p}{M-p}\left(\frac 3{M-1}-\frac{4p}{M-p}+\frac{p^2}{M-p^2}\right)-2M(M-1)^2p\\ &\quad\times\left(\frac{2}{(M-1)(M^2-p)}-\frac{4p}{(M-p)(M^2-p^2)}+\frac{p^2}{(M-p^2)(M^2-p^3)}\right), \end{align*}

and therefore

\begin{align*} \overline{\mathcal V}_0(F) &=1-E_1+(M-1)^2\left({-}\frac{2p}{M^2-p}+\frac{4p^2}{M^2-p^2}-\frac{p^3}{M^2-p^3}\right)\!.\end{align*}

Combining some of the terms gives the formula stated in Theorem 1.2 for $\overline{\mathcal V}_0(F)$.

To complete the proof, it remains to verify the equations (5.14) and (5.19). To understand the structure of $F_n^1\cap F_n^4$, it is enough to study the intersection of two independent one-dimensional fractal percolations $K^{(1)}$ and $K^{(2)}$ defined on a common interval [0, 1] (with the same parameters M and p as F). For $n\in {\mathbb N}$ and $i=1,2$, let $K_n^{(i)}$ denote the nth steps of their construction. Similarly as in (4.2), let $\widetilde K_n^{(i)}$, $i=1,2$, be the random set which equals $K_n^{(i)}$ with probability p and is empty otherwise; i.e. we add an additional 0th step to decide whether the set $K_n^{(i)}$, $n\in{\mathbb N}$, is kept or discarded. This is to account for the first step of the construction of F (in which the cubes $J_i$ are discarded with probability $1-p$). Then, for each n, we have the following equality in distribution:

(5.20) \begin{align} F_n^1\cap F_n^4=\psi \big(\widetilde K_{n-1}^{(1)}\cap \widetilde K_{n-1}^{(2)}\big),\end{align}

where $\psi\,:\,{\mathbb R}\to{\mathbb R}^2, x\mapsto (t/M) a+ ((1-t)/M) b$ is the similarity, which maps [0, 1] to the segment $J_1\cap J_4$ with endpoints a and b. Since intrinsic volumes are independent of the ambient space dimension, motion-invariant, and homogeneous, this implies in particular that

(5.21) \begin{align} {\mathbb E} V_k\big(F_n^1\cap F_n^4\big)&={\mathbb E} V_k\big(\psi \big(\widetilde K_{n-1}^{(1)}\cap \widetilde K_{n-1}^{(2)}\big)\big)\\ &=r^k p^2 {\mathbb E} V_k\big(K_{n-1}^{(1)}\cap K_{n-1}^{(2)}\big).\notag\end{align}

Now the claims (5.14) and (5.19) follow from combining (5.21) with Proposition 5.3.

Remark 5.2. From the proof of Theorem 1.2, we also get explicit expressions for the expected intrinsic volumes of the approximation sets $F_{n}$ for each $n\in{\mathbb N}$. To determine $\overline{v}_k(m)\,:\!=r^{m(D-k)}{\mathbb E} V_k(F_{m})$, it is enough to truncate all the sums in the formula (5.12) after the mth step and compute the resulting finite geometric sums. This yields, for $k=0$ and $n\in{\mathbb N}$,

\begin{align*} \overline{v}_0(n)&=1-\frac{2p(M-1)^2}{M-p}\left(\frac 3{M-1}-\frac{4p}{M-p}+\frac{p^2}{M-p^2}\right)\left[1-\left(\frac pM\right)^n\right]\\[4pt] &\quad+\frac{2p(M^2-1)}{M^2-p}\left[1-\left(\frac{p}{M^2}\right)^n\right]-\frac{4p^2(M-1)^2}{(M-p)^2}\bigg[1-\left(\frac{p^2}{M^2}\right)^n\bigg]\\[4pt] &\quad+\frac{p^3(M-1)^2(M+p^2)}{(M-p^2)(M^2-p^3)}\bigg[1-\left(\frac{p^3}{M^2}\right)^n\bigg].\end{align*}

It is easy to see that this sequence converges exponentially fast to $\overline{\mathcal V}_0(F)$ as $n\to\infty$. More precisely, we have

\begin{equation*}\overline{v}_0(n)-\overline{\mathcal V}_0(F)\sim c\,(p/M)^n \quad \text{ as } n\to \infty\end{equation*}

(i.e. the quotient of the left- and the right-hand side converges to 1) with the constant

\begin{equation*}c\,:\!=\frac{2p(M-1)^2}{M-p}\left(\frac 3{M-1}-\frac{4p}{M-p}+\frac{p^2}{M-p^2}\right)\end{equation*}

being positive for each $p\in(0,1]$ and $M\in{\mathbb N}_{\geq 2}$. Moreover, the sequence $(\overline{v}_0(n))_n$ is eventually strictly decreasing, i.e. strictly decreasing from some index $n_0\in {\mathbb N}$. This exemplifies that the convergence $\overline{v}_k(n)\to \overline{\mathcal V}_k(F)$ is extremely fast and that the functionals $\overline{\mathcal V}_k(F)$ can be approximated well by the $\overline{v}_k(n)$. This was also observed in simulations, where already for small n (e.g. $n=8$, even for $M=2$; see Figure 4), $\overline{v}_k(n)$ is virtually indistinguishable from the limit $\overline{\mathcal V}_k(F)$; see also Remark 5.3 below. Fast convergence can also be expected for the limits $\overline{\mathcal V}_k(K)$ of other random self-similar sets K, for which no exact formula may be available. It is another intriguing question whether a similar speed of convergence can be expected for the percolation probabilities of $F_n$.

We simulate realizations of finite approximations for $M=2$, 3, and 4. For each value of M, we choose three levels n of the approximation: $n=32/(2^M)$, $n=32/(2^M)+2$, or ${n=32/(2^M)+4}$. Since the rate of convergence increases with M, for larger M smaller values of n are sufficient. For each chosen value of the probability of survival $p=0.11, 0.13, \ldots, 0.99$, we simulate 75000, 5000, or 2500 samples for $M=2$, 3, or 4, respectively. Only in the case of $M=2, p\leq 0.31$ for $F_n$, the number of samples is increased by a factor 10 for improved statistics.

The mean values are unbiasedly estimated by the arithmetic mean of the Euler characteristics of the samples. The error bars in the plots represent the sample standard deviations. The simulation results, shown in Figure 4, are in excellent agreement with the analytic curves; see Remarks 5.2 and 6.2. The code is freely available via GitHub [Reference Klatt and Winter14].

6. Approximation of F by the closed complements of $\textbf{(\textit{F}}_{\textbf{\textit{n}}}\textbf{)}_{\textbf{\textit{n}}}$

Now we consider the closed complements $C_{n}\,:\!=\overline{J\setminus F_n}$, $n\in{\mathbb N}_0$, of the construction steps $F_n$ of the fractal percolation process inside the unit cube $J=[0,1]^d$. Note that $C_0=\emptyset$, since $F_0=J$. The random sets $C_{n}$ are also given by

\begin{equation*}C_{n}=\bigcup_{\sigma\in\Sigma_n\setminus\mathcal{T}_n} J_\sigma\end{equation*}

(cf. Section 4), implying in particular that each realization of $C_{n}$ consists of a finite number of closed cubes and is thus polyconvex. Hence intrinsic volumes are well defined. The set $C_{n}$ consists of those subcubes $J_\sigma$ of level n for which at least one of the cubes $J_{\sigma|i}$, $i\in\{1,\ldots, n\}$, was discarded. We also introduce, for each $j\in\{1,\ldots,M^d\}$ (and each $n\in{\mathbb N}_0$), the set

\begin{equation*}C_{n}^j\,:\!=\bigcup_{{\sigma\in\Sigma_n\setminus\mathcal{T}_n},{\sigma|1=j}} J_\sigma,\end{equation*}

as the union of those cubes of level n which are contained in $J_j\cap C_{n}$.

We are interested in the expected intrinsic volumes ${\mathbb E} V_k(C_{n})$, $k=0,\ldots, d$, and in particular in the limiting behavior as $n\to\infty$, for which we have the following general formula analogous to (1.4) in Theorem 1.1.

Theorem 6.1. Let $k\in\{0,\ldots,d-1\}$, and let F be a fractal percolation in ${\mathbb R}^d$ with parameters $M\in{\mathbb N}_{\geq 2}$ and $p\in(r^{d-k},1]$. Let D be the Minkowski dimension of F (see (1.2)). Then the limit

\begin{equation*} \overline{\mathcal V}^c_k(F)\,:\!=\lim_{n\to\infty} r^{n(D-k)} {\mathbb E} V_k(C_{n}) \end{equation*}

exists and is given by the expression

(6.1) \begin{align} q_{d,k}\frac{M^{d-k}(1-p)}{M^{d-k}p-1} +\sum_{T\subset\{1,\ldots,M^d\},|T|\geq 2}({-}1)^{|T|-1} \sum_{n=1}^\infty r^{n(D-k)} {\mathbb E} V_k\bigg(\bigcap_{j\in T} C_{n}^j\bigg), \end{align}

where, as before, $q_{d,k}=V_k([0,1]^d)$.

