2. General terminology and notation

We write e for the identity element of a group.

For finite sets K and F of a group G, we define the K-boundary of F by

\[\partial_K F = \{ t\in G : Kt \cap F\neq\emptyset \text{ and } Kt \cap (G\setminus F)\neq\emptyset \} . \]

For $\delta \gt 0$ , we say that F is (left) $(K,\delta )$ -invariant if $|\partial_K F| \leq\delta |F|$ . By the Følner characterization of amenability, the group G is amenable if and only if it admits a nonempty finite $(K,\delta )$ -invariant set for every finite set $K\subseteq G$ and $\delta \gt 0$ .

Let $G\curvearrowright X$ be a free continuous action of a countable group on a compact metric space (we only consider free actions in this paper). By a tower we mean a pair (S,V) where S is a finite subset of G (the shape) and V is a subset of X (the base) such that the sets sV for $s\in S$ (the levels) are pairwise disjoint. The tower is open if V is open and clopen if V is clopen. The footprint of the tower is the set SV.

A castle is a finite collection $\{ (S_i ,V_i ) \}_{i\in I}$ of towers such that the sets $S_i V_i$ for $i\in I$ are pairwise disjoint. The castle is open if each of the towers is open and clopen if each of the towers is clopen. The footprint of the castle is the set $\bigsqcup_{i\in I} S_i V_i$ .

Let A and B be subsets of X. We say that A is subequivalent to B and write $A\prec B$ if for every closed set $C \subseteq A$ there are finitely many open sets $U_1 , \dots , U_n$ which cover C and elements $s_1 , \dots , s_n \in G$ such that the sets $s_i U_i$ for $i=1,\dots , n$ are pairwise disjoint and contained in B. For a nonnegative integer m we write $A\prec_m B$ if for every closed set $C\subseteq A$ there are a finite collection $\mathscr{U}$ of open subsets of X which cover C, an $s_U \in G$ for every $U \in \mathscr{U}$ , and a partition of $\mathscr{U}$ into subcollections $\mathscr{U}_0 , \dots , \mathscr{U}_m$ such that for every $i = 0,\ldots,m$ the sets $s_U U$ for $U \in \mathscr{U}_i$ are pairwise disjoint and contained in B.

The action $G\curvearrowright X$ has comparison if $A\prec B$ for all nonempty open sets $A,B\subseteq X$ which satisfy $\mu (A) \lt \mu (B)$ for every G-invariant Borel probability measure $\mu$ on X. It has m-comparison for a nonnegative integer m if $A \prec_m B$ for all nonempty open sets $A,B \subseteq X$ which satisfy $\mu (A) \lt \mu (B)$ for all G-invariant Borel probability measures $\mu$ on X. In these definitions one can equivalently take A to range over closed sets instead of open ones [Reference KerrKer20, Proposition 3.4].

The action $G\curvearrowright X$ is almost finite if for every $n\in\mathbb{N}$ , finite set $K\subseteq G$ , and $\delta \gt 0$ there exist:

1. (i) an open castle $\{ (S_i , V_i ) \}$ each of whose shapes is $(K,\delta )$ -invariant and each of whose levels has diameter less than $\delta$ ; and

2. (ii) sets $S_i' \subseteq S_i$ such that $|S_i' | \leq |S_i |/n$ and $X\setminus \bigsqcup_{i\in I} S_i V_i \prec \bigsqcup_{i\in I} S_i' V_i$ .

Note that the shape condition in part (i) implies that G is amenable. In the case that G is finite, the action $G\curvearrowright X$ is almost finite if and only if X is zero-dimensional (note that for $n \gt |G|$ the sets $S_i'$ in the above definition will have to be empty).

It is worth noting (although we will not need this fact) that when X is zero-dimensional we can characterize almost finiteness by the existence, for every finite set $K\subseteq G$ and $\delta \gt 0$ , of an open castle whose shapes are $(K,\delta )$ -invariant and whose footprint is the entire space X (see [Reference KerrKer20, Theorem 10.2]).

When G is amenable, the upper and lower densities (or G-densities if we wish to emphasize the acting group) of a set $A\subseteq X$ are defined by

\begin{gather*}\overline{D}_G (A) = \inf_F \sup_{x\in X} \frac{1}{|F|} \sum_{s\in F} 1_A (sx)\quad \text{and}\quad\underline{D}_G (A) = \sup_F \inf_{x\in X} \frac{1}{|F|} \sum_{s\in F} 1_A (sx),\end{gather*}

where F ranges in both cases over the nonempty finite subsets of G. Writing $M_G (X)$ for the set of all G-invariant Borel probability measures on X, we can alternatively express the upper density as $\sup_{\mu\in M_G (X)} \mu (A)$ when A is closed and the lower density as $\inf_{\mu\in M_G (X)} \mu (A)$ when A is open (see [Reference Kerr and SzabóKS20, Proposition 3.3]).

Almost finiteness in measure for the action $G\curvearrowright X$ is defined in the same way as almost finiteness except that condition (ii) is replaced by the requirement that $X\setminus \bigsqcup_{i\in I} S_i V_i$ have upper density less than $\delta$ (uniform smallness in measure). By [Reference Kerr and SzabóKS20, Theorem 5.6], the action $G\curvearrowright X$ is almost finite in measure if and only if it has the small boundary property, which asks that X have a basis of open sets whose boundaries are null for every G-invariant Borel probability measure on X. By Theorem 6.1 of [Reference Kerr and SzabóKS20] the following are equivalent:

1. (i) the action is almost finite;

2. (ii) the action is almost finite in measure and has comparison;

3. (iii) the action is almost finite in measure and has m-comparison for some m.

