Notation 2.2. We sometimes also use a more intuitive, multiplicative notation for the valuation,

\begin{equation*} K \to \mathbb {L}^{\mathbb {Z}} \cup \{0\}, \quad a \mapsto |a| := \mathbb {L}^{-\mathrm {val}(a)}, \end{equation*}

and the multiplicative value group $\mathbb {L}^{\mathbb {Z}}$ is equipped with the order satisfying $\mathbb {L}^r \le \mathbb {L}^s$ if and only if $r \le s$ . Note that this notation suggests a natural map from $\mathbb {L}^{\mathbb {Z}}$ into the set of constructible motivic functions $\mathcal {C}(\{pt\})$ at the point (see Notation 4.1).

Notation 2.3. For $x \in K^n$ and $\lambda \in \mathbb {L}^{\mathbb {Z}}$ , we write

\begin{equation*} B(x,\lambda ) = \{y \in K^n; |y - x| \le \lambda \} \end{equation*}

for the closed ball of radius $\lambda$ around $x$ . By a ball, we mean such a closed valuative ball. We denote the radius $\lambda$ of $B(x,\lambda )$ by $\operatorname {rad} B(x,\lambda )$ .

Remark 2.6. All conditions of Definition 2.5 are preserved under adding constants from $K$ , $k$ or $\mathbb {Z}$ , so anything which we prove about $\emptyset$ -definable sets and/or maps also holds for $A$ -definable sets/maps, when $A$ is a subset of $K\cup k \cup \mathbb {Z}$ . We will use this implicitly on various occasions.

Remark 2.7. [Reference Cluckers, Halupczok and Rideau-KikuchiCHR22, Proposition 2.6.12] states that under the assumption of $1$ -h-minimality, the $\mathrm {RV}$ -sort (which, in our setting, is just $k \times \mathbb {Z}$ ) is stably embedded in a strong sense. Combining this with orthogonality of the value group and the residue field, one deduces that the residue field is also stably embedded in that strong sense. One way to formulate this is as follows: if $X \subset K^n \times k^m \times \mathbb {Z}^r$ and $Z \subseteq X \times k^{m'}$ are $\emptyset$ -definable, for some $n, m, r, m' \in \mathbb {N}$ , then there exist an $m'' \in \mathbb {N}$ , an $\emptyset$ -definable map $\rho \colon X \to k^{m''}$ and a $\emptyset$ -definable set $\hat Z \subseteq k^{m''} \times k^{m'}$ such that for every $x \in X$ , the fibre $Z_x$ is equal to the fibre $\hat Z_{\rho (x)}$ . (To get from the version of stable embeddedness of [Reference Cluckers, Halupczok and Rideau-KikuchiCHR22, Proposition 2.6.12] to the one given here, one uses a standard compactness argument: first, one applies stable embeddedness to each fibre $Z_x$ individually, and then compactness shows that the resulting $\rho (x)$ and $\hat Z_{\rho (x)}$ can be taken to be definable uniformly in $x$ .)

Remark 3.2. Clearly, risometries are always injective. Moreover, in our setting, a risometry from a ball $B \subseteq K^n$ to itself is always surjective. Indeed, since $K = k(\!(t)\!)$ is spherically complete (meaning that any descending chains of balls have non-empty intersection), surjectivity can be deduced using a version of the Banach fixed point theorem in such fields (see [Reference Bradley-Williams and HalupczokBH22, Lemma 3.3.4]). That surjectivity is crucial for the existence of $\mathrm {rtsp}_B$ in Definition 3.3 below.

Remark 3.4. One easily sees that if a set $X \subseteq K^n$ is $V$ -riso-trivial on a ball $B \subseteq K^n$ , for some $V \subseteq k^n$ , then $X$ is also $V'$ -riso-trivial on any sub-ball $B' \subseteq B$ for any sub-vector space $V' \subseteq V$ .