Proof. We follow the lines of the proof of Theorem 1.1. Let

\begin{equation*}\overline{v}^c_k(n)\,:\!=r^{n(D-k)} {\mathbb E} V_k(C_{n}), \quad n\in{\mathbb N}_0.\end{equation*}

Since $C_0=\emptyset$, we have $\overline{v}^c_k(0)={\mathbb E} V_k(\emptyset)=0$. Setting

\begin{equation*}w_k(n)\,:\!= \overline{v}^c_k(n)-\overline{v}^c_k(n-1), \quad n\in{\mathbb N},\end{equation*}

we observe that, similarly as in (5.2) above,

(6.2) \begin{align} \overline{\mathcal V}^c_k(F)=\lim_{n\to\infty} \overline{v}^c_k(n)=\sum_{n=1}^\infty w_k(n).\end{align}

By definition of $w_k$, we have

(6.3) \begin{align} w_k(n)&= \overline{v}^c_k(n)-\overline{v}^c_k(n-1) = r^{n(D-k)}\big( {\mathbb E} V_k(C_{n})- M^dp \, r^{k} {\mathbb E} V_k(C_{n-1})\big)\!.\end{align}

Now recall from (4.2) that, for each $j\in\{1,\ldots,M^d\}$, $F_n^j$ survives the first construction step with probability p (in which case it is distributed like $\phi_j(F_{n-1})$), and it is empty otherwise. Thus, for the closed complements $C_{n}^j$, we get

\begin{align*} {\mathbb E} V_k(C_{n}^j)&=p {\mathbb E} V_k(\phi_j(C_{n-1}))+(1-p) {\mathbb E} V_k(J_j)\\ &=p r^k {\mathbb E} V_k(C_{n-1})+(1-p)r^k q_{d,k},\end{align*}

and therefore

\begin{equation*}\sum_{j=1}^{M^d} {\mathbb E} V_k(C_{n}^j)=M^d p r^k {\mathbb E} V_k(C_{n-1})+M^d(1-p)r^k q_{d,k}. \end{equation*}

Plugging this into (6.3) and recalling that $r^{-D}=M^dp$ yields

\begin{align*} w_k(n)&= r^{n(D-k)}\left( {\mathbb E} V_k(C_{n})- \sum_{j=1}^{M^d} {\mathbb E} V_k(C_{n}^j)\right)+r^{(n-1)(D-k)}\frac{1-p}p q_{d,k}.\end{align*}

Since $C_{n}=\bigcup_{j=1}^{M^d} C_{n}^j$, by the inclusion–exclusion principle, this can be expressed in a more convenient form: for each $n\in{\mathbb N}$ and each $k\in\{0,\ldots,d\}$,

(6.4) \begin{align} w_k(n)&= r^{(n-1)(D-k)}\frac{1-p}p q_{d,k}+\sum_{\begin{array}{@{}c@{}} {\scriptstyle T\subset\{1,...,M^d\}}\\[-1mm]{\scriptstyle |T|\geq 2} \end{array}}({-}1)^{|T|-1} r^{n(D-k)} {\mathbb E} V_k\bigg(\bigcap_{j\in T} C_{n}^j\bigg).\end{align}

Note that this is again a finite sum with a fixed number of terms (independent of n) and that all the intersections appearing in this formula are at most $(d-1)$-dimensional. The summation over n can be shown to converge for each summand separately (see Proposition 6.1 below; this is where the hypothesis $p>r^{d-k}$ is used).

Inserting the representation (6.4) for $w_k$ into (6.2) yields

(6.5) \begin{align} \overline{\mathcal V}^c_k(F) \notag&=\sum_{n=1}^\infty \left(r^{(n-1)(D-k)}\frac{1-p}p q_{d,k}+\sum_{\begin{array}{@{}c@{}} {\scriptstyle T\subset\{1,...,M^d\}}\\[-1mm]{\scriptstyle |T|\geq 2} \end{array}}({-}1)^{|T|-1} r^{n(D-k)} {\mathbb E} V_k\bigg(\bigcap_{j\in T} C_{n}^j\bigg)\right)\notag\\ &=q_{d,k}\frac{1-p}p\sum_{n=0}^\infty r^{n(D-k)}+\sum_{\begin{array}{@{}c@{}} {\scriptstyle T\subset\{1,...,M^d\}}\\[-1mm]{\scriptstyle |T|\geq 2} \end{array}}({-}1)^{|T|-1} \sum_{n=1}^\infty r^{n(D-k)} {\mathbb E} V_k\bigg(\bigcap_{j\in T} C_{n}^j\bigg), \end{align}

where the convergence of the geometric series in the first term is due to the assumption $p>r^{d-k}$, which implies $D>k$. The convergence of the series in the last expression for each index set T is ensured by Proposition 6.1 just below (for which condition $p>r^{d-k}$ is needed again), justifying in particular the interchange of the summations and showing the existence of the limit of the $\overline{v}^c_k(n)$ as $n\to\infty$. Now the formula (6.1) follows easily from computing the series in the first term and recalling that $r^{-D}=M^dp$.

Proposition 6.1. Let $k\in\{0,\ldots,d-1\}$ and F be a fractal percolation in $[0,1]^d$ with parameters $M\in{\mathbb N}_{\geq 2}$ and $p\in(r^{d-k},1]$. Then, for each $T\subset\{1,\ldots, M^d\}$ with $|T|\geq 2$,

\begin{equation*} \sum_{n=1}^\infty r^{n(D-k)} {\mathbb E} C_k^\textsf{var}\bigg(\bigcap_{j\in T} C_{n}^j\bigg)<\infty, \end{equation*}

where, as before, $C_k^{\textsf{var}}(K)$ denotes the total mass of the total variation measure of the kth curvature measure of a polyconvex set K. In particular, the sums

\begin{equation*} \sum_{n=1}^\infty r^{n(D-k)} {\mathbb E} V_k\bigg(\bigcap_{j\in T} C_{n}^j\bigg) \end{equation*}

converge absolutely.

We postpone the proof of Proposition 6.1 to the last section.

The case $\textbf{\textit{d}}=\textbf{1}$. In order to derive explicit formulas for the limits $\overline{\mathcal V}^c_k(F)$ in ${\mathbb R}^2$ it is again necessary to discuss these functionals in ${\mathbb R}$ first. We start with a general formula to determine the intrinsic volumes of a polyconvex set $C\subset{\mathbb R}$ from the intrinsic volumes of its closed complement. This will be used to derive expressions for ${\mathbb E} V_k\big(D_n^{(1)}\big)$ and ${\mathbb E} V_k\big(D_n^{(1)}\cap D_n^{(2)}\big)$ from the ones already obtained in Section 5 for ${\mathbb E} V_k\big(K_n^{(1)}\big)$ and ${\mathbb E} V_k\big(K_n^{(1)}\cap K_n^{(2)}\big)$, where $D_n^{(i)}\,:\!=\overline{I\setminus K_n^{(i)}}$.

Lemma 6.1. Let $I\,:\!=[0,1]\subset {\mathbb R}$ be the unit interval and let $K\subset I$ be polyconvex (i.e. a finite union of intervals). Then the closed complement $C\,:\!=\overline{I\setminus K}$ of K within I is polyconvex, $V_1(C)=1-V_1(K)$, and

\begin{align*} V_0(C)=1+V_0(K) -\textbf{1}_K(0)-\textbf{1}_K(1)-N(K), \end{align*}

where $\textbf{1}_A$ denotes the indicator function of a set A, and N(A) is the number of isolated points in A. Moreover, if $K'\subset I$ is a second polyconvex set and $C'\,:\!=\overline{I\setminus K'}$, then

\begin{align*} V_1(C\cap C')&=1-V_1(K)-V_1(K')+V_1(K\cap K') \quad \textit{ and }\\ V_0(C\cap C')&=1+V_0(K)+V_0(K')-V_0(K\cap K') -\textbf{1}_{K\cup K'}(0)\\&\phantom{=}-\textbf{1}_{K\cup K'}(1)-N(K)-N(K')+N(K\cap K'). \end{align*}

Proof. The first formula is an easy consequence of the additivity of $V_1$, noting that $I=K\cup C$, $V_1(I)=1$, and $V_1(C\cap K)=0$. The second formula for $V_1$ follows from the first one and additivity by noting that