3. Finite extensions

Let G be a finite extension of a countable group H. Let $G\curvearrowright X$ be a free continuous action on compact metrizable space.

Proof. Since the action $H\curvearrowright X$ is almost finite, by the results recalled in § 2 it has the small boundary property. Since every G-invariant Borel probability measure on X is also H-invariant, it follows that the action $G\curvearrowright X$ has the small boundary property. Thus to show that $G\curvearrowright X$ is almost finite it suffices, again by the discussion in § 2, to prove that it has m-comparison for some m. Suppose that for some open sets $A, B \subseteq X$ we have $\mu(A) \lt \mu(B)$ for every G-invariant Borel probability measure $\mu$ on X. Let $g_1, \dots, g_n$ be representatives for the left cosets of H in G with $g_1 = e$ . Since the action $H\curvearrowright X$ is almost finite, it follows from the proof of [Reference Kerr and SzabóKS20, Lemma 7.4] that the set A can be covered by $n+1$ open sets $A_1 , \dots , A_{n+1}$ such that $\nu(A_i) \lt ({1}/{n})\nu(A)$ for every i and every H-invariant Borel probability measure $\nu$ on X (to construct the $(n+1)$ th set take the closed complement of the footprint of the open castle in the proof of [Reference Kerr and SzabóKS20, Lemma∼7.4] and enlarge it to an open set whose measure is only slightly larger for every H-invariant Borel probability measure, as is possible by [Reference KerrKer20, Lemma 3.3]). Given such a measure $\nu$ , the Borel probability measure

\[\overline{\nu}(D) = \frac{1}{n}(\nu(g_1D) + \nu(g_2D) + \cdots + \nu(g_nD)) \]

is G-invariant, and for every $i = 1,\dots ,n+1$ we have

\begin{align*}\nu(A_i) \le n\overline{\nu}(A_i) \lt \overline{\nu}(A)< \overline{\nu}(B) &=\frac{1}{n}(\nu(g_1B) + \cdots + \nu(g_nB)) \\&\le \nu(g_1B \cup \cdots \cup g_nB).\end{align*}

Given a closed subset C of A and taking closed sets $C_i \subseteq A_i$ such that $C = \bigcup_{i=1}^{n+1} C_i$ , the fact that the action of H has comparison (by virtue of being almost finite) thus yields, for every i, pairwise disjoint open sets $U_{i, 1}, \dots, U_{i, k_i}\subseteq g_1B \cup \cdots \cup g_nB$ and $h_{i,1} , \dots , h_{i,k_i} \in H$ such that $C_i \subseteq\bigcup_{k=i}^{k_i} h_{i,k} U_{i,k}$ . For every $1 \le i \le n+1$ and $1 \le j \le n$ the sets $W_{i,j,k} := g_j^{-1}(g_jB \cap U_{i, k})$ for $k=1,\dots ,k_i$ are pairwise disjoint and contained in B, and we have $C \subseteq \bigcup_{i=1}^{n+1} \bigcup_{j=1}^n \bigcup_{k=1}^{k_i} h_{i,k} g_j W_{i,j,k}$ . This shows that $G\curvearrowright X$ has $n(n+1)$ -comparison.

4. Two lemmas

We collect here two lemmas that will be needed for the proof of Theorem 5.4. The first concerns conditions under which the join of finitely many disjoint collections of subsets of a group, when restricted to an ambient Følner set S, will mostly consist of Følner sets, where the degree of approximate invariance is prescribed but necessarily much lower than that of S, and ‘mostly’ is understood in the sense that the exceptional sets will have collective size proportionally small relative to $|S|$ . The second lemma is a version of the implication (ii) $\Rightarrow$ (i) in [Reference Kerr and SzabóKS20, Theorem 6.1] in which the hypotheses are relativized to a subgroup.

Let $\mathscr{F}$ be a collection of subsets of G. We say that a set $A\subseteq G$ is $\mathscr{F}$ -tileable if there is a $T\subseteq G$ and sets $F_t \in \mathscr{F}$ for $t\in T$ such that the sets $F_t t$ for $t\in T$ form a partition of A.

Lemma 4.1. Let $n\in\mathbb{N}$ . Let K be a finite subset of G and $\delta \gt 0$ . Let $\mathscr{F}$ be a finite collection of $(K,\delta^3 /(8|K|^2 n))$ -invariant finite subsets of G, and writing $D = (\bigcup\mathscr{F} )(\bigcup\mathscr{F} )^{-1} $ let S be a $(D^2 ,\delta^2 /(4|K|))$ -invariant finite subset of G. For each $i=1,\dots ,n$ let $\{ B_{i,1} , \dots , B_{i,m_i} \}$ be a finite disjoint collection of $\mathscr{F}$ -tileable finite subsets of G. For every $I\subseteq \{ 1,\dots ,n \}$ set $\Omega_I = \prod_{i\in I} \{ 1,\dots , m_i \}$ and for every $\omega\in\Omega_I$ set

\[B_\omega = \bigg( S \cap \bigcap_{i\in I} B_{i,\omega_i } \bigg)\setminus \bigg( \bigcup_{i\in I^\textrm{c}} \bigsqcup_{j=1}^{m_i} B_{i,j} \bigg). \]

Then the set $\Omega_0$ of all $\omega \in\Omega := \bigsqcup_{I\subseteq \{ 1,\dots ,n\}} \Omega_I$ such that $B_\omega$ fails to be $(K,\delta )$ -invariant satisfies $|\bigcup_{\omega\in\Omega_0} B_\omega | \leq \delta |S|$ .