Remark 3.5. In Definition 3.3(1), the choice of the lift $\tilde V$ does not matter, i.e. if there exist a lift and a risometry $\Phi$ witnessing $V$ -riso-triviality, then for any other lift $\tilde V'$ of $V$ , there exists a risometry $\Phi '$ such that $(\tilde V', \Phi ')$ also witness $V$ -riso-triviality.

Remark 3.9. The sets $B_1, \dots, B_\ell$ are uniquely determined by $X$ , up to permutation; these are:

1. (1) those balls $B$ for which $X$ is not riso-trivial on $B$ but $1$ -riso-trivial on any proper sub-ball of $B$ ; and

2. (2) those singletons $\{b\}$ such that $X$ is not riso-trivial on every ball containing $b$ .

Remark 3.10. By Theorem 3.7, the characterization of the $B_i$ given in Remark 3.9 is definable: if $(X_{z})_{z\in Z}$ is a $\emptyset$ -definable familiy of sets $X_z \subseteq K^n$ and $C_z$ is the union of the points and balls $B_{z,1}, \dots, B_{z,\ell _z}$ corresponding to $X_z$ , then $(C_z)_z$ is also a $\emptyset$ -definable family. In addition, for any such family, there exists $N \in \mathbb {N}$ bounding the cardinalities $\ell _z$ for all $z \in Z$ . (This follows by applying [Reference Bradley-Williams and HalupczokBH22, Lemma 3.4.3] in a language with a constant symbol for $z$ added.)

Example 3.11. Set $X := \mathcal {O}_K \times \{0\} \subset K^2$ . Then clearly $X$ is $1$ -riso-trivial on $\mathcal {O}_K \times \mathcal {O}_K$ but not on $B := t^{-1}\mathcal {O}_K \times t^{-1}\mathcal {O}_K$ (recall that $t \in \mathcal {O}_K$ is a generator of the maximal ideal of $K$ ). In this way, one obtains that the finitely many sets $B_i$ provided for $X$ by Theorem 3.8 consist only of $B$ .

Example 3.12. Let $X \subset K^2$ be the graph of the function $x \mapsto x^2$ . Then one can verify that $X$ is not riso-trivial on $\mathcal {O}_K \times \mathcal {O}_K$ but $1$ -riso-trivial on each proper sub-ball of $\mathcal {O}_K \times \mathcal {O}_K$ . This shows that $B := \mathcal {O}_K \times \mathcal {O}_K$ is among the balls provided by Theorem 3.8. One then also checks that $X$ is $1$ -riso-trivial on every ball disjoint from $B$ , so $B$ is the only ball given by Theorem 3.8 for $X$ .

Example 3.13. Let $X \subset K^2$ be the union of the lines $K \times \{0\}$ and $\{(x,tx); x \in K\}$ (see Figure 2). Then $X$ is not riso-trivial on every ball $B$ containing the origin. If $B' \subset K^2$ is a ball not containing the origin, then $X$ is clearly $1$ -riso-trivial on $B'$ if $B'$ meets only one of the two lines, but even if $B'$ meets both lines, $X$ is $1$ -riso-trivial on $B'$ , since the two lines can be made parallel using a risometry. Overall, we obtain that Theorem 3.8 yields only the set $\{(0,0)\}$ .

Notation 4.1. For a valued field $K$ with residual field $k$ and value group $\mathbb {Z}$ as in § 1, let us first recall the definition given in [Reference Cluckers and LoeserCL08] of the ring of constructible motivic functions on some $\emptyset$ -definable set $X\subseteq K^n\times k^m\times \mathbb {Z}^r$ . For this purpose consider $\mathrm {RDef}_X$ the set of $\emptyset$ -definable subsets of $X\times k^p$ , for $p\in \mathbb {N}$ , equipped with their natural projection on $X$ . One denotes by $\mathcal {Q}_+(X)$ the Grothendieck semiring of $\emptyset$ -definable isomorphism classes $[Y]$ (more accurately classes of projections from $Y$ to $X$ ), $Y\in \mathrm {RDef}_X$ , with the following relations for any $\emptyset$ -definable sets $Y,Z\in \mathrm {RDef}_X$ :

1. – $[\emptyset ]=0$ ;

2. – $[Y\cup Z]+[Y\cap Z]=[Y]+[Z]$ ;

3. – $[Y\times Z]=[Y]\cdot [Z]$ .