(6.6) \begin{align} C\cup C'=\overline{I\setminus (K\cap K')}. \end{align}

In ${\mathbb R}$ the Euler characteristic $V_0$ equals the number of connected components of a polyconvex set. If $K\subset I$ has k connected components, $k\in{\mathbb N}_0$, then $K^c={\mathbb R}\setminus K$ has $k+1$ (including the two unbounded ones), and so $I\cap K^c$ has $k+1-\textbf{1}_K(0)-\textbf{1}_K(1)$, since an unbounded connected component of $K^c$ contributes a component to $I\cap K^c$ only if it has nonempty intersection with I (that is, if $0 \notin K$ or $1 \notin K$). Finally, taking the closure of $I\setminus K$ leaves the number of connected components unchanged, provided there are no isolated points in K. Any isolated point, however, reduces the number of connected components in C by one, since it causes the two connected components of $I\setminus K$ adjacent to this point to merge to one component of C. This proves the first formula for $V_0$. The second formula follows from the first one, taking into account (6.6):

\begin{align*} V_0&(C\cap C')=V_0(C)+V_0(C')-V_0(C\cup C')\\ &=1+ V_0(K) -\textbf{1}_{K}(0)-\textbf{1}_{K}(1)-N(K)\\ &\quad+1+V_0(K') -\textbf{1}_{K'}(0)-\textbf{1}_{K'}(1)-N(K')\\ &\quad-1-V_0(K\cap K') +\textbf{1}_{K\cap K'}(0)+\textbf{1}_{K\cap K'}(1)+N(K\cap K')\\ &=1+V_0(K)+V_0(K')- V_0(K\cap K')-\textbf{1}_{K\cup K'}(0)-\textbf{1}_{K\cup K'}(1)\\ &\quad-N(K)-N(K')+N(K\cap K'), \end{align*}

where we have used the additivity of the indicator function, implying $\textbf{1}_{K}+\textbf{1}_{K'}=\textbf{1}_{K\cap K'}+\textbf{1}_{K\cup K'}$. This completes the proof of the last formula.

It is clear that corresponding formulas hold for the expected intrinsic volumes of random polyconvex subsets K and Kʹ of [0, 1] and their closed complements. Note that the functional N counting the number of isolated points is not additive. Below we always have the situation that K and Kʹ have no isolated points, while isolated points may appear in the intersection $K\cap K'$; see Figure 8 for an example.

Figure 8: In the intersection of two unions of intervals (or of unions of cubes in neighboring rows) isolated points appear, which need to be taken into account in the formulas.

Corollary 6.1. Let $K^{(1)}\kern-0.5pt, K^{(2)}$ be two independent fractal percolations on the interval $I=[0,1]$, both with the same parameters $M\in{\mathbb N}_{\geq 2}$ and $p\in(0,1]$. For $n\in{\mathbb N}_0$, let $K_n^{(i)}$ denote the nth step of the construction of $K_i$, $i=1,2$, and let $D_n^{(i)}\,:\!=\overline{I\setminus K_n^{(i)}}$. Then, for any $n\in{\mathbb N}_0$,

\begin{align*} {\mathbb E} V_1\big(D_n^{(1)}\cap D_n^{(2)}\big)&=1-2{\mathbb E} V_1\big(K_n^{(1)}\big)+{\mathbb E} V_1\big(K_n^{(1)}\cap K_n^{(2)}\big) \qquad \textit{ and }\\ {\mathbb E} V_0\big(D_n^{(1)}\cap D_n^{(2)}\big)&=2{\mathbb E} V_0\big(K_n^{(1)}\big)-{\mathbb E} V_0\big(K_n^{(1)}\cap K_n^{(2)}\big)+{\mathbb E} N\big(K_n^{(1)}\cap K_n^{(2)}\big) \\&\phantom{=}+1-4p^n+2p^{2n}. \end{align*}

Moreover, we have ${\mathbb E} V_1\big(D_n^{(1)}\big)=1-{\mathbb E} V_1\big(K_n^{(1)}\big)$ and

\begin{align*} {\mathbb E} V_0\big(D_n^{(1)}\big)= {\mathbb E} V_0\big(K_n^{(1)}\big)+1-2p^n. \end{align*}

Proof. Applying Lemma 6.1 to realizations C, Cʹ of the random sets $D_n^{(1)}$, $D_n^{(2)}$, respectively, and taking expectations, for $k=1$ we obtain directly the formula stated above, while for $k=0$ we obtain

\begin{align*} {\mathbb E} V_0\big(D_n^{(1)}\cap D_n^{(2)}\big)&=1+ {\mathbb E} V_0\big(K_n^{(1)}\big)+{\mathbb E} V_0\big(K_n^{(2)}\big)-{\mathbb E} V_0\big(K_n^{(1)}\cap K_n^{(2)}\big)\\ &\quad -{\mathbb P}\big(0\in K_n^{(1)}\cup K_n^{(2)}\big)-{\mathbb P}\big(1\in K_n^{(1)}\cup K_n^{(2)}\big)\\ &\quad -{\mathbb E} N\big(K_n^{(1)}\big)-{\mathbb E} N\big(K_n^{(2)}\big)+{\mathbb E} N\big(K_n^{(1)}\cap K_n^{(2)}\big). \end{align*}

Now observe that almost surely the set $K_n^{(i)}$ contains no isolated points, implying that ${\mathbb E} N(K_n^{(i)})=0$. Moreover,

\begin{align*} {\mathbb P}\big(0\in K_n^{(1)}\cup K_n^{(2)}\big) &={\mathbb P}\big(0\in K_n^{(1)}\big)+{\mathbb P}\big(0\in K_n^{(2)}\big)-{\mathbb P}\big(0\in K_n^{(1)}\cap K_n^{(2)}\big)\\&=2p^n-p^{2n},\end{align*}

and similarly for the point 1 instead of 0. This proves the second formula. The third formula is a direct application of the first formula in Lemma 6.1 to the realizations C of the random set $D_n^{(1)}$. Similarly, the last formula follows from applying the second formula in Lemma 6.1 to the realizations C of $D_n^{(1)}$ and taking expectations:

\begin{align*} {\mathbb E} V_0\big(D_n^{(1)}\big)&=1+{\mathbb E} V_0\big(K_n^{(1)}\big)-{\mathbb P}\big(0\in K_n^{(1)}\big)-{\mathbb P}\big(1\in K_n^{(1)}\big)-{\mathbb E} N\big(K_n^{(1)}\big)\\ &={\mathbb E} V_0\big(K_n^{(1)}\big)+1 -2p^n,\end{align*}

since $ {\mathbb P}\big(0\in K_n^{(1)}\big)={\mathbb P}\big(1\in K_n^{(1)}\big)=p^n$ and ${\mathbb E} N\big(K_n^{(1)}\big)=0$.

To get more explicit expressions for ${\mathbb E} V_k\big(D_n^{(1)}\big)$ and ${\mathbb E} V_k\big(D_n^{(1)}\cap D_n^{(2)}\big)$ from Corollary 6.1, we can employ Propositions 5.2 and 5.3, where formulas for ${\mathbb E} V_k\big(K_n^{(1)}\big)$ and ${\mathbb E} V_k\big(K_n^{(1)}\cap K_n^{(2)}\big)$ have been derived. The missing piece is an explicit expression for the expected number ${\mathbb E} N\big(K_n^{(1)}\cap K_n^{(2)}\big)$ of isolated points.