For every $I\subseteq \{ 1,\dots ,n\}$ set $\Gamma_I = \prod_{i\in I} \{ (j,t) : 1\leq j\leq m_i ,\, t\in T_{i,j}'' \}$ . For each $\gamma = ((j_i , t_i ))_{i\in I} \in\Gamma_I$ define

\[E_\gamma = \bigg(\bigcap_{i\in I} F_{i,j_i ,t_i} t_i \bigg)\setminus \bigg(\bigcup_{i\in I^\textrm{c}} \bigsqcup_{j=1}^{m_i} B_{i,j} \bigg) \subseteq S\setminus \partial_D S \]

and note that these sets over all $\gamma\in \Gamma := \bigsqcup_{I\subseteq\{ 1,\dots ,n\}} \Gamma_I$ are pairwise disjoint. By the observation at the end of the first paragraph, for every $\gamma\in \Gamma$ we have

\[E_\gamma = \bigg(\bigcap_{i\in I} F_{i,j_i ,t_i} t_i \bigg)\cap \bigg(\bigcup_{i\in I^\textrm{c}} \bigsqcup_{j=1}^{m_i}\bigsqcup_{t\in T_{i,j}'} (G\setminus F_{i,j,t} t)\bigg), \]

and therefore

(4.1) \begin{gather}\partial_K E_\gamma \subseteq \bigcup_{i=1}^n \bigsqcup_{j=1}^{m_i} \bigsqcup_{t\in T_{i,j}'} \partial_K F_{i,j,t} t .\end{gather}

Write $\Gamma_0$ for the set of all $\gamma\in\Gamma$ such that $E_\gamma$ fails to be $(K,\delta /2)$ -invariant. Since the sets $E_\gamma$ are pairwise disjoint, each element of G belongs to $\partial_K E_\gamma$ for at most $|K|$ many $\gamma$ , and so

(4.2) \begin{gather}\sum_{\gamma\in\Gamma} |\partial_K E_\gamma |\leq |K| \bigg|\bigcup_{\gamma\in\Gamma} \partial_K E_\gamma \bigg| .\end{gather}

Also, since the sets $F_{i,j,t} t$ for $t\in T_{i,j}'$ are subsets of S and each element of S is contained in at most n of them, we have

(4.3) \begin{gather}\sum_{i=1}^n \sum_{j=1}^{m_i} \sum_{t\in T_{i,j}'} |F_{i,j,t} t | \leq n|S| .\end{gather}

We therefore obtain

Set $S_1 = S_0 \cup \bigsqcup_{\gamma\in\Gamma_0} E_\gamma$ . Then

\[|S_1 | \leq |S_0 | + \sum_{\gamma\in\Gamma_0} |E_\gamma |\leq \frac{\delta^2}{4|K|} |S| + \frac{\delta^2}{4|K|} |S|= \frac{\delta^2}{2|K|} |S| . \]

Writing $\Omega_1$ for the set of all $\omega\in\Omega$ such that $|B_\omega \cap S_1 | \gt \delta (2|K|)^{-1} |B_\omega |$ , we thereby obtain

\begin{gather*}\bigg|\bigsqcup_{\omega\in\Omega_1} B_\omega \bigg|= \sum_{\omega\in\Omega_1} |B_\omega |< \frac{2|K|}{\delta} \sum_{\omega\in\Omega_1} |B_\omega \cap S_1 |\leq \frac{2|K|}{\delta} |S_1 |\leq \delta |S| .\end{gather*}

To complete the proof it is therefore enough to verify that $\Omega_0 \subseteq\Omega_1$ .

Let $\omega\in\Omega\setminus\Omega_1$ . Then there is a $\Gamma_\omega \subseteq \Gamma$ such that $(S\setminus S_0 )\cap B_\omega = (S\setminus S_0 )\cap\bigsqcup_{\gamma\in\Gamma_\omega} E_\gamma$ and $E_\gamma \subseteq B_\omega$ for all $\gamma\in\Gamma_\omega$ , in which case we can write $B_\omega$ as the union of $B_\omega \cap S_1$ and $\bigsqcup_{\gamma\in\Gamma_\omega \setminus\Gamma_0} E_\gamma$ , so that

\begin{align*}|\partial_K B_\omega |&\leq |\partial_K (B_\omega \cap S_1 )|+ \sum_{\gamma\in\Gamma_\omega \setminus\Gamma_0} |\partial_K E_\gamma | \\&\leq |K||B_\omega \cap S_1 |+ \frac{\delta}{2} \sum_{\gamma\in\Gamma_\omega \setminus\Gamma_0} |E_\gamma | \\&\leq |K|\cdot \frac{\delta}{2|K|} |B_\omega | + \frac{\delta}{2} |B_\omega | \\&= \delta |B_\omega | .\end{align*}

This shows that $\omega\notin\Omega_0$ and hence that $\Omega_0 \subseteq\Omega_1$ .