We denote by the formal symbol $\mathbb {L}$ , the same symbol as used in Notation 2.2 to define the norm, the class $[X\times k]\in \mathcal {Q}_+(X)$ (actually the class of the projection of $X\times k$ on $X$ ), and by $\mathcal {Q}(X)$ the associated abelian Grothendieck ring.

Still with the same symbol $\mathbb {L}$ we consider on the other hand the ring

\begin{equation*} A:=\mathbb {Z}\left[\mathbb {L}, \mathbb {L}^{-1}, \left (\frac {1}{1-\mathbb {L}^i}\right )_{i\gt 0}\right], \end{equation*}

and $A_+$ its semigroup of nonnegative elements, where positivity is defined by evaluations $\mathbb {L}\mapsto r$ at $r$ , for any real number $r\gt 1$ . One defines the ring $\mathcal {P}(X)$ of Presburger functions, as a subring of the ring of functions $X\to A$ , as generated by constant functions, $\emptyset$ -definable functions $X\to \mathbb {Z}$ and functions $\mathbb {L}^\beta$ , with $\beta : X\to \mathbb {Z}$ $\emptyset$ -definable. The semiring of functions of $\mathcal {P}(X)$ with nonnegative values (i.e. with values in $A_+$ ) is denoted $\mathcal {P}_+(X)$ .

We denote by $\mathcal {P}_+^0(X)$ (respectively, $\mathcal {P}^0(X)$ ) the sub-semiring (respectively, sub-ring) of $\mathcal {P}_+(X)$ (respectively, $\mathcal {P}$ (X)) generated by all characteristic functions $\mathbf {1}_Y$ , with $Y\subseteq X$ , and the constant function $\mathbb {L}-1$ . We can send $\mathcal {P}_+^0(X)$ (respectively, $\mathcal {P}(X)$ ) in $\mathcal {Q}_+(X)$ (respectively, $\mathcal {Q}(X)$ ) by $\mathbf {1}_Y \mapsto [Y]$ and $\mathbb {L}-1\mapsto \mathbb {L}-1$ , to form the semiring $\mathcal {C}_+(X)$ of nonnegative constructible motivic functions on $X$ as follows:

\begin{equation*} \mathcal {C}_+(X):=\mathcal {Q}_+(X)\otimes _{\mathcal {P}_+^0(X)} \mathcal {P}_+(X). \end{equation*}

Likewise, we can sendthe ring $\mathcal {C}(X)$ of constructible motivic functions on $X$ as follows:

\begin{equation*} \mathcal {C}(X):=\mathcal {Q}(X)\otimes _{\mathcal {P}^0(X)} \mathcal {P}(X). \end{equation*}

Remark 4.2. One can define (see [Reference Cluckers and LoeserCL08, Sections 4.2, 12.2]) a partial order on $\mathcal {P}(X)$ , denoted $\le$ , defined by

\begin{equation*}\forall \varphi, \psi \in \mathcal {P}(X), \quad \varphi \le \psi \Longleftrightarrow \psi -\varphi \in \mathcal {P}_+(X).\end{equation*}

Similarly, one can define a partial preorder on $\mathcal {C}(X)$ , still denoted $\leq$ , by replacing $\mathcal {P}(X)$ by $\mathcal {C}(X)$ and $\mathcal {P}_+(X)$ by $\mathcal {C}_+(X)$ in the above definition. In both situations, when $\varphi \le \psi$ , one says that $\varphi$ is bounded by $\psi$ .