Proposition 6.2. Let $K^{(1)}, K^{(2)}$ be independent fractal percolations on the interval [0, 1], both with the same parameters M and p. Then, for any $n\in{\mathbb N}_0$,

\begin{align*} {\mathbb E} &N\big(K_n^{(1)}\cap K_n^{(2)}\big)\\ &=(Mp^2)^n\left(2-2 M^{-n}-4p \frac{M-1}{M-p}\left[1-\left(\frac pM\right)^n\right]+2p^2\frac{M-1}{M-p^2}\left[1-\left(\frac {p^2}M\right)^n\right]\right)\!. \end{align*}

Proof. First observe that for $n=0$ both sides of the formula equal zero and thus the formula holds in this case. Let $N(n)\,:\!=N\big(K_n^{(1)}\cap K_n^{(2)}\big)$, and let $N_j(n)\,:\!=N(K_n^{(1)}\cap K_n^{(2)}\cap O_j)$, $j=1,\ldots,M$, be the number of those isolated points contained in the open subinterval $O_j\,:\!=\text{int}(J_j)=((j-1)/M,j/M)$. Then obviously

(6.7) \begin{align} N(n)=\sum_{j=1}^M N_j(n) +\sum_{j=1}^{M-1} \textbf{1}\big\{j/M \text{ isolated in } K_n^{(1)}\cap K_n^{(2)}\big\}.\\[-12pt] \end{align}

Because of the self-similarity, $N_j(n)$ has the same distribution as $\widetilde{N}(n-1)$, where $\widetilde{N}(n-1)$ is the random variable which equals $N(n-1)$ with probability $p^2$ and is zero otherwise (which accounts for the effect that $J_j$ may be discarded in the first construction step of $K^{(1)}$ or $K^{(2)}$, in which case there are no isolated points generated). Moreover, by symmetry, the indicator variables in the second sum all have the same distribution, given by

\begin{align*} {\mathbb P}\big(\big\{1/M \text{ isolated in } K_n^{(1)}\cap K_n^{(2)}\big\}\big)=2 p^{2n}(1-p^n)^2=\!:\,q_n(p). \end{align*}

Indeed, in order for $1/M$ to be isolated, either both $K_n^{(1),1}$ and $K_n^{(2),2}$ (with $K_n^{(i),j}$ as defined in (4.1)) need to have a nonempty intersection with $1/M$ (which happens with probability $p^{2n}$) while at the same time neither $K_n^{(1),2}$ nor $K_n^{(2),1}$ intersects $1/M$ (which happens with probability $(1-p^n)^2$); or we must have the same situation exactly reversed, i.e. $K_n^{(1),2}$ and $K_n^{(2),1}$ intersect $1/M$ while $K_n^{(1),1}$ and $K_n^{(2),2}$ do not. Taking expectations in (6.7), we get

\begin{align*} {\mathbb E} N(n)&=\sum_{j=1}^M p^2 {\mathbb E} N(n-1) +(M-1) {\mathbb P}\big(\big\{1/M \text{ isolated in } K_n^{(1)}\cap K_n^{(2)}\big\}\big)\\ &= Mp^2 {\mathbb E} N(n-1)+(M-1)q_n(p), \end{align*}

which is a recursion relation for the sequence $(\gamma_n)_{n\in{\mathbb N}}$ with $\gamma_n\,:\!={\mathbb E} N(n)$. By induction, we infer that

\begin{align*} \gamma_n=(M-1)q_n(p)+ Mp^2\gamma_{n-1} =\ldots =(M-1)\sum_{s=1}^{n} (Mp^2)^{n-s} q_{s}(p).\\[-12pt] \end{align*}

Since $q_s(p)=2 p^{2s} (1-2p^s+p^2s)=2 (p^{2s}-2p^{3s}+p^{4s})$, we conclude that

\begin{align*} \gamma_n&=2(M-1)(Mp^2)^{n}\sum_{s=1}^{n} (Mp^2)^{-s} (p^{2s}-2p^{3s}+p^{4s})\\ &=2(M-1)(Mp^2)^{n}\sum_{s=1}^{n} \left((1/M)^s-2 (p/M)^s+(p^2/M)^s\right)\\ &= 2(M-1)(Mp^2)^{n} \Bigg( \left[1-\left(\frac1M\right)^n\right]-\frac{2p}{M-p}\left[1-\left(\frac pM\right)^n\right] +\frac{p^2}{M-p^2}\bigg[1-\left(\frac{p^2}M\right)^n\bigg] \Bigg), \end{align*}

from which the expression stated in Proposition 6.2 easily follows.

Now we are ready to derive explicit expressions for ${\mathbb E} V_k\big(D_n^{(1)}\big)$ and ${\mathbb E} V_k\big(D_n^{(1)}\cap D_n^{(2)}\big)$ from Corollary 6.1.

Theorem 6.2. Let $K^{(1)}, K^{(2)}$ be independent fractal percolations on the interval $I=[0,1]$, both with the same parameters $M\in{\mathbb N}_{\geq 2}$ and $p\in(0,1]$. For $n\in{\mathbb N}_0$, let $K_n^{(i)}$ be the nth step of the construction of $K_i$, $i=1,2$, and let $D_n^{(i)}\,:\!=\overline{I\setminus K_n^{(i)}}$.

Then, for any $n\in{\mathbb N}_0$, ${\mathbb E} V_1\big(D_n^{(1)}\cap D_n^{(2)}\big)=1-2p^n+p^{2n}$ and

\begin{align*} {\mathbb E} V_0\big(D_n^{(1)}\cap D_n^{(2)}\big)\,=\,&2 (Mp)^n\left(1-p\frac{M-1}{M-p}\left[1-\left(\frac pM\right)^n\right]\right)+1-4p^n+2p^{2n}\\ &+(Mp^2)^n\left({-}1+p^2\frac{M-1}{M-p^2}\left[1-\left(\frac {p^2}M\right)^n\right]\right)\!. \end{align*}

Moreover, we have ${\mathbb E} V_1\big(D_n^{(1)}\big)=1-p^n$ and

\begin{align*} {\mathbb E} V_0\big(D_n^{(1)}\big)= (Mp)^n\left(1-p\frac{M-1}{M-p}\left[1-\left(\frac {p}M\right)^n\right]\right)+1-2p^n. \end{align*}

Proof. Combine Corollary 6.1 with Propositions 5.2, 5.3, and 6.2. The two formulas for $V_1$ and also the one for ${\mathbb E} V_0\big(D_n^{(1)}\big)$ follow at once. In case of ${\mathbb E} V_0\big(D_n^{(1)}\cap D_n^{(2)}\big)$ observe that, for any $n\in{\mathbb N}_0$,

\begin{align*}{\mathbb E} &N\big(K_n^{(1)}\cap K_n^{(2)}\big)-{\mathbb E} V_0\big(K_n^{(1)}\cap K_n^{(2)}\big)\\ &=(M p^2)^n\left(2-2 M^{-n}-4p \frac{M-1}{M-p}\left[1-\left(\frac pM\right)^n\right]+2p^2\frac{M-1}{M-p^2}\left[1-\left(\frac {p^2}M\right)^n\right]\right)\\ &\quad -(Mp^2)^n\left(3-2 M^{-n}-4p \frac{M-1}{M-p}\left[1-\left(\frac pM\right)^n\right]+p^2\frac{M-1}{M-p^2}\left[1-\left(\frac {p^2}M\right)^n\right]\right)\\&=(Mp^2)^n\left({-}1+ p^2\frac{M-1}{M-p^2}\left[1-\left(\frac {p^2}M\right)^n\right]\right)\\&=(Mp^2)^n\left(\frac{M(p^2-1)}{M-p^2}-\frac{(M-1)p^2}{M-p^2}\left(\frac {p^2}M\right)^n\right), \end{align*}

and thus ${\mathbb E} V_0\big(D_n^{(1)}\cap D_n^{(2)}\big)$ equals

\begin{align} 2{\mathbb E} V_0\big(K_n^{(1)}\big)+1-4p^n+2p^{2n} +(Mp^2)^n\left({-}1+ p^2\frac{M-1}{M-p^2}\left[1-\left(\frac {p^2}M\right)^n\right]\right)\!. \end{align}

It is now easy to derive explicit expressions for the limit functionals $\overline{\mathcal V}^c_k(K)=\lim_{n\to\infty} r^{(D-k)n}{\mathbb E} V_k(D_{n})$ of fractal percolation K in ${\mathbb R}$ for all possible parameters.

Corollary 6.2. Let K be a fractal percolation on the interval $I=[0,1]$ with parameters $M\in{\mathbb N}_{\geq 2}$ and $p\in(0,1]$. If $K_n$ is the nth construction step and $D_{n}\,:\!=\overline{I\setminus K_{n}}$, then

\begin{align*} \lim_{n\to\infty} {\mathbb E} V_1(D_{n})=\begin{cases}1\quad \textit{ for } p\in(0,1) & \textit{(while } \overline{\mathcal V}^c_1(K)= \infty \textit{),}\\ 0\quad \textit{ for } p=1 &\textit{(which equals } \overline{\mathcal V}^c_1(K) \textit{ in this case).} \end{cases} \end{align*}

Moreover,

\begin{align*} \overline{\mathcal V}^c_0(K)=\begin{cases}\frac{M(1-p)}{M-p} & \textit{ for } p\in(1/M,1] \quad \textit{(which equals } \overline{\mathcal V}_0(K); \textit{ cf.\ Corollary \linkref{cor:Vk-dim1}{5.1}),}\\[4pt] \frac{2M+1}{M+1} & \textit{ for } p=1/M,\\[4pt] \infty & \textit{ for } p\in(0,1/M).\end{cases} \end{align*}

Proof. Since here

\begin{equation*}D=\frac{\log Mp}{\log M}\end{equation*}

(cf. (1.2)), which implies $Mp=r^{-D}$, we infer from Theorem 6.2 that, for any $n\in{\mathbb N}$, $r^{(D-1)n}{\mathbb E} V_1(D_{n})=p^{-n}-1$ and

\begin{align*} r^{Dn}{\mathbb E} V_0(D_{n})= 1-p\frac{M-1}{M-p}\left[1-\left(\frac {p}M\right)^n\right]+(Mp)^{-n}(1-2p^n). \end{align*}

Letting now $n\to\infty$, the stated limits follow at once.