Proof. If H is finite then every subset of X of upper H-density less than $|H|^{-1}$ is empty, so that when $\delta \lt |H|^{-1}$ the castles in part (ii) are clopen and have footprint equal to X, from which we deduce that $G\curvearrowright X$ is almost finite. We may thus assume that H is infinite. Let K be a finite subset of G and $0 \lt \delta \lt 1$ . Choose a finite set $e \in K' \subseteq H$ with $|K'| \gt 1/\delta$ . By assumption, there is an open castle $\{ (S_i ,V_i ) \}_{i\in I}$ for the G-action whose shapes are $(K \cup K',\delta )$ -invariant and the complement of whose footprint has upper H-density at most $\delta$ . In particular, it satisfies the first condition in the definition of almost finiteness with respect to K and $\delta$ . Choose a set R of representatives for the right cosets of H in G. For each $i \in I$ partition $S_i$ into subsets of right cosets of H, i.e.

\[S_i = \bigsqcup\limits_{g \in R}B_{i, g}g, \]

where each $B_{i, g}$ is contained in H. Note that left translation by K’ preserves the right cosets of H. If $B_{i, g}$ for some $g\in R$ has cardinality less than $1/\delta$ then all of its elements belong to $\partial_{K'} B_{i, g}$ and so $| \partial_{K'} B_{i, g}| \geq |B_{i, g}|$ . Writing L for the set of all $g\in R$ such that $0 \lt |B_{i, g}| \lt 1/\delta$ , it follows that

\[\sum_{g\in L} |B_{i,g} |\leq \sum_{g\in L} | \partial_{K'} B_{i, g}|\leq | \partial_{K'} S_i |\leq \delta |S_i|, \]

i.e. most elements of $S_i$ share a coset with at least $1/\delta$ other elements. For each $i \in I$ and $g \in R$ choose a set $B_{i, g}' \subseteq B_{i, g}$ with cardinality equal to $\lceil{\delta}/({1-\delta})|B_{i, g}|\rceil$ . Set $S_i' = \bigsqcup_{g \in R} B_{i, g}' g$ and note that when $|B_{i,g} | \geq 1/\delta$ we have $|B_{i, g}'|\leq {\delta}/({1-\delta})|B_{i, g}| + 1 \leq 2\delta |B_{i,g} |$ , so that

\[|S_i' |\leq \sum_{g\in L} |B_{i,g} | + \sum_{g\in R\setminus L} |B_{i,g}' |\leq 3\delta |S_i |. \]

Let $\mu$ be any H-invariant Borel probability measure on X. By construction, the set $\bigsqcup_{i\in I} S_i' V_i$ has $\mu$ -measure at least ${\delta}/({1-\delta})\mu(\bigsqcup_{i\in I} S_i V_i)$ , which is greater than or equal to $\delta$ . On the other hand, since the closed set $X\setminus \bigsqcup_{i\in I} S_i V_i$ has upper H-density less than $\delta$ its $\mu$ -measure is less than $\delta$ , and so our hypothesis that the H-action has comparison yields

\[X\setminus \bigsqcup_{i\in I} S_i V_i \prec \bigsqcup_{i\in I} S_i^\prime V_i. \]

Since we can take $\delta$ to be as small as we wish, this shows that the action $G\curvearrowright X$ is almost finite.

5. Extensions by $\mathbb{Z}$

Our goal here is to prove Theorem 5.4. We will need a version of the Ornstein–Weiss quasitiling theorem [Reference Ornstein and WeissOW87], which we record as Corollary 5.2. The statement is a simple consequence of the following more usual version, which we reproduce in the form presented in [Reference Kerr and LiKL16]. A finite subset K of a group G is said to be $\varepsilon$ -quasitiled by a finite collection $\mathscr{F} = \{ F_1 , \dots , F_n \}$ of finite subsets of G if there are sets $C_1 , \dots , C_n \subseteq G$ and $F_{i,c} \subseteq F_i$ with $|F_{i,c} | \geq (1-\varepsilon )|F_i|$ for every $i=1,\dots , n$ and $c\in C_i$ such that (i) the union $\bigcup_{i=1}^n F_i C_i$ is contained in K and has cardinality at least $(1-\varepsilon )|K|$ , and (ii) the collection $\{ F_{i,c}c : 1\leq i\leq n, \, c\in C_i \}$ is disjoint. As in § 4, we say that K is $\mathscr{F}$ -tileable if there are sets $C_1 , \dots , C_n \subseteq G$ such that the collection $\{ F_i c : 1\leq i\leq n, \, c\in C_i \}$ partitions K.

Let H be a countable group and $\alpha$ an automorphism of H, and form the corresponding semidirect product $H\rtimes\mathbb{Z}$ . Inside $H\rtimes\mathbb{Z}$ we view $\mathbb{Z}$ multiplicatively as the group $\langle g \rangle$ with generator g satisfying $g^m tg^{-m} = \alpha^m (t)$ for all $m\in\mathbb{Z}$ and $t\in H$ . When we say an interval in $\langle g \rangle$ we mean a set of the form $\{ g^m , g^{m-1} , \dots , g^n \}$ for some integers $m\leq n$ , and by the length of this interval we mean $n-m$ .

Proof. For every $A\subseteq X$ we have, with F ranging over the nonempty finite subsets of H,

\begin{align*}\overline{D}_H (A)= \inf_F \sup_{x\in X} \frac{1}{|F|}\sum_{s\in F} 1_A (sx)&= \inf_F \sup_{x\in X} \frac{1}{|F|}\sum_{s\in F} 1_{gA} (gsx) \\&= \inf_F \sup_{x\in X} \frac{1}{|\alpha (F)|}\sum_{s\in F} 1_{gA} (\alpha (s)gx) \\&= \inf_F \sup_{x\in X} \frac{1}{|\alpha (F)|}\sum_{t\in \alpha (F)} 1_{gA} (tgx) \\&= \overline{D}_H (gA) .\\[-32pt]\end{align*}

