2.1 Non-Archimedean valued fields and normed spaces

A valuation (or absolute value) on a field $\mathbb {K}$ is a function $\lvert {\mspace {1mu}\cdot \mspace {1mu}}\rvert \colon\ \mathbb {K}\to [0,\infty )$ satisfying, for all $x,y\in \mathbb {K}$ , the following conditions:

1. (1) $\lvert x \rvert =0 \Leftrightarrow x=0$ ,

2. (2) $\lvert xy \rvert =\lvert x \rvert \lvert y \rvert $ , and

3. (3) $\lvert x+y \rvert \leq \lvert x \rvert +\lvert y \rvert $ .

The pair $(\mathbb {K},\lvert {\mspace {1mu}\cdot \mspace {1mu}}\rvert )$ , written simply as $\mathbb {K}$ if the valuation is known from the context, is then called a valued field. If we replace condition (3) with the stronger one (3’): $\lvert x+y\rvert \leq \max \{\lvert x\rvert ,\lvert y\rvert \}$ , then the valuation $\lvert {\mspace {1mu}\cdot \mspace {1mu}}\rvert $ is called non-Archimedean.

If $(X,d)$ is a metric space, $x_0\in X,$ and $r>0$ , we put

$$\begin{align*}\begin{aligned} B[x_0,r] &:= \{x\in X: d(x,x_0)\leq r\}, \\ B(x_0,r) &:= \{x\in X: d(x,x_0)<r\}, \\ S[x_0,r] &:= \{x\in X: d(x,x_0)= r\}. \end{aligned} \end{align*}$$

A metric $d$ on a set $X$ is called an ultrametric if it satisfies the strong triangle inequality: $d(x,z) \leq \max \{d(x,y), d(y,z)\}$ for all $x,y,z \in X$ . The pair $(X,d)$ is then called an ultrametric space. Any ultrametric space $(X,d)$ has the isosceles property: If $x,y,z \in X$ and $d(x,y)\neq d(y,z)$ , then $d(x,z) = \max \{d(x,y), d(y,z)\}$ . Note that, in an ultrametric space, any point belonging to a ball (open or closed) of a given radius may play the role of its Moreover, each of the sets $B[x_0,r]$ , $B(x_0,r)$ , $S[x,r]$ is closed and open in the topology of $(X,d)$ .

Each valuation $\lvert {\mspace {1mu}\cdot \mspace {1mu}}\rvert $ on a field $\mathbb {K}$ induces a metric $d\colon\ \mathbb {K}\times \mathbb {K} \to [0, \infty )$ by $d(x,y):=\lvert x-y \rvert $ . If $\lvert {\mspace {1mu}\cdot \mspace {1mu}}\rvert $ is non-Archimedean, $d$ is an ultrametric. We say that $(\mathbb {K},\lvert {\mspace {1mu}\cdot \mspace {1mu}}\rvert )$ is complete if $(\mathbb {K},d)$ is a complete metric space.

Let $(\mathbb {K},\lvert {\mspace {1mu}\cdot \mspace {1mu}}\rvert )$ be a fixed non-Archimedean valued field. We introduce the following notation: $D_{\mathbb {K}}= B[0,1]$ , $\mathfrak {m}_{\mathbb {K}}= B(0,1)$ , and $D_{\mathbb {K}}^{\times } = S[0,1].$ The ball $D_{\mathbb {K}}$ is a subring of $\mathbb {K}$ called the ring of integers (or valuation ring) of $\mathbb {K}$ ; $\mathfrak {m}_{\mathbb {K}}$ is the unique maximal ideal of $D_{\mathbb {K}}$ and $D_{\mathbb {K}}^{\times }$ is the group of units (i.e., invertible elements) of $D_{\mathbb {K}}$ (cf. [Reference Schneider14, Lemma 1.2]). The quotient ring $D_{\mathbb {K}}/\mathfrak {m}_{\mathbb {K}}$ is a field called the residue field of $(\mathbb {K},\lvert {\mspace {1mu}\cdot \mspace {1mu}}\rvert )$ .