The case $\textbf{\textit{d}}=\textbf{2}$. In ${\mathbb R}^2$, the formula (6.1) in Theorem 6.1 reduces to

(6.8) \begin{align} \overline{\mathcal V}^c_k(F) =q_{2,k} \frac{M^{2-k}(1-p)}{M^{2-k}p-1}-E_1-E_2+E_3-E_4, \end{align}

where

\begin{align*} E_1&\,:\!=2M(M-1)\sum_{n=1}^\infty r^{n(D-k)} {\mathbb E} V_k\big(C_n^1\cap C_n^4\big),\notag\\ E_2&\,:\!=2(M-1)^2 \sum_{n=1}^\infty r^{n(D-k)} {\mathbb E} V_k\big(C_n^1\cap C_n^2\big), \notag \\ E_3&\,:\!=4(M-1)^2 \sum_{n=1}^\infty r^{n(D-k)} {\mathbb E} V_k\big(C_n^1\cap C_n^2\cap C_n^3\big), \notag \\ E_4&\,:\!=(M-1)^2 \sum_{n=1}^\infty r^{n(D-k)} {\mathbb E} V_k\bigg(\bigcap_{j=1}^4 C_{n}^j\bigg) .\notag \end{align*}

Here the sets $C_n^1, \ldots, C_n^4$ are four of the $M^2$ sets $C_{n}^j=\overline{J_j\setminus F_n^j}$, chosen so that the corresponding sets $J^j$, $j=1,\ldots,4$, intersect in a point x and are numbered as indicated in Figure 7. The factor in front of the summation in each $E_i$ indicates how many times this particular intersection configuration occurs in the union

\begin{equation*}C_{n}=\bigcup_{j=1}^{M^2} C_{n}^j,\end{equation*}

taking into account all symmetries.

Theorem 6.1 asserts that, for $k=0$, the formula (6.8) is valid for all $p\in(1/M^2,1]$, and for $k=1$, for all $p\in(1/M,1]$. While for $E_1$ we will again employ the one-dimensional case, the last three summands $E_2, E_3$, and $E_4$ vanish for $k=1$, since the intersections involved contain at most one point. For $k=0$ these terms can be obtained by direct inspection of the intersections of the $C_{n}^j$.

Lemma 6.2. Suppose $p>1/M^2$. Then, for $k=0$, the terms $E_2, E_3$, and $E_4$ in (6.8) are given by

\begin{align*} E_2&= 2(M-1)^2\left(\frac{1}{M^2p-1}-\frac 2{M^2-1}+\frac {p}{M^2-p}\right),\\ E_3&=4(M-1)^2\left(\frac{1}{M^2p-1}-\frac 3{M^2-1}+\frac {3p}{M^2-p}-\frac{p^2}{M^2-p^2}\right),\end{align*}

and

\begin{align*} E_4&= (M-1)^2\left(\frac{1}{M^2p-1}-\frac 4{M^2-1}+\frac {6p}{M^2-p}-\frac{4p^2}{M^2-p^2}+\frac{p^3}{M^2-p^3}\right)\!. \end{align*}

Proof. In all three intersection configurations of the sets $C_{n}^j$ considered here, the intersection contains at most one point, x; cf. Figure 7. For each of the sets $C_{n}^j$ to contain x, it is necessary that in at least one of the sets $F^j_k$, $k=1,\ldots, n$, the kth-level subsquare intersecting x is discarded (which happens with probability $1-p^n$). Thus, by independence, we obtain for the intersections of $\ell=2$, 3, or 4 of these sets

\begin{align*} {\mathbb E} V_0\bigg(\bigcap_{j=1}^\ell C_{n}^j\bigg)={\mathbb P}\bigg(\bigcap_{j=1}^\ell C_{n}^j=\{x\}\bigg)=(1-p^n)^\ell, \end{align*}

and therefore

(6.9) \begin{align} \sum_{n=1}^\infty r^{nD} {\mathbb E} V_0\bigg(\bigcap_{j=1}^\ell C_{n}^j\bigg)&=\sum_{k=0}^\ell \binom{l}{k} ({-}1)^k\frac{p^{k-1}}{M^2-p^{k-1}}. \end{align}

Indeed, employing the binomial theorem and the relation $r^{-D}=M^2p$, we get

\begin{align*} \sum_{n=1}^\infty r^{nD} {\mathbb E} V_0\bigg(\bigcap_{j=1}^\ell C_{n}^j\bigg)&=\sum_{n=1}^\infty (M^2p)^{-n}(1-p^n)^\ell=\sum_{k=0}^\ell \binom{l}{k}({-}1)^k \sum_{n=1}^\infty \left(\frac{p^{k-1}}{M^2}\right)^{n}, \end{align*}

where in the last expression all the geometric series converge thanks to the assumption $p>1/M^2$. Computing these series yields the expression stated in (6.9). Finally, the assertion of the lemma follows from plugging (6.9) into the expressions for $E_2$, $E_3$, and $E_4$ given by (6.8).

The expressions derived in Theorem 6.2 for intersections of fractal percolations in one dimension will now be used to compute the expected intrinsic volumes of the (one-dimensional) intersections $C_n^1\cap C_n^4$ appearing in the term $E_1$ in the formula (6.8) for fractal percolation in ${\mathbb R}^2$.

Proposition 6.3. Let F be a fractal percolation in ${\mathbb R}^2$ with parameters $M\in{\mathbb N}_{\geq 2}$ and $p\in(0,1]$. Then, for any $n\in{\mathbb N}$,

\begin{align*} {\mathbb E} V_1\big(C_n^1\cap C_n^4\big)&=\frac 1M\left(1-2p^{n}+p^{2n}\right)\end{align*}

and

\begin{align*} {\mathbb E} V_0\big(C_n^1\cap C_n^4\big)&=2(Mp)^n\left(\frac{1-p}{M-p}+\frac{M-1}{M-p}\left(\frac {p}M\right)^{n}\right)+1-4p^{n}+2p^{2n}\\&\quad-(Mp^2)^n\left(\frac{1-p^2}{M-p^2}+\frac{M-1}{M-p^2}\left(\frac {p^2}M\right)^{n}\right)\!. \end{align*}

Proof. Let $K^{(1)}, K^{(2)}$ be two independent fractal percolations on the interval $I=[0,1]$ with the same parameters M and p as F, and independent of F. For $n\in{\mathbb N}_0$, let $K_n^{(i)}$ be the nth step of the construction of $K^{(i)}$, $i=1,2$, and let $D_n^{(i)}\,:\!=\overline{I\setminus K_n^{(i)}}$ (just as in Corollary 6.1). Denote by $\widetilde K_n^{(i)}$, $i=1,2$, the random set which equals $K_n^{(i)}$ with probability p and is empty otherwise. Recalling from (5.20) that in distribution

\begin{equation*} F_n^1\cap F_n^4=\psi \big(\widetilde K_{n-1}^{(1)}\cap \widetilde K_{n-1}^{(2)}\big), \end{equation*}

we infer that, since $C_{n}^j$ is determined by $F_n^j$ and similarly $D_{n-1}^{(i)}$ is determined by $K_{n-1}^{(i)}$, the following equation also holds in distribution: for each $n\in{\mathbb N}$,

(6.10) \begin{align} C_n^1\cap C_n^4=\psi \big(\hat D_{n-1}^{(1)}\cap \hat D_{n-1}^{(2)}\big), \end{align}

where $\hat D^{(i)}_n$ is the random set which equals $D^{(i)}_n$ with probability p and I with probability $1-p$. This implies that for each $n\in{\mathbb N}_0$,

\begin{align*} {\mathbb E} V_0\big(C^1_{n+1}\cap C^4_{n+1}\big)&={\mathbb E} V_0\big(\hat D_n^{(1)}\cap \hat D_n^{(2)}\big)\\ &=p^2 {\mathbb E} V_0\big(D_n^{(1)}\cap D_n^{(2)}\big)+ p(1-p) {\mathbb E} V_0\big(D_n^{(1)}\cap I\big)\\ &\quad +p(1-p) {\mathbb E} V_0\big(I\cap D_n^{(2)}\big)+(1-p)^2 {\mathbb E} V_0(I\cap I)\\ &=p^2 {\mathbb E} V_0\big(D_n^{(1)}\cap D_n^{(2)}\big)+2 p(1-p) {\mathbb E} V_0\big(D_n^{(1)}\big)+(1-p)^2. \end{align*}