Set $K' = \bigcup_{m=-4r}^{4r} \alpha^m (K)$ and $K'' = \bigcup_{m=-4r}^{4r} \alpha^m (K') = \bigcup_{m=-8r}^{8r} \alpha^m (K)$ . By Corollary 5.2 there is a finite collection $\mathscr{F}$ of $(K'',\varepsilon^{15} /(8|K'|^2 (2r+1)r^{15}))$ -invariant finite subsets of H containing e such that for every $(\bigcup\mathscr{F} ,\varepsilon /(8r))$ -invariant finite set $F\subseteq H$ there exists an $\mathscr{F}$ -tileable $F' \subseteq F$ satisfying $|F' | \geq (1-{\varepsilon}/{2r} )|F|.$

Since $H\curvearrowright X$ is almost finite it is almost finite in measure, we can find an open castle $\{ (S_k , V_k ) \}_{k=1}^n$ for this action whose shapes are $(((\bigcup\mathscr{F} )(\bigcup\mathscr{F} )^{-1} )^2,\varepsilon^{10} /(4|K'|r^{10})))$ -invariant and whose footprint $\bigsqcup_{k=1}^n S_k V_k$ has lower H-density at least $1-{\varepsilon}/{2r}$ . The proof of [Reference Kerr and SzabóKS20, Theorem 5.6] shows that we may assume the boundary of each level of each tower in the castle to have zero upper H-density. By [Reference Kerr and SzabóKS20, Theorem 5.5], for any $k\in \{ 1,\dots , n\}$ we can find a finite disjoint collection $\mathscr{U}$ of open subsets of $V_k$ whose diameters are as small as we wish such that the set $V_k \setminus \bigcup\mathscr{U}$ has zero upper H-density. Since the action $H\rtimes\mathbb{Z} \curvearrowright X$ is free, we may therefore furthermore assume, by replacing each tower $(S_k ,V_k )$ with a collection of towers with shape $S_k$ whose bases are the members of a suitable collection of open subsets of $V_k$ of the type just described, that $(A^2 S_k , V_k )$ is a tower for every $k=1,\dots ,n$ . For each $k=1,\dots ,n$ , since the shape $S_k$ is $(\bigcup\mathscr{F} ,\varepsilon /(8r))$ -invariant we can find, by the previous paragraph, an $\mathscr{F}$ -tileable $B_k \subseteq S_k$ satisfying $|B_k | \geq (1-{\varepsilon}/{2r} )|S_k |$ .

Set $\mathscr{F}' = \{ \alpha^m (F) : -4r\leq m\leq 4r, \, F\in\mathscr{F} \}$ . Since each member of $\mathscr{F}$ is $(K'',\varepsilon^{15}/(8|K'|(8r+1)r^{15}))$ -invariant, each member of $\mathscr{F}'$ is $(K',\varepsilon^{15}/(8|K'|(8r+1)r^{15}))$ -invariant, as is easily checked using the boundary commutation formula $\partial_{K'} \alpha^i (F) = \alpha^m (\partial_{\alpha^{-m} (K')} F)$ for $m\in\mathbb{Z}$ and finite sets $F\subseteq H$ . Moreover, the fact that each $B_k$ is $\mathscr{F}$ -tileable implies that for every $k=1,\dots ,n$ and $m=-4r,\dots ,4r$ the set $\alpha^m (B_k )$ is $\mathscr{F}'$ -tileable.

We now carry out a recursive disjointification process over k. This will be similar to the Ornstein–Weiss-type argument in [Reference Conley, Jackson, Kerr, Marks, Seward and Tucker-DrobCJK+18, Reference Kerr and SzabóKS20] for producing dynamical tilings, but with two significant differences: (i) we carry out the disjointification procedure in one single recursion without having to repeat it across different scales, and (ii) the decision to retain a new piece at a given stage depends on whether its proportion in the ambient tile is greater than $\varepsilon$ (actually $\varepsilon^4$ in our case) instead of greater than $1-\varepsilon$ (the latter is what makes the recursion over different scales in the Ornstein–Weiss setting necessary, since at every scale only a small part of the space gets covered).

Write $\mathscr{T}_k$ for the collection of all subsets of $AB_k$ . Write $\mathscr{T}_k^+$ for the collection of all $T\in\mathscr{T}_k$ such that $|T|\geq \varepsilon^4 |AB_k |$ . We will recursively construct sets $Z_1 , \dots , Z_n \subseteq X$ and (not necessarily open) castles $\{ (T,V_{k,T} ) \}_{T\in\mathscr{T}_k^+}$ for $k=1,\dots , n$ such that $Z_k = Z_{k-1} \sqcup \bigsqcup_{T\in\mathscr{T}_k^+} TV_{k,T}$ for $k=2,\dots , n$ . Many of the sets $V_{k,T}$ will be empty, in part because nonemptiness forces some extra structure on the corresponding T (as explained below), but we will worry about this after the construction so as to not complicate notation.

We begin by setting $V_{1,T} = V_1$ for $T=AB_1$ and $V_{1,T} = \emptyset$ for $T\neq AB_1$ . We also put $Z_1 = AB_1 V_1$ .