Remark 4.3 [Reference Cluckers and LoeserCL08, Theorem 10.1.1 A0 (c)]. One has $\varphi \in \mathrm {I}_X\mathcal {C}_+(Y)$ if and only if $\varphi \in \mathrm {I}_Z\mathcal {C}_+(Y)$ and $f_{!}(\varphi )\in \mathrm {I}_X\mathcal {C}_+(Z)$ .

Remark 4.4. A normalization of the volume is made in order to obtain uniqueness of the functor of motivic integration, leading in particular to $\mu _1(\mathcal {O}_K)=1$ and $\mu _1(\mathcal {M}_K)=\mathbb {L}^{-1}$ (see [Reference Cluckers and LoeserCL08, Theorem 10.1.1, A7]).

Remark 4.5. Let $X,Y$ be $\emptyset$ -definable subsets of $K^n\times k^m \times \mathbb {Z}^r$ , of dimension $d$ , say, such that $X\subset Y$ . Then, according to Remark 4.2, $\mathbf {1}_X$ is bounded by $\mathbf {1}_Y$ in $\mathcal {P}_+(Y)$ (or in $\mathcal {C}_+(Y)$ ), since $\mathbf {1}_Y-\mathbf {1}_X=\mathbf {1}_{Y\setminus X}\in \mathcal {P}_+(Y)$ , and thus integrating, by linearity of the integral and preservation of nonnegativity, one gets $\mu _d(X)\le \mu _d(Y)$ .

Notation 4.6. For $n,m,r\in \mathbb {N}$ , $Y,Z\subseteq K^n\times k^m\times \mathbb {Z}^r$ $\emptyset$ -definable sets, $f:Y\to Z$ a $\emptyset$ -definable map and $\varphi \in \mathcal {C}(Z)$ we denote by $f^*(\varphi )\in \mathcal {C}(Y)$ the pull-back of $\varphi$ along $f$ , which should be understood as the composition $\varphi \circ f$ . For $f$ , given by a commutative diagram

for $\lambda \in \Lambda$ , we denote by $f_{\lambda }:Y_\lambda \to Z_\lambda$ the restriction of $f$ to the fibres of $Y$ and $Z$ above $\lambda$ . We also denote, with a slight abuse of notation, by $i_\lambda$ the inclusion of $Y_\lambda$ (respectively, of $Z_\lambda$ ) in $Y$ (respectively, in $Z$ ).

Remark 4.7. By [Reference Cluckers and LoeserCL08, Proposition 14.2.1] one has for any $\lambda \in \Lambda$ ,

\begin{equation*}i^*_\lambda (f_{!\Lambda }(\varphi ))=f_{\lambda !}(i^*_\lambda (\varphi )).\end{equation*}

This is why we will often denote $f_{!\Lambda }(\varphi )$ by $(f_{\lambda !}(\varphi _\lambda ))_{\lambda \in \Lambda }$ ; that is to say, we will consider the constructible motivic function $f_{!\Lambda }(\varphi )$ as a family of measures over $\Lambda$ .

Remark 4.8. In the situation where $\pi : X\times Y\to X$ is the standard projection, then $\pi _!=\pi _{!X}$ , and thus $\pi _!(\varphi )\in \mathrm {I}_X\mathcal {C}_+(X)=\mathcal {C}_+(X)$ is the family $\left (\mu (\varphi _x)\right )_{x\in X}=\left (\int _{Y} \varphi (x,y) \ {\rm d}y\right )_{x\in X}$ of motivic integrals of $\varphi$ in the fibres $\{x\}\times Y$ , for $x\in X$ (see [Rai20, Remarque 5.3.4.6.1]).