We denote by $\mathbb {K}^{\times }$ the multiplicative group of $\mathbb {K}$ . The set $\lvert \mathbb {K}^{\times } \rvert :=\{\lvert x \rvert \colon x\in \mathbb {K}^{\times }\}$ is a subgroup of the multiplicative group $\mathbb {R}_{+}$ and is called the value group of $\mathbb {K}$ . If $\lvert \mathbb {K}^{\times } \rvert =\{1\}$ , the valuation $\lvert {\mspace {1mu}\cdot \mspace {1mu}}\rvert $ is called trivial. If the set $\lvert \mathbb {K}^{\times }\rvert \cap (0,1)$ has a greatest element $c$ , then $\lvert \mathbb {K}^{\times } \rvert =\{c^n: n\in \mathbb {Z}\}$ and the valuation is called discrete. In the remaining case, $\lvert \mathbb {K}^{\times } \rvert $ is a dense subset of $[0,\infty )$ and the valuation $\lvert {\mspace {1mu}\cdot \mspace {1mu}}\rvert $ is called dense.

Observe that if $\lvert \mathbb {K}^{\times }\rvert =\{c^n: n\in \mathbb {Z}\}$ , where $c\in (0,1)$ , then $B(x_0, c^k)=B[x_0,c^{k+1}]$ for all $x_0\in \mathbb {K}$ , $k\in \mathbb {Z}$ . Therefore, in the case of a discretely valued $\mathbb {K}$ , every ball is a closed one.

It is known that a non-Archimedean nontrivially valued field $(\mathbb {K}, \lvert {\mspace {1mu}\cdot \mspace {1mu}}\rvert )$ is locally compact if and only if it is complete, the valuation $\lvert {\mspace {1mu}\cdot \mspace {1mu}}\rvert $ is discrete, and the residue field $D_{\mathbb {K}}/\mathfrak {m}_{\mathbb {K}}$ is finite [Reference Schikhof13, Corollary 12.2].

A standard example of a non-Archimedean locally compact field is the field $(\mathbb {Q}_p,\lvert {\mspace {1mu}\cdot \mspace {1mu}}\rvert _p)$ of p-adic numbers for any prime $p$ . It is defined as the completion of $\mathbb {Q}$ with respect to the p-adic valuation, that is:

$$\begin{align*}\lvert 0 \rvert_p:= 0, \; \left\lvert p^k \frac{m}{n}\right \rvert_p:=p^{-k} \quad \text{for } k,m,n\in\mathbb{Z}, \quad p\nmid m, p\nmid n. \end{align*}$$

From now on, we will mostly consider fields $\mathbb {K}$ that are complete with respect to a discrete non-Archimedean valuation $\lvert {\mspace {1mu}\cdot \mspace {1mu}}\rvert $ . Such a $\mathbb {K}$ will be briefly called a complete discretely valued field. Sometimes it is useful to distinguish an element $\pi \in \mathbb {K}$ satisfying $\lvert \pi \rvert =c$ , where $c\in (0,1)$ is the generator of $\lvert \mathbb {K}^{\times }\rvert $ ; such a fixed $\pi $ is called a uniformizer.

Let $X$ be a linear space over a non-Archimedean valued field $(\mathbb {K},\lvert {\mspace {1mu}\cdot \mspace {1mu}}\rvert )$ . A function $\lVert {\mspace {1mu}\cdot \mspace {1mu}} \rVert \colon X\to [0, \infty )$ is called a non-Archimedean norm (briefly: norm) on $X$ if, for all $x,y \in X$ and $\alpha \in \mathbb {K}$ , we have the following conditions:

1. (1) $\lVert x \rVert =0 \Leftrightarrow x=0$ ,

2. (2) $\lVert \alpha x \rVert = \lvert \alpha \rvert \lVert x \rVert $ , and

3. (3) $\lVert x+y \rVert \leq \max \{\lVert x \rVert ,\lVert y \rVert \}$ .

The pair $(X,\lVert {\mspace {1mu}\cdot \mspace {1mu}} \rVert )$ is then called a non-Archimedean normed space over $\mathbb {K}$ . If additionally $\left \{\lVert x\rVert \colon x\in X\setminus \{0\}\right \}=\lvert \mathbb {K}^{\times }\rvert $ , the normed space $(X,\lVert {\mspace {1mu}\cdot \mspace {1mu}}\rVert )$ is called strongly solid. Any non-Archimedean norm $\lVert {\mspace {1mu}\cdot \mspace {1mu}} \rVert $ induces an ultrametric $d$ on $X$ , namely, $d(x,y):=\lVert x-y \rVert $ . If $X$ is complete with respect to $d$ , we call $X$ a non-Archimedean Banach space.