Suppose now that $1 \lt k \le n$ and that we have constructed $Z_1 , \dots , Z_{k-1}$ and the castles $\{ (T,V_{j,T} ) \}_{T\in\mathscr{T}_j^+}$ for $j=1,\dots , k-1$ . For each $T\in\mathscr{T}_k$ set

(5.1) \begin{gather}V_{k,T} = V_k \cap \bigg(\bigcup_{s\in AB_k \setminus T} s^{-1} Z_{k-1} \bigg)\cap \bigg(\bigcup_{s\in T} (X\setminus s^{-1} Z_{k-1} ) \bigg) .\end{gather}

Then $\{ (T,V_{k,T} ) \}_{T\in\mathscr{T}_k^+}$ is a castle and we define $Z_k$ to be the union of the two sets $Z_{k-1}$ and $\bigsqcup_{T\in\mathscr{T}_k^+} TV_{k,T}$ , which are disjoint (in this construction we only care about $V_{k,T}$ for $T\in\mathscr{T}_k^+$ , but for later use we have defined $V_{k,T}$ above for all $T\in\mathscr{T}_k$ ). This completes the recursion. A very important observation is that, setting $V_k^+ = \bigsqcup_{T\in\mathscr{T}_k^+} V_{k,T} \subseteq V_k$ , we have

(5.2) \begin{align}Z_k = \bigcup_{j=1}^k AB_jV_j^+.\end{align}

Write Q for the remainder $X\setminus \bigsqcup_{k=1}^n B_kV_k$ of the castle $\{ (B_k ,V_k )\}_{k=1}^n$ . It will be convenient to have the following dual picture of the tower partition $Q\sqcup \bigsqcup_{k=1}^nB_kV_k$ in terms of the partial orbits within each tower. We define the equivalence relation E on X as the smallest under which xEgx whenever x is in some base $V_k$ and g is in the corresponding shape $B_k$ . Each equivalence class $[x]_E$ either belongs to a unique tower $B_kV_k$ or is a singleton in the remainder Q. For $m\in\mathbb{Z}$ we have

\[X = g^m \bigg(Q\sqcup \bigsqcup_{k=1}^nB_kV_k \bigg) = g^m Q \sqcup \bigsqcup_{k=1}^n (g^mB_kg^{-m})g^mV_k, \]

that is, X is also partitioned by the castle $\{(g^mB_kg^{-m}, g^mV_k )\}_{k=1}^n$ together with its remainder $g^mQ$ . We denote the corresponding equivalence relation by $(g^m)_*E$ . Note that $[x]_{(g^m)_*E} = g^m([x]_E)$ and therefore $x(g^m)_*Ey$ if and only if $g^{-m}xEg^{-m}y$ . For $q\in\mathbb{N}$ define

\begin{gather*}E^q = \bigcap_{m=-q}^q (g^m)_*E.\end{gather*}

Observe that for any $1 \leq k \leq n$ it follows from (5.2) that if $xE^ry$ then

\[x \in Z_k \Leftrightarrow y \in Z_k. \]

Now let $x \in V_k$ for some $1\leq k\leq n$ and consider the set $AB_kx \setminus Z_{k-1} = AB_kx \setminus \bigcup_{j=1}^{k-1}AB_jV_j^+$ . For a fixed element $h \in B_k$ the set $Ahx \setminus Z_{k-1}$ has the form $A_h hx$ where $A_h$ is a subinterval of A. Moreover, if $h^\prime x \in [hx]_{E^{2r}}$ then $g^mh^\prime x \in [g^mhx]_{E^{r}}$ for every $m=-r, \ldots, r$ and therefore $A_{h^\prime} = A_h$ . In other words, the interval $A_h$ only depends on the $E^{2r}$ -equivalence class of hx.

Consider now an equivalence relation $E^\prime$ defined by setting $h_0xE^\prime h_1x$ if $A_{h_0} = A_{h_1}$ . By the previous observation, every $E^\prime$ -equivalence class is a disjoint union of $E^{2r}$ -equivalence classes. The set of points in $AB_kx$ that are not covered by $Z_{k-1}$ , which is equal to Tx where $T\in\mathscr{T}_k$ is such that $x\in V_{k,T}$ , can be partitioned into ‘rectangles’ of the form $A_B Bx$ where Bx is an $E^\prime$ -equivalence class and $A_B$ is equal to $A_h$ for any h with $h\in B$ . By identifying T with Tx via $t\mapsto tx$ this yields a partition of T itself into sets of the form $A_B B$ , and this partition is the same for all x in $V_{k,T}$ .

Write $\mathscr{N}_k$ for the collection of all $T\in\mathscr{T}_k$ for which $V_{k,T}$ is nonempty. Let $T\in\mathscr{N}_k$ . Write $\mathscr{B}_{k,T}$ for the collection of all sets B appearing in the common partition of T into rectangles of the form $A_B B$ for points in $V_{k,T}$ , as described above. We can thus express T as $\bigsqcup_{B\in\mathscr{B}_{k,T}} A_B B$ . Write $\mathscr{C}_{k,T}$ for the collection of all $B\in\mathscr{B}_{k,T}$ such that B is $(K',\varepsilon )$ -invariant. Set $C_{k,T} = \bigsqcup \mathscr{C}_{k,T}$ . Since $S_k$ is $(((\bigcup\mathscr{F} )(\bigcup\mathscr{F} )^{-1} )^2,\varepsilon^{10} /(4|K'|r^{10}))$ -invariant, the members of $\mathscr{F}'$ are $(K',\varepsilon^{15} /(8|K'|(8r+1)r^{15}))$ -invariant, and the sets $\alpha^m (B_k )$ for $m=-4r,\dots ,4r$ and $k=1,\dots , n$ are $\mathscr{F}'$ -tileable, we can apply Lemma 4.1 (taking there $n=8r+1$ , $\delta = (\varepsilon/r)^5$ , $K=K'$ and the sets $B_{i,j}$ to be the right translates of $\alpha^i (B_k )$ which are at play in the description of an intersection pattern within an orbit) to see that

(5.3) \begin{align}|C_{k,T} | \geq (1-(\varepsilon/r)^5 ) |B_k | \gt (1-\varepsilon^5 ) |B_k | .\end{align}

Set $\mathscr{N}_k^+ = \mathscr{T}_k^+ \cap \mathscr{N}_k$ and $\mathscr{N}_k^- = \mathscr{N}_k \setminus \mathscr{T}_k^+$ .