Employing now the formulas derived in Theorem 6.2 for ${\mathbb E} V_0\big(D_n^{(1)}\cap D_n^{(2)}\big)$ and ${\mathbb E} V_0\big(D_n^{(1)}\big)$, we obtain for each $n\in{\mathbb N}_0$

\begin{align*} {\mathbb E} &V_0\left(C^1_{n+1}\cap C^4_{n+1}\right)\\ &=p^2 \left(\vphantom{\left(\frac{p^2}{M}\right)^n} 2 (Mp)^n\left(1-p\frac{M-1}{M-p}\left[1-\left(\frac pM\right)^n\right]\right)+1-4p^n+2p^{2n}\right.\\ &\quad\qquad \left. +\,(Mp^2)^n\left({-}1+p^2\frac{M-1}{M-p^2}\left[1-\left(\frac {p^2}M\right)^n\right]\right) \right)\\ &\quad +2p(1-p)\left((Mp)^n\left(1-p\frac{M-1}{M-p}\left[1-\left(\frac {p}M\right)^n\right]\right)+1-2p^n \right) +(1-p)^2\\ &=2p (Mp)^n\left(\vphantom{\left(\frac{p^2}{M}\right)^n}1-p\frac{M-1}{M-p}\left[1-\left(\frac {p}M\right)^n\right]\right)+1-4p^{n+1}+2p^{2(n+1)}\\ &\qquad\qquad\quad +p^2 (Mp^2)^n\left({-}1+p^2\frac{M-1}{M-p^2}\left[1-\left(\frac {p^2}M\right)^n\right]\right),\end{align*}

where we combined some of the terms to get to the last expression. Replacing $n+1$ by n, this simplifies to

\begin{align*} {\mathbb E} V_0\left(C_n^1\cap C_n^4\right)&=\frac 2M (Mp)^n\left(1-p\frac{M-1}{M-p}\left[1-\left(\frac {p}M\right)^{n-1}\right]\right)+1-4p^{n}+2p^{2n}\\ &\quad +\frac 1M (Mp^2)^n\left({-}1+p^2\frac{M-1}{M-p^2}\left[1-\left(\frac {p^2}M\right)^{n-1}\right]\right)\\ &=2(Mp)^n\left(\frac{1-p}{M-p}+\frac{M-1}{M-p}\left(\frac {p}M\right)^{n}\right)+1-4p^{n}+2p^{2n}\\ &\quad-(Mp^2)^n\left(\frac{1-p^2}{M-p^2}+\frac{M-1}{M-p^2}\left(\frac {p^2}M\right)^{n}\right),\end{align*}

for any $n\in{\mathbb N}$, completing the proof of the formula for $V_0$ in Proposition 6.3. The formula for $V_1$ follows similarly from (6.10) using the corresponding formulas from Theorem 6.2:

\begin{align*} {\mathbb E} V_1&\big(C^1_{n+1}\cap C^4_{n+1}\big)={\mathbb E} V_1\big(\psi\big(\hat D_n^{(1)}\cap \hat D_n^{(2)}\big)\big)=\frac 1M {\mathbb E} V_1\big(\hat D_n^{(1)}\cap \hat D_n^{(2)}\big)\\ &=\frac 1M\left( p^2 {\mathbb E} V_1\big(D_n^{(1)}\cap D_n^{(2)}\big)+2 p(1-p) {\mathbb E} V_1\big(D_n^{(1)}\big)+(1-p)^2\right)\\ &=\frac 1M\left(p^2\left[1-2p^n+p^{2n}\right]+2 p(1-p) \left[1-p^n\right]+(1-p)^2\right)\\ &=\frac 1M\big(1-2p^{n+1}+p^{2(n+1)}\big)\!. \end{align*}

Corollary 6.3. If $p>1/M^2$, then for $k=0$ the term $E_1$ in (6.8) is given by

\begin{align*} E_1&=\frac{2M(M-1)}{M-p}\left(\frac{2(1-p)}{M-1}+\frac{2(M-1)p}{M^2-p}-\frac{p(1-p^2)}{M-p^2}\right)\\ &\quad-\frac{2M(M-1)^2 p^3}{(M-p^2)(M^2-p^3)} +\frac{2M(M-1)}{M^2p-1}-\frac {8M}{M+1}+\frac{4M(M-1)p}{M^2-p}. \end{align*}

Similarly, if $p>1/M$, then for $k=1$,

\begin{equation*}E_1=2(M-1)\left(\frac 1{Mp-1}-\frac 2{M-1}+\frac p{M-p}\right)\!.\end{equation*}

Proof. To determine $E_1$ for $k=0$, we multiply the expression derived in Proposition 6.3 for ${\mathbb E} V_0\left(C_n^1\cap C_n^4\right)$ by $r^{Dn}=(M^2p)^{-n}$ and sum over n to obtain

(6.11) \begin{align} E_1&=2M(M-1)\sum_{n=1}^\infty (M^2p)^{-n}\left[\vphantom{\left(\frac{p^2}{M}\right)^n}2(Mp)^n\left(\frac{1-p}{M-p}+\frac{M-1}{M-p}\left(\frac {p}M\right)^{n}\right)\right.\end{align}

\begin{align} &\notag \qquad\qquad\qquad\qquad\quad\qquad\quad\ \left.-\,(Mp^2)^n\left(\frac{1-p^2}{M-p^2}+\frac{M-1}{M-p^2}\left(\frac {p^2}M\right)^{n}\right)+1-4p^{n}+2p^{2n}\right],\end{align}

and the expression stated above is then derived by computing the various geometric series (which do all converge, thanks to the assumption $p>1/M^2$, justifying thus in particular the above interchange of summations) and combining some of the terms. For $k=1$, the stated expression for $E_1$ follows similarly by multiplying the expression for ${\mathbb E} V_1\!\left(C_n^1\cap C_n^4\right)$ from Proposition 6.3 by $r^{(D-1)n}=(Mp)^{-n}$ and summing over n. The series involved converge due to the assumption $p>1/M$.

Now we are ready to prove Theorem 1.3. For convenience, we repeat the statement here, and we add a corresponding formula for the limit $\overline{\mathcal V}^c_1(F)$ of the rescaled boundary lengths.

Proposition 6.4. Let F be a fractal percolation in ${\mathbb R}^2$ with parameters $M\in{\mathbb N}_{\geq 2}$ and $p\in [0,1]$. Then, for any $p>1/M^2$,

\begin{align*} \overline{\mathcal V}^c_0(F)&= M^2(1-p)\frac{p^3+(M-1)p^2+(M-1)p-M}{(M^2-p^3)(M-p)}. \end{align*}

Moreover, for any $p>1/M$,

\begin{align*} \overline{\mathcal V}^c_1(F)&= 2M \frac{1-p}{M-p}\quad\left(=\overline{\mathcal V}_1(F); \quad \text{ cf. {\rm(\linkref{thm:Vk-limit-dim2}{1.2})}}\right)\!. \end{align*}

Proof of Proposition 6.4 and thus in particular of Theorem 1.3. All one has to do is to insert the expressions for $E_1,\ldots, E_4$ obtained in Lemma 6.2 and Corollary 6.3 into the formula (6.8) for $\overline{\mathcal V}^c_k(F)$. For $k=0$, we obtain (recalling that $q_{2,0}=V_0(J)=1$)

(6.12) \begin{align} \overline{\mathcal V}^c_0(F)&=\frac{M^{2}(1-p)}{M^{2}p-1}-E_1-E_2+E_3-E_4\notag \\ &=\frac{M^{2}(1-p)}{M^{2}p-1}-\frac{2M(M-1)}{M-p}\left(\frac{2(1-p)}{M-1}+\frac{2(M-1)p}{M^2-p}-\frac{p(1-p^2)}{M-p^2}\right)\\ \notag &\quad+\frac{2M(M-1)^2 p^3}{(M-p^2)(M^2-p^3)} -\frac{2M(M-1)}{M^2p-1}+\frac {8M}{M+1}-\frac{4M(M-1)p}{M^2-p}\\ \notag &\quad -E_2+E_3-E_4, \end{align}

where

\begin{align*} -E_2+E_3-E_4&=(M-1)^2\left(\frac{1}{M^2p-1}-\frac 4{M^2-1}+\frac {4p}{M^2-p}-\frac{p^3}{M^2-p^3}\right)\!.\end{align*}

Fortunately, this can be simplified to the expression stated above. Similarly, for $k=1$ (taking into account that $E_2=E_3=E_4=0$ in this case), we get

\begin{align*} \overline{\mathcal V}^c_1(F)&= 2M\frac{1-p}{Mp-1}-2(M-1)\left(\frac{1}{Mp-1}-\frac{2}{M-1}+\frac{p}{M-p}\right), \end{align*}

which simplifies to the expression stated above. This completes the proof.