We note an important property of non-Archimedean Banach spaces. Namely, a series $\sum _{n=1}^{\infty } x_n$ , where $x_n\in X$ , is convergent if and only if $x_n \to 0$ (see [Reference Perez-Garcia and Schikhof11, Theorem 2.1.3]).

If $g\colon X\to X$ is an affine mapping $g(x):=L(x)+\tau $ with linear part $L\colon X\to X$ and translation part $\tau \in X$ , we will write $g=(L,\tau )$ . Clearly, $g$ is an isometry of $(X,\lVert {\mspace {1mu}\cdot \mspace {1mu}}\rVert )$ if and only if $L$ is.

Let $(\mathbb {K},\lvert {\mspace {1mu}\cdot \mspace {1mu}}\rvert )$ be a non-Archimedean valued field and $n\ge 1$ . Define

$$\begin{align*}\lVert {\mspace{1mu}\cdot\mspace{1mu}} \rVert_{\infty}\colon\ \mathbb{K}^n \to [0,\infty), \quad (x_1,\dots,x_n)\mapsto \max\{\lvert x_1\rvert, \dots, \lvert x_n\rvert\}. \end{align*}$$

The function $\lVert {\mspace {1mu}\cdot \mspace {1mu}}\rVert _{\infty }$ is a non-Archimedean norm on $\mathbb {K}^n$ and is called the maximum norm. The normed space $(\mathbb {K}^n,\lVert {\mspace {1mu}\cdot \mspace {1mu}} \rVert _{\infty })$ is obviously strongly solid. It is a Banach space if $\mathbb {K}$ is complete.

In analogy with the Euclidean norm on $\mathbb {R}^n$ , one can define $\lVert {\mspace {1mu}\cdot \mspace {1mu}} \rVert _{2}\colon\ \mathbb {K}^n \to [0,\infty )$ by $(x_1,\dots ,x_n)\mapsto (\lvert x_1\rvert ^2 + \dots + \lvert x_n\rvert ^2)^{1/2}$ . However, this function is not a non-Archimedean norm on $\mathbb {K}^n$ since it does not satisfy the defining condition (3) – the induced metric is not an ultrametric.

According to the characterization of linear isometries of $(\mathbb {K}^n,\lVert {\mspace {1mu}\cdot \mspace {1mu}}\rVert _{\infty })$ stated in [Reference Orzechowski9, Theorem 2.2] (cf. [Reference van Rooij19, Corollary 1.7]), a linear mapping $L\colon\ \mathbb {K}^2 \to \mathbb {K}^2$ with matrix $L=\left (\begin {smallmatrix} \alpha & \beta \\ \gamma & \delta \end {smallmatrix}\right )$ in the canonical basis of $\mathbb {K}^2$ is an isometry of $(\mathbb {K}^2, \lVert {\mspace {1mu}\cdot \mspace {1mu}}\rVert _{\infty })$ if and only if $L\in {\mathrm {GL}}(2,D_{\mathbb {K}})$ , that is, $\lvert \det {L}\rvert =1$ and $\lvert \alpha \rvert ,\lvert \beta \rvert ,\lvert \gamma \rvert ,\lvert \delta \rvert \le 1$ . Let ${\mathrm {SL}}(2,\mathbb {K}):=\{A \in M_2(\mathbb {K}): \det {A}=1\}$ denote the special linear group over $\mathbb {K}$ .

In [Reference Orzechowski10, Definition 2.1], we introduced certain specific families of groups of linear and affine transformations of $\mathbb {K}^2$ . We recall two of them, which will be useful later. Namely, for $\varepsilon \in (0,1)$ , let

(1) $$ \begin{align} {\mathrm{SL}}(2,D_{\mathbb{K}},\varepsilon)&:=\left\{\begin{pmatrix} \alpha & \beta\\ \gamma & \delta \end{pmatrix} \in {\mathrm{SL}}(2,\mathbb{K}): \lvert \alpha -1\rvert, \lvert \delta -1\rvert, \lvert \beta\rvert, \lvert \gamma\rvert \le \varepsilon\right\},\\ {\mathrm{SA}}(2,D_{\mathbb{K}},\varepsilon)&:=\{(L,\tau)\in {\mathrm{SL}}(2,D_{\mathbb{K}},\varepsilon)\times \mathbb{K}^{2}: \quad \lVert\tau\rVert_{\infty} \le \varepsilon\}.\nonumber \end{align} $$