Given a $T\in\mathscr{N}_k^+$ , let us verify that it is $(K\cup \{ g \} ,2\varepsilon )$ -invariant. Define $F_{k,T,1} = \bigsqcup_{h\in C_{k,T}} A_h h$ where $A_h$ is such that $A_h h V_{k,T} = Ah V_{k,T} \setminus Z_{k-1}$ (in accordance with our orbitwise discussion above), and set $F_{k,T,0} = T \setminus F_{k,T,1}$ . By the definition of $C_{k,T}$ and the boundary commutation formula $\partial_K (g^m F) = g^{-m} \partial_{\alpha^{-m} (K)} F$ for $m\in\mathbb{Z}$ and finite sets $F\subseteq H$ , the set $F_{k,T,1}$ is $(K,\varepsilon )$ -invariant, and since

we thus see that the set $T = F_{k,T,0} \sqcup F_{k,T,1}$ is $(K,2\varepsilon )$ -invariant. On the other hand, defining $D_{k,T,0}$ to be the set of all $h\in B_k$ such that $|A_h | \leq \varepsilon^5 r$ and setting $D_{k,T,1} = B_k \setminus D_{k,T,0}$ , we have

\[\bigg|\bigsqcup_{h\in D_{k,T,0}} A_h h \bigg| \leq \varepsilon^5 r |B_k | = \varepsilon^5 \cdot \frac{r}{2r+1} |AB_k |\leq \varepsilon^5\cdot \frac{1}{3\varepsilon^4}|T| \lt \varepsilon |T| \]

while $\bigsqcup_{h\in D_{k,T,1}} A_h h$ is $(\{ g \} , \varepsilon )$ -invariant by our choice of r, so that $T = \bigsqcup_{h\in D_{k,T,0}} A_h h \sqcup \bigsqcup_{h\in D_{k,T,1}} A_h h$ is $(\{ g \} , 2\varepsilon )$ -invariant.

Now let $T\in\mathscr{N}_k^-$ and define $B_{k,T,0}$ to be the set of all $h\in B_k$ such that $|A_h | \leq \varepsilon r$ (with $A_h$ having the same meaning as above with respect to k and T) and set $B_{k,T,1} = B_k \setminus B_{k,T,0}$ . Then

\begin{align*}|B_{k,T,1} | \cdot \varepsilon r \leq |T| \lt \varepsilon^4 |AB_k | = \varepsilon^4 (2r+1)|B_k |,\end{align*}

and hence

(5.4) \begin{align}|B_{k,T,1} | \lt 3\varepsilon^3 |B_k |,\end{align}

a fact that we will use later. Recall that $Z_n = \bigsqcup_{k=1}^n \bigsqcup_{T\in\mathscr{N}_k^+} TV_{k,T}$ (the footprint of the castle we have recursively constructed) and set

\[W = \bigsqcup_{k=1}^n B_kV_k \setminus Z_n. \]

We will next carry out a process by which we donate some of the points of W to the towers in our recursively constructed castle. By splitting up the towers according to how their shapes have been amplified orbit by orbit through this donation process, we obtain a new castle with a much larger collection of towers. The donation process will be carried out so that the amplified shapes are proportionally not much larger and hence still approximately invariant, and it will be sufficiently algorithmic so that the boundaries of the levels of the new towers will still have zero upper H-density. Once we have done this, it will remain to shave off the boundaries of the tower levels so that the towers become open, which will involve discarding a set of zero upper H-density, and then finally check that the remainder of the resulting castle has small H-density, for then we can apply Lemma 4.2 to finish.

To set up the donation operation, suppose that we have a subset of W of the form Jx for some $x \in X$ and some (possibly infinite) interval J in $\langle g \rangle$ which is maximal with respect to the inclusion of Jx in W. We will either donate none of Jx or all of it.

If $|J| \gt \varepsilon r$ we do not donate Jx. Assume then that $|J| \le \varepsilon r$ . By replacing x if necessary, we may assume that J has the form $\{ g^0 , g^{1} , \dots , g^q \}$ . Consider the point $g^{-1}x$ . Since it is not in W, it is either in $Q = X \setminus \bigsqcup_{k=1}^n B_kV_k$ or in $Z_n$ . In the first case, we do not donate Jx and only note that $Jx \subseteq \{g^1, \ldots, g^{\lceil \varepsilon r \rceil + 1}\}Q$ . Suppose now that $g^{-1}x \in Z_n$ , that is, $g^{-1}x = ty$ for some $y \in V_{k, T}$ and $t \in T \in \mathscr{N}_k^+$ . Consider the equivalence class $[ty]_{E^{2r}}$ . If it is $(K^\prime, \varepsilon)$ -invariant then we donate Jx to the tower containing Ty, thereby enlarging the partial orbit Ty within this tower. Otherwise, we again do not donate Jx.