Remark 6.2. (On the speed of convergence.) From the proof of Proposition 6.4, we also get explicit expressions for the expected intrinsic volumes of the approximation sets $C_m$ for each $m\in{\mathbb N}$. To determine $\overline{v}^c_k(m)\,:\!=r^{m(D-k)}{\mathbb E} V_k(C_m)$, $m\in{\mathbb N}$, it is enough to truncate all the sums in the formula (6.8) after the mth term (including the very first one, which appears already in summed form in (6.8); cf. (6.5)). For $k=0$, we obtain for any $m\in{\mathbb N}$

\begin{align*} \overline{v}^c_0(m)&=\frac{1-p}p\sum_{n=0}^m r^{nD}-E_1(m)-2E_2(m)+4 E_3(m)-E_4(m),\end{align*}

where $E_1(m)$, the truncated term corresponding to $E_1$, can be read off from the equation (6.11) in the proof of Corollary 6.3:

\begin{align*} E_1(m)&\,:\!= 2M(M-1)\sum_{n=1}^m \left[\vphantom{\left(\frac{p^2}{M}\right)^n}2\frac{1-p}{M-p}\left(\frac 1M\right)^n+2\frac{M-1}{M-p}\left(\frac {p}{M^2}\right)^{n}-\frac{1-p^2}{M-p^2}\left(\frac pM\right)^n\right.\\ &\qquad\qquad\qquad\qquad\left.-\frac{M-1}{M-p^2}\left(\frac {p^3}{M^2}\right)^{n}+\left(\frac 1{M^2p}\right)^n - 4 \left(\frac 1{M^2}\right)^n +2\left(\frac p{M^2}\right)^n\right]. \end{align*}

Similarly, $E_\ell(m)$, $\ell=2,3,4$, are derived by truncating the corresponding sums $E_\ell$ and computing the resulting finite geometric sums (cf. (6.9)):

\begin{align*} E_\ell(m)&\,:\!=(M-1)^2 \sum_{n=1}^m (M^2p)^{-n}(1-p^n)^\ell\\&=(M-1)^2\sum_{k=0}^\ell \binom{\ell}{k} ({-}1)^k\frac{p^{k-1}}{M^2-p^{k-1}}\left[1-\left(\frac{p^{k-1}}{M^2}\right)^m\right]. \end{align*}

Computing all the finite geometric sums, we get for each of the terms in the equation (6.12) a corresponding one for $\overline{v}^c_0(m)$ with a factor of the form $\left(1-q^m\right)$ for a suitable q (just as in the last line of the formula above). The constant terms add up to $\overline{\mathcal V}^c_0(F)$, so that we end up with the following exact expansion in m:

(6.13) \begin{align} \overline{v}^c_0(m)&=\overline{\mathcal V}^c_0(F)+\frac{4M(1-p)}{M-p} M^{-m} - \frac{2M(M-1)p(1-p^2)}{(M-p)(M-p^2)} M^{(D-3)m} + M^{-Dm}\notag\\ &\quad -4 M^{-2m}+\frac{4p(M-1)}{M-p} M^{(D-4)m}+ \tilde c M^{(3D-8)m},\end{align}

where

\begin{equation*}\tilde c\,:\!=\frac{(M-1)p^3}{M^2-p^3}\frac{(M-1)(M-p^2)-2M(M-p)}{M-p^2}.\end{equation*}

It is easy to see that this sequence again converges exponentially fast to $\overline{\mathcal V}^c_0(F)$ as $m\to\infty$.

Remark 6.3. (On fractal subdimensions.) Multiplying (6.13) by $M^{Dm}$, we obtain an exact expansion for ${\mathbb E} V_0(C_m)$:

\begin{align*} {\mathbb E} V_0(C_m)&=\overline{\mathcal V}^c_0(F) M^{Dm}+\frac{4M(1-p)}{M-p} M^{(D-1)m} - \frac{2M(M-1)p(1-p^2)}{(M-p)(M-p^2)} M^{(2D-3)m}\\ &\quad + 1-4 M^{(D-2)m}+\frac{4p(M-1)}{M-p} M^{(2D-4)m}+ \tilde c M^{(4D-8)m}.\end{align*}

Since $D<2$ for $p<1$, the last three terms vanish as $m\to\infty$. The remaining terms determine the subdimensions of F in the sense of [Reference Schönhöfer and Mecke26]. We obtain

\begin{align*} {\mathbb E} V_0(C_m) &=\overline{\mathcal V}^c_0(F) M^{Dm}+c_2 M^{D_2 m} +c_3 M^{D_3 m}+1+o(1)\end{align*}

as $m\to\infty$, where

\begin{equation*}c_2\,:\!=\frac{4M(1-p)}{M-p}\end{equation*}

is the amplitude of the first subdimension $D_2\,:\!=D-1$, and

\begin{equation*}c_3\,:\!=- \frac{2M(M-1)p(1-p^2)}{(M-p)(M-p^2)}\end{equation*}

is the amplitude of second subdimension

\begin{equation*}D_3\,:\!=2D-3=\frac{\log Mp^2}{\log M}.\end{equation*}

Recall from Proposition 5.4 and Remark 5.4 that $D_3$ (which is positive for $p>1/\sqrt{M}$) is the dimension of the intersection of two copies of F constructed in neighboring squares sharing a common side. Similarly $D_2$ (which is positive for $p>1/M$) is the dimension of a fractal percolation on an interval with the same parameters as F, or equally the dimension of $F\cap \partial[0,1]^2$. Hence two subdimensions appear for these random fractals, as suggested by [Reference Knüfing, Schollmeyer, Riegler and Mecke15, Reference Schönhöfer and Mecke26], and they carry geometric meaning as in the deterministic setting studied there.

7. Proof of Propositions 5.1 and 6.1

Let $n\in{\mathbb N}$ and let $W_1, W_2,\ldots$ be unions of subcubes of $J=[0,1]^d$ of level n. More precisely, if $\Omega^{1},\Omega^{2},\ldots$ are arbitrary subsets of $\{1,\ldots,M^d\}^n$, then we let

(7.1) \begin{align} W_i\,:\!=\bigcup_{\sigma\in \Omega^i} J_\sigma,\qquad i\in{\mathbb N}.\end{align}

Our first aim is to establish a general bound on the curvature of the intersection $W_1\cap W_2\cap\ldots \cap W_\ell$ for an arbitrary number $\ell\in{\mathbb N}_{\ge 2}$ of these sets. For this we will employ an estimate from [Reference Winter29]. Recall from [Reference Winter29] that for any finite family $\mathcal{X}=\{X_1,\ldots, X_m\}$ of sets, the intersection number $\Gamma=\Gamma(\mathcal{X})$ is defined by

\begin{align*} \Gamma\,:\!=\max_{i\in\{1,\ldots,m\}}|\{j\,:\, X_j\cap X_i\neq\emptyset\}|.\end{align*}

If $\Gamma$ is small compared to m, then the following estimate, which is a special case of [Reference Winter29, Corollary 3.0.5] for a family of convex sets, is particularly useful.

Lemma 7.1. Let $\{X_1,\ldots, X_m\}$ be a family of compact, convex subsets of ${\mathbb R}^d$ and let $\Gamma$ be its intersection number. Then, for any $k\in\{0,\ldots,d\}$,

\begin{equation*}C^\textsf{var}_k\bigg(\bigcup_{j=1}^m X_j\bigg)\le m 2^\Gamma b_k,\end{equation*}

where $b_k\,:\!=\max\{C_k(X_j)\,:\,j=1,\ldots,m\}$.

Proof. Since the $X_i$ are convex, any intersection $X_I\,:\!=\bigcap_{i\in I} X_i$, $I\subseteq\{1,\ldots,m\}$ is also convex and contained in each of the sets $X_i$, $i\in I$. Therefore, the monotonicity and positivity of the intrinsic volumes implies that

\begin{equation*}C_k^{\textsf{var}}(X_I)=C_k(X_I)\leq \max_{i\in I} C_k(X_i)\leq b_k.\end{equation*}

Thus (for $B\,:\!={\mathbb R}^d$) the assumptions of [Reference Winter29, Corollary 3.0.5] are satisfied with $b\,:\!=b_k$ and the assertion follows.

Recall that $r=1/M$.