Remark 4.11. The notions of boundedness and increasingness can equivalently be defined directly using $\mathcal {C}_+$ , that is, without passing through $\mathcal {P}_+$ (see also Remark 4.2): $\psi \in \mathcal {C}_+(X \times Y)$ is bounded by $\psi ' \in \mathcal {C}_+(X)$ if there exists $\phi \in \mathcal {C}_+(X \times Y)$ such that $\psi + \phi = \psi '$ , and $\psi \in \mathcal {C}_+(X \times \mathbb {N})$ is increasing if there exists $\phi \in \mathcal {C}_+(X \times \mathbb {N})$ such that $\psi + \phi = \psi '$ , where $\psi '(x,r) = \psi (x,r+1)$ .

Notation 4.12. For $n,m,r\in \mathbb {N}$ , $X\subseteq K^n\times k^m\times \mathbb {Z}^r$ an $\emptyset$ -definable set and $\phi,\psi \in \mathcal {C}_+(X)$ , we write $\phi \le \psi$ if $\phi$ is bounded by $\psi$ in the sense of Definition 4.10.

Remark 4.13. The limit $\psi '$ , if it exists, is well defined; that is, it does not depend on the choice of the representation of $\psi$ in the form (9). To see this, denote by $\mathcal {P}^{\mathrm {lim}}_+(X \times \mathbb {N}) \subset \mathcal {P}_+(X \times \mathbb {N})$ and $\mathcal {C}^{\mathrm {lim}}_+(X \times \mathbb {N}) \subset \mathcal {C}_+(X \times \mathbb {N})$ the sub-semirings of converging nonnegative Presburger and constructible motivic functions, where the convergence is defined in Definition 4.10. By definition of $\mathcal {C}^{\mathrm {lim}}_+(X \times \mathbb {N})$ , we have

\begin{equation*}\mathcal {C}^{\mathrm {lim}}_+(X \times \mathbb {N})= \mathcal {Q}_+(X)\otimes _{\mathcal {P}_+^0(X)} \mathcal {P}^{\mathrm {lim}}_+(X \times \mathbb {N}),\end{equation*}

where we use again the same isomorphism as in (9), provided by [Reference Cluckers and LoeserCL08, Section 5.3] or [Reference Cluckers and LoeserCL15, Proposition 7.5], to omit the $\mathbb {N}$ from the left-hand side of the tensor product and from the $\mathcal {P}_+^0$ .

The map $h\colon \mathcal {P}^{\mathrm {lim}}_+(X \times \mathbb {N}) \to \mathcal {P}_+(X)$ sending a function to its limit induces the identity on $\mathcal {P}_+^0(X)$ , so we can apply the functor $S \mapsto \mathcal {Q}_+(X)\otimes _{\mathcal {P}_+^0(X)} S$ to that map $h$ to define the limit on $\mathcal {C}^{\mathrm {lim}}_+(X \times \mathbb {N})$ in a way which is independent of how we write a function $\psi \in \mathcal {C}^{\mathrm {lim}}_+(X \times \mathbb {N})$ as in (9).

Remark 4.15 (Monotone convergence). A $\emptyset$ -definable subset $X$ of $K^n\times \mathbb {N}$ with projection $\pi : X \mapsto \mathbb {N}$ , and fibres $\pi ^{-1}(p)=X_p$ , such that $(X_p)_{p\in \mathbb {N}}$ is uniformly bounded with respect to $p\in \mathbb {N}$ , and which is an increasing sequence with respect to the inclusion relation, gives rise to the bounded increasing function $\varphi =\pi _{!\mathbb {N}}(\mathbf {1}_{X})=(\mu (X_p))_{p\in \mathbb {N}}\in \mathcal {C}_+(\mathbb {N})$ . It follows that this function converges as $p$ goes to $+\infty$ , and by definition of the motivic integral, that its limit is $\mu \left ( \bigcup _{i=0}^{+\infty } X_i\right )$ (see [Reference ForeyFor17, Lemma 5.1] for more details).