Note that the sets $W, Q, TV_{k, T}$ , and $Z_n$ are $E^r$ -invariant, as can be easily seen from (5.2). Hence, if the interval Jx was donated to a tower via the partial orbit Ty, then, in fact, the whole set $Jg[g^{-1}x]_{E^{2r}}$ was donated to this tower via Ty. Through this procedure we obtain a new collection of towers $(T^\sharp , V_{k, T^\sharp} )$ . These clearly form a castle, which we denote by $\mathscr{L}_0$ . By our definition of K’ and the boundary commutation formula $\partial_K (g^m F) = g^{-m} \partial_{\alpha^{-m} (K)} F$ for all $m\in\mathbb{Z}$ and finite sets $F\subseteq H$ , we see that each new shape $T^\sharp$ is still $(K\cup \{ g \} , \varepsilon )$ -invariant (the invariance in the $\mathbb{Z}$ -direction only improves since the $\langle g \rangle$ cross-sections can only become longer intervals). Given that the set $H \cup \{ g \}$ generates $H\rtimes\mathbb{Z}$ , a standard exercise then shows that we can make the shape as left invariant as we wish by making an appropriate choice of K and taking $\varepsilon$ small enough.

By Lemma 5.3, the collection of subsets of X with upper H-density zero is $H\rtimes\mathbb{Z}$ -invariant, and so the sets $g(V_k \setminus V_k^\circ )$ for $k=1,\dots ,n$ and $g \in H\rtimes\mathbb{Z}$ all belong to this collection given that each $V_k \setminus V_k^\circ$ does. Since this collection is also an algebra we therefore deduce, in view of the algorithmic way in which the above construction proceeded based on intersection patterns, that the boundaries of the levels of the tower in $\mathscr{L}_0$ all have zero H-density. In view of the uniform continuity of the individual homeomorphisms making up the action of $H\rtimes\mathbb{Z}$ , we can also make the levels of the towers in $\mathscr{L}_0$ have as small a diameter as we wish by taking the diameters of the bases $V_1 , \dots , V_n$ of the initial towers to be sufficiently small.

Write $\mathscr{L}$ for the open castle consisting of the towers $(T^\sharp , V_{k,T^\sharp}^\circ )$ . This castle will be our witness for the almost finiteness of the $H\rtimes\mathbb{Z}$ -action. It remains to check that its remainder satisfies the smallness condition in the definition of almost finiteness, and for this it suffices to show that it is small in upper H-density, since the restriction action $H\curvearrowright X$ has comparison and we can apply Lemma 4.2.

More precisely, we will verify that the remainder of the castle $\mathscr{L}$ has upper H-density at most $7\varepsilon$ . By construction, it is contained in the union of the sets:

1. (i) $g(V_{k} \setminus V_{k}^\circ)$ for all k and $g \in H\rtimes\mathbb{Z}$ ;

2. (ii) $\{g^1, \ldots, g^{\lceil \varepsilon r \rceil + 1}\}Q$ ;

3. (iii) the union of all intervals of the form Jx in W with $|J| \gt \varepsilon r$ , denoted $W^\prime$ ; and

4. (iv) the union of sets of the form $\{g^1, \ldots, g^{\lceil \varepsilon r \rceil + 1}\}[x]_{E^{2r}}$ for all $[x]_{E^{2r}}$ that are not $(K^\prime, \varepsilon)$ -invariant.

We already observed that the sets in part (i) have zero upper H-density. For part (ii), recall that $X \setminus \bigsqcup_{k=1}^n S_kV_k$ has upper H-density less than $1 - {\varepsilon}/{2r}$ and that $|B_k| \gt (1 - {\varepsilon}/{2r})|S_k|$ . It follows that $\overline{D}_H(Q) \lt \varepsilon/r$ and therefore

\[\overline{D}_H(\{g^1, \ldots, g^{\lceil \varepsilon r \rceil + 1}\}Q) \le \lceil \varepsilon r \rceil\overline{D}_H(Q) \lt \varepsilon. \]

To estimate part (iii), consider a point $x \in V_{k, T}$ for some $T \in \mathscr{N}_k$ and let $hx \in B_kx \cap W^\prime$ . As hx is not in $Z_n$ , we conclude that $T \in \mathscr{N}_k^-$ . Moreover, it is clear that $|A_h| \gt \varepsilon r$ and thus $h \in B_{k, T, 1}$ . Applying (5.4) then yields

\[|W^\prime \cap B_kx| \lt 3\varepsilon^3|B_k| \]

for any $k=1, 2, \ldots, n$ and any $x \in V_k$ . Thus,

\[\overline{D}_H(W^\prime) \le 3\varepsilon^3\overline{D}_H\bigg(\bigsqcup_{k=1}^nB_kV_k\bigg) + \overline{D}_H(Q) \lt \varepsilon. \]

Finally, to estimate part (iv), let $x \in X \setminus \bigcup_{m=-2r}^{2r}g^mQ$ and suppose that $[x]_{E^{2r}}$ is not $(K', \varepsilon)$ -invariant. Clearly, $x = hy$ for some $y \in V_k$ and $h \in B_k$ and Lemma 4.1 shows (see (5.3) for a similar calculation) that there are at most $(\varepsilon/r)^5|B_k|$ such points x in the set $B_ky$ . Thus, the set in item (iv) has upper H-density at most

\[(4r + \lceil \varepsilon r \rceil + 1)\overline{D}_H(Q) + \lceil \varepsilon r \rceil (\varepsilon/r)^5 \lt 5\varepsilon. \]

Since upper H-density is monotone with respect to inclusions and subadditive with respect to finite unions, we conclude from the above estimates that the remainder of the castle $\mathscr{L}$ has upper H-density at most $7\varepsilon$ . By hypothesis the restricted action $H\curvearrowright X$ is almost finite and hence has comparison, and so it follows by Lemma 4.2 that the action $H\rtimes\mathbb{Z} \curvearrowright X$ is almost finite, as desired.