Notice that the conditions $\lvert \alpha - 1\rvert , \lvert \delta -1\rvert \le \varepsilon < 1$ imply $\lvert \alpha \rvert =\lvert \delta \rvert =1$ , so ${\mathrm {SL}}(2,D_{\mathbb {K}},\varepsilon )$ is a subgroup of ${\mathrm {GL}}(2,D_{\mathbb {K}})$ for any $\varepsilon \in (0,1)$ . Therefore, all elements of ${\mathrm {SL}}(2,D_{\mathbb {K}},\varepsilon )$ and ${\mathrm {SA}}(2,D_{\mathbb {K}},\varepsilon )$ are isometries of $(\mathbb {K}^2,\lVert {\mspace {1mu}\cdot \mspace {1mu}}\rVert _{\infty })$ .

2.2 Laurent sequences

Let $R$ be a set of cardinality at least two with a distinguished element $0\in R$ . By a Laurent sequence over $R,$ we mean any function $\varphi \colon\ \mathbb {Z} \to R$ such that its support ${\mathrm {supp}}\ {\varphi }:=\{n\in \mathbb {Z} \colon \varphi (n)\ne 0\}$ is well-ordered. We denote by ${\mathrm {LS}}{(R)}$ the set of all Laurent sequences over $R$ .

For any number $c\in (0,1)$ , the formula

$$\begin{align*}d_c(\varphi,\psi):=\begin{cases} c^{\min\{n\in \mathbb{Z} \colon \varphi(n)\ne \psi(n)\}} & \text{if } \varphi\ne \psi,\\ 0 & \text{if } \varphi= \psi \end{cases} \end{align*}$$

defines an ultrametric on ${\mathrm {LS}}{(R)}$ . We denote by ${\mathrm {LS}}{(R,c)}$ the resulting ultrametric space. Note that if $R_1$ and $R_2$ are sets of equal cardinality and $c_1=c_2$ , then the ultrametric spaces ${\mathrm {LS}}{(R_1,c_1)}$ and ${\mathrm {LS}}{(R_2,c_2)}$ are isometric.

If $R=\mathbb {F}$ is a field, then a Laurent sequence over $\mathbb {F}$ is called a (formal) Laurent series over $\mathbb {F}$ , and we write $\mathbb {F}((T))$ for ${\mathrm {LS}}{(\mathbb {F})}$ . A Laurent series $\varphi \in \mathbb {F}((T))$ is traditionally written as the formal sum

(2) $$ \begin{align} \varphi = \sum_{n=-\infty}^{\infty} \varphi(n) T^n, \end{align} $$

with an indeterminate symbol $T$ . Addition and multiplication are defined in the usual way in $\mathbb {F}((T))$ (see [Reference Sambale12, Definition 7.1]). Thus, $\mathbb {F}((T))$ becomes a field [Reference Sambale12, Theorem 7.2], into which $\mathbb {F}$ is canonically embedded. For a given $c\in (0,1)$ , we introduce the following non-Archimedean valuation on $\mathbb {F}((T))$ :

(3) $$ \begin{align} \left\lvert \sum_{n=-\infty}^{\infty} \varphi(n) T^n \right\rvert:= c^{\min {\mathrm{supp}}\ {\varphi}} \, \text{ if } \varphi(k)\ne 0, \quad \lvert 0\rvert:=0. \end{align} $$

In this way, $\mathbb {F}((T))$ becomes a complete discretely valued field with the valuation ring $\mathbb {F}[[T]]:=\{\varphi \in \mathbb {F}((T))\colon {\mathrm {supp}}\ {\varphi }\subseteq \mathbb {N}_0\}$ and the residue field isomorphic to $\mathbb {F}$ (cf. [Reference Serre15, Example 2 on p. 6]). The metric induced by the valuation (3) is precisely the one of ${\mathrm {LS}}{(\mathbb {F},c)}$ .