Lemma 7.2. There is a constant $c_{d,k}$ such that for any $n\in{\mathbb N}$ and $\ell\in{\mathbb N}_{\geq 2}$ and any collection $W_1,\ldots,W_\ell$ of unions of cubes of level n as defined in (7.1), the following estimate holds:

(7.2) \begin{align} C_k^{\textsf{var}}(W_1\cap \ldots \cap W_\ell)\leq c_{d,k} |\Omega^{1}| r^{kn}, \end{align}

where $|\Omega^{1}|$ is the number of cubes in $W_1$.

Note that $|\Omega^{1}|\leq M^{dn}=r^{-dn}$, which implies that the right-hand side of (7.2) is always bounded from above by $c_{d,k} r^{(k-d)n}$.

Proof. First let $\ell=2$. We write the intersection $W_1\cap W_2$ as a union of convex sets. It is clear that each set $J_\sigma$ in the union $W_1$ intersects at most $3^d$ cubes (the neighboring ones) from the union $W_2$. Let ${\Omega}^{2}_{\sigma}\subset \Omega^{2}$ be the set of indices of the cubes from $W_2$ intersecting $J_\sigma$. Then $|\Omega^{2}_\sigma|\leq 3^d$ and we have

\begin{align*} W_1\cap W_2=\bigcup _{\sigma\in \Omega^{1}} \bigg(J_\sigma\cap \bigcup _{\omega\in \Omega^{2}_\sigma} J_\omega\bigg) =\bigcup _{\sigma\in \Omega^{1}} \bigcup _{\omega\in \Omega^{2}_\sigma} \left(J_\sigma\cap J_\omega\right)\!. \end{align*}

In this way we have represented $W_1\cap W_2$ as a union of at most $|\Omega^{1}|\cdot 3^d\leq(M^d)^n 3^d$ convex sets $R_{\sigma,\omega}\,:\!=J_{\sigma}\cap J_{\omega}$. Note that each of the sets $R_{\sigma,\omega}$ is the intersection of two cubes and thus a k-face of some cube of level n (of some dimension $k\in\{0,\ldots,d\}$). We may reduce the number of sets in this representation by deleting the double occurrences of any face without changing the union set. Then the reduced family $\mathcal{F}\subset\{R_{\sigma,\omega}\}$ has an intersection number $\Gamma=\Gamma(\mathcal{F})$ bounded from above by $3^d$ times the number of faces of a cube in ${\mathbb R}^d$ (which also equals $3^d$). Indeed, each set $R\in\mathcal{F}$ is contained in a cube of dimension d, and any other set $R'\in\mathcal{F}$ intersecting R must be a face of the same cube or of one of the neighboring cubes. Note also that each of the sets $R\in\mathcal{F}$ is convex and contained in a cube of side length $r^n$. Therefore,

\begin{align*} C_k^{\textsf{var}}(R)= C_k(R)\leq C_k(r^n J) = r^{kn} C_k(J)=r^{kn} q_{d,k}, \end{align*}

where we have used the monotonicity, motion-invariance, and homogeneity of the intrinsic volumes. Now we can apply Lemma 7.1 to the family $\mathcal{F}$ consisting of $m\leq(M^d)^n 3^d$ sets and satisfying $b_k\,:\!=\max\{C_k(R)\,:\,R\in \mathcal{F}\}\leq r^{kn} q_{d,k}$. We obtain

\begin{align*} C_k^{\textsf{var}}(W_1\cap W_2)\leq |\Omega^{1}| 3^d 2^{3^{2d}} r^{kn} q_{d,k}= c_{d,k} |\Omega^{1}|r^{kn}, \end{align*}

where the constant

\begin{equation*}c_{d,k}\,:\!= 3^d 2^{3^{2d}} q_{d,k}\end{equation*}

is independent of n. This proves the case $\ell=2$. For the general case, fix some $\ell>2$ and note that $W_1\cap\ldots\cap W_\ell$ can be represented by

(7.3) \begin{align} W_1\cap \ldots \cap W_\ell =\bigcup _{\sigma\in \Omega^{1}} \bigcup _{\omega_2\in \Omega^{2}_\sigma} \ldots \bigcup _{\omega_\ell\in \Omega^{\ell}_\sigma} J_\sigma\cap J_{\omega_2}\cap \ldots\cap J_{\omega_\ell}, \end{align}

where, similarly as before, $\Omega^{j}_\sigma$ is the family of those words $\omega\in\Omega^{j}$ for which $J_\sigma \cap J_{\omega} \neq\emptyset$, $j=2,\ldots,\ell$. Now observe that $J_\sigma\cap J_{\omega_2}\cap \ldots\cap J_{\omega_\ell}$ is a finite intersection of cubes of the grid and thus a k-face of $J_\sigma$ (of some dimension $k\in\{0,\ldots,d\}$)—if not empty. Hence, for fixed $\sigma$, there are at most $3^d$ distinct sets in the union corresponding to the faces of $J_\sigma$. Deleting all multiplicities so that no set appears more than once in the union on the right of (7.3), we again end up with a representation of $W_1\cap \ldots \cap W_\ell$ by at most $|\Omega^{1}|\cdot 3^d$ convex sets, and as before, the intersection number of the reduced family will not exceed $3^{2d}$. Hence (7.2) follows again from Lemma 7.1 with the same constant $c_{d,k}$ as before. This completes the proof for arbitrary integers $\ell\geq 2$.

Remark 7.1. If the union set $W_1$ contains many ‘interior’ cubes of the intersection $\bigcap_{j=1}^\ell W_j$, then the above estimate can be improved. Recall that, for $k\leq d-1$, the kth curvature measure of any set is concentrated on its boundary. Therefore, in order to bound the total curvature variation of the set $\bigcap_{j=1}^\ell W_j$, it is enough to represent this set near its boundary by a union of cubes. To this end, let $\Omega_\partial^1\subseteq \Omega^1$ be the set of those cubes in $\Omega^1$ which have a nonempty intersection with the boundary $\partial\bigcap_{j=1}^\ell W_j$. Let A be the set defined by the right-hand side of (7.3) when we replace the set $\Omega^1$ in the first union by $\Omega^1_\partial$. Then A is obviously a union of cubes and a subset of $\bigcap_{j=1}^\ell W_j$. Moreover, it has the property that $\partial\bigcap_{j=1}^\ell W_j\subset \partial A$ and that any point $x\in\partial A$ that is not in $\partial\bigcap_{j=1}^\ell W_j$ has positive distance (at least $r^n$) to $\partial\bigcap_{j=1}^\ell W_j$. Therefore, since curvature measures are locally determined, we conclude that (for any $k\in\{0,\ldots, d-1\}$)

\begin{align*} C_k^\textsf{var}(W_1\cap\ldots \cap W_\ell)\leq C_k^\textsf{var}(A)\leq c_{d,k} |\Omega^{1}_\partial| r^{kn}, \end{align*}

where the second inequality follows from applying the same argument to A that we applied in the proof of Lemma 7.2 to the set on the right-hand side of (7.3).

With Lemma 7.2 and Remark 7.1 in hand, we are now in a position to prove Propositions 5.1 and 6.1.

Proof of Propositions 5.1 and 6.1. First note that in both statements the second assertion is an immediate consequence of the first one, which is due to the fact that $|V_k(K)|\leq C_k^\textsf{var}(K)$ for any polyconvex set K. In order to prove the first assertions in the two propositions, we fix a set $T\subseteq\{1,\ldots,M^d\}$, $|T|\geq 2$, and let $U\,:\!=\bigcap_{j\in T} J_j$ (where $J_j$ is the cube of side length $r=1/M$ containing $F^j$). U is a cube of some dimension $u\in\{0,\ldots, d-1\}$, and the intersection $\bigcap_{j\in T} F_n^j$ is contained in U (and similarly $\bigcap_{j\in T} C_{n}^j\subseteq U$). Let H be the affine hull of U, which is a u-dimensional affine space. Since intrinsic volumes are independent of the dimension of the ambient space, it is enough to study the intersection of the sets $F_n^j\cap H$ (or $C_{n}^j\cap H$, respectively), for $j\in T$, in the space H. The sets $F^j\cap H$ can be modeled by fractal percolations on u-dimensional cubes. Let $K^{(j)}$, $j\in T$, be independent fractal percolations on $[0,1]^u$ with the same parameters p and M as F. Denote by $\widetilde{K}^{(j)}$ the random set which equals $K^{(j)}$ with probability p and is empty otherwise. Then we have, for each $n\in{\mathbb N}$, $F_n^j\cap H=\psi\big(\widetilde{K}_n^{(j)}\big)$, $j\in T$, in distribution, where $\psi\,:\,H\to{\mathbb R}^{u}$ is one of the similarities (with factor $1/r$) mapping U to $[0,1]^{u}$. (In fact, it is possible to couple $F^j$ and $\widetilde{K}^{(j)}$ in such a way that this distributional relation becomes an almost sure one. But we do not need this here for our argument.) In particular, it follows that