2.3 Group actions and paradoxical decompositions

By a group action $G{\mspace {1mu}\curvearrowright \mspace {1mu}} X$ of a group $G$ on a set $X,$ we mean a function $G\times X \ni (g,x)\mapsto g.x\in X$ such that $(gh).x=g.(h.x)$ and $1.x=x$ for all $g,h\in G$ , $x\in X$ . An isomorphism of group actions $G{\mspace {1mu}\curvearrowright \mspace {1mu}} X$ and $H {\mspace {1mu}\curvearrowright \mspace {1mu}} Y$ is a pair $(f,\Phi )$ consisting of a bijection $f\colon X \to Y$ and a group isomorphism $\Phi \colon G\to H$ such that $\Phi (g).f(x)=f(g.x)$ for all $g\in G$ , $x\in X$ .

Let $X,Y$ be any sets and $G$ a group of bijections of $X$ . For any bijection $f\colon X\to Y$ , let $fGf^{-1}:=\{f\circ g\circ f^{-1}\colon g\in G\}$ , which is a group of bijections of $Y$ . Define $G \ni g \mapsto \Phi (g):=f\circ g \circ f^{-1}$ . Then $(f,\Phi )$ is clearly an isomorphism of group actions $G{\mspace {1mu}\curvearrowright \mspace {1mu}} X$ and $fGf^{-1}{\mspace {1mu}\curvearrowright \mspace {1mu}} Y$ , we call it the isomorphism induced by $f$ . In particular, if $X,Y$ are metric spaces and ${\mathrm {Iso}}\ {X},{\mathrm {Iso}}\ {Y}$ are their (full) isometry groups, then every isometry ${f\colon X\to Y}$ induces an isomorphism between the group actions ${\mathrm {Iso}}\ {X}{\mspace {1mu}\curvearrowright \mspace {1mu}} X$ and ${\mathrm {Iso}}\ {Y}{\mspace {1mu}\curvearrowright \mspace {1mu}} Y$ .

A group action $G{\mspace {1mu}\curvearrowright \mspace {1mu}} X$ is called locally commutative if, for all $x\in X$ , the stabilizer ${\mathrm {Stab}}_G{x}:=\{g\in G: g.x=x\}$ of $x$ is an Abelian subgroup of $G$ . The action is free if each of the stabilizers is trivial. Suppose that $(f,\Phi )$ is an isomorphism of group actions $G{\mspace {1mu}\curvearrowright \mspace {1mu}} X$ and $H{\mspace {1mu}\curvearrowright \mspace {1mu}} Y$ . Since ${\Phi ({\mathrm {Stab}}_G{x})={\mathrm {Stab}}_H{f(x)}}$ for all $x\in X$ , the properties of local commutativity and freeness are preserved by isomorphisms of group actions.

Let $G{\mspace {1mu}\curvearrowright \mspace {1mu}} X$ be a group action. We say that a subset $E\subseteq X$ is G-paradoxical using r pieces [Reference Tomkowicz and Wagon18, Definition 5.1] if, for some positive integers $m$ , $n$ with $m+n=r$ , there exist subsets $A_1,\dots ,A_m,B_1,\dots ,B_n$ of $E$ and elements $g_1,\dots ,g_m,h_1,\dots ,h_n\in G$ such that

(4) $$ \begin{align} E=A_1 \sqcup \dots \sqcup A_m \sqcup B_1 \sqcup \dots \sqcup B_n =\bigsqcup_{i=1}^{m} g_i.A_i = \bigsqcup_{j=1}^{n} h_j.B_j, \end{align} $$

where the symbols $\sqcup $ and $\bigsqcup $ indicate that the components of the respective unions are pairwise disjoint. In such a case, the decomposition (4) of $E$ is called a G-paradoxical decomposition. The empty set is obviously $G$ -paradoxical (using two pieces). In general, we do not require $E$ to be $G$ -invariant; consider, for example, the classical Banach–Tarski paradox, where the unit ball in $\mathbb {R}^3$ is not invariant under the isometry group of $\mathbb {R}^3$ (the invariance is violated by translations).

We say that two sets $A, B\subseteq X$ are G-equidecomposable [Reference Tomkowicz and Wagon18, Definition 3.4] if, for some $n\in \mathbb {N}$ , there exist partitions

$$ \begin{align*} A=\bigsqcup_{i=1}^n A_i, \quad B=\bigsqcup_{i=1}^n B_i, \end{align*} $$

and elements $g_1,\dots ,g_n\in G$ such that $g_i.A_i=B_i$ for all $1\leq i\leq n.$ Then we write $A\sim _{G} B$ , and thus obtain an equivalence relation $\sim _{G}$ on the power set of $X$ .

It is known that, for a group action $G{\mspace {1mu}\curvearrowright \mspace {1mu}} X$ , the set $X$ is $G$ -paradoxical using four pieces if and only if $G$ contains a free subgroup $F_2$ of rank two such that the restricted action $F_2 {\mspace {1mu}\curvearrowright \mspace {1mu}} X$ is locally commutative. More precisely, $X$ has a decomposition $X=A_1 \sqcup A_2 \sqcup A_3 \sqcup A_4 = A_1 \sqcup a.A_2 = A_3 \sqcup b.A_4$ for $a,b \in G$ if and only if $\{a,b\}$ is a basis of a free subgroup of rank two $F_2\le G$ whose action on $X$ is locally commutative [Reference Tomkowicz and Wagon18, Theorems 5.5 and 5.8].

The following result will be used in the proof of Theorem 11.

If $E\subseteq X$ is $G$ -paradoxical, then $E\sim _{G}A$ for any $A\subseteq E$ such that $E\subseteq g_1 A \cup \dots \cup g_n A$ for some $n\in \mathbb {N}$ , $g_1,\dots ,g_n \in G$ . Hence, any two such subsets $A', A"$ of $E$ are $G$ -equidecomposable and $G$ -paradoxical [Reference Tomkowicz and Wagon18, Corollary 10.22].

2.4 Baire paradoxical decompositions

We recall some notions from descriptive set theory. Let $X$ be a topological space. A subset $A\subseteq X$ has the Baire property (in $X$ ) if there exists an open subset $U$ of $X$ such that the symmetric difference $A\,\triangle \, U$ is a meager set in $X$ , that is, a countable union of nowhere dense sets in $X$ . The family $\mathcal {B}(X)$ of all subsets of $X$ with the Baire property is a $\sigma $ -algebra [Reference Kechris5, Proposition 8.22]. Moreover, if $E\in \mathcal {B}(X)$ and $A\in \mathcal {B}(E)$ , then $A\in \mathcal {B}(X)$ (see [Reference Kuratowski and French7, Section 11.V.1, p. 60]).

Let $E\in \mathcal {B}(X)$ and let $G$ act on $X$ by homeomorphisms. We say that $E$ is Baire G-paradoxical using $r$ pieces if $E$ is $G$ -paradoxical using $r$ pieces and all of the pieces involved in the decomposition (4) have the Baire property in $X$ .

Dougherty and Foreman provided a sufficient condition under which a Polish space (i.e., a separable completely metrizable topological space) $X$ is Baire $G$ -paradoxical using six pieces, for a group $G$ acting on $X$ by homeomorphisms. Namely, it suffices that the action $G {\mspace {1mu}\curvearrowright \mspace {1mu}} X$ is locally commutative, free on a comeager set (i.e., on the complement of a meager set), and $G$ contains a free subgroup of rank two [Reference Dougherty and Foreman3, Theorem 5.9]. More recently, Marks and Unger proved a general theorem that establishes the existence of Baire $G$ -paradoxical decomposition, provided that $G$ acts on $X$ by Borel automorphisms and $X$ admits a usual “set-theoretical” $G$ -paradoxical decomposition [Reference Marks and Unger8, Theorem 1.1]. However, the cited result does not give an explicit upper bound of the number of pieces. We also mention that Wehrung proved that six is the minimal number of pieces needed for a Baire paradoxical decomposition of a compact metric space with respect to isometries [Reference Wehrung20, Corollary].

In the proof of Theorem 11, we will use a criterion from [Reference Grabowski, Máthé and Pikhurko4], which contains powerful results on equidecomposability using pieces with some special properties (Borel, measurable, Baire – among others). The criterion can be adapted to our setting as follows.

Let $G{\mspace {1mu}\curvearrowright \mspace {1mu}} X$ be a group action on a locally compact Polish space $X$ by homeomorphisms. Assume that any two relatively compact subsets of $X$ with nonempty interiors are $G$ -equidecomposable. Then, any two relatively compact sets $A,B\in \mathcal {B}(X)$ with nonempty interiors are Baire $G$ -equidecomposable.

The statement is essentially a simplified version of [Reference Grabowski, Máthé and Pikhurko4, Proposition 1.8(ii)] for homeomorphic actions.