Remark 4.16. Let $X \subset K^n$ be a bounded $\emptyset$ -definable set of dimension $d$ and, for $r \in \mathbb {Z}$ , let $T_r(X)=\{x\in K^n;\exists y\in X, \mathrm {val}(y-x)\ge r\}$ be the corresponding tubular neighbourhood. Then, for $r \ge 0$ , $\mu _n(T_r(X))$ is bounded by $(\mathbb {L}^{-r})^{n-d}\cdot C$ for some constant $C \in \mathcal {C}_+(\{pt\})$ depending only on $X$ (i.e. $(\mathbb {L}^{-r})^{n-d}\cdot C - \mu _n(T_r(X)) \in \mathcal {C}_+(\mathbb {N})$ ). Indeed, by cell decomposition with $1$ -Lipschitz centres, $X$ can be partitioned into a finite $\emptyset$ -definable family of sets $X_\xi$ each of which is, up to a permutation of coordinates, the graph of a $1$ -Lipschitz function $f_\xi$ on a subset $\bar X_\xi$ of $K^d$ (for more details about this kind of cell decomposition, see Step 1 of the proof of Theorem 7.1). This implies that $\mu _n(T_r(X_\xi ))$ is bounded by $\mu _d(T_0(\bar X_\xi ))\cdot (\mathbb {L}^{-r})^{n-d}$ . Finally note that, applying Remark 4.5, the volume of the tubular neighbourhood of $X$ is bounded by the sum of the volume of the tubular neighbourhoods of the $X_\xi$ .

Remark 5.2. Note that in (13), $\varphi$ and $f^*_X\varphi$ are indeed integrable along $\pi _X$ :. For $\varphi$ , this follows from the assumption $\varphi \in \mathrm {I}\mathcal {C}_+(Y \times X)$ (which implies $\varphi \in \mathrm {I}_{X}\mathcal {C}_+(Y \times X)$ by Remark 4.3. For $f^*_X\varphi$ , note that since $f$ has finite fibres, $\varphi \in \mathrm {I}_X\mathcal {C}_+(Y \times X)$ implies $f^*_X\varphi \in \mathrm {I}_X\mathcal {C}_+(Z \times X)$ .

Example 5.4. If $k=\mathbb {C}$ , for instance, one can give an example of a constructible motivic function in $\mathrm {I}\mathcal {C}_{\succcurlyeq 0}(X)$ but not in the image of $\mathrm {I}\mathcal {C}_+(X)$ in $\mathrm {I}\mathcal {C}(X)$ . Indeed, let us consider the constructible motivic function $\varphi =[Z]-1$ for $Z = \{\pm \sqrt 2\} \subset \mathbb {C}$ . In view of Lemma 5.11, since there is a $\emptyset$ -definable surjection from the $\emptyset$ -definable set $Z$ to the point, we have $\varphi \in \mathrm {I}\mathcal {C}_{\succcurlyeq 0}(X)$ .

Let us now show that $\varphi$ does not lie in the image of $\mathrm {I}\mathcal {C}_+(X)$ . This amounts to showing that there is no $\emptyset$ -definable set $Y \subset \mathbb {C}^m$ (for some $m$ ) and injective $\emptyset$ -definable map $f\colon \{pt\} \sqcup Y \to Z \sqcup Y$ . Suppose the existence of $Y$ and $f$ . Without loss of generality, they are defined by quantifier-free formulas, so that we also obtain a $\emptyset$ -definable injection between the restrictions of those sets to any subfield $F \subset \mathbb {C}$ . We pick $F$ to be a pseudo-finite field not containing the square roots of $2$ . Then $f$ further induces a bijection in some sufficiently big finite field not containing the square roots of $2$ . However, this is not possible for cardinality reasons.

We typically need this kind of comparison allowed by our preorder in equation (24) of the proof of Theorem 8.6.

Remark 5.7. For $m,n,r\in \mathbb {N}$ and $X\subseteq K^n\times k^m\times \mathbb {Z}^r$ a $\emptyset$ -definable set, when $F\in \mathrm {I}\mathcal {C}_{\succcurlyeq 0}(X)$ and $G\in \mathcal {C}_+(X)$ , then $F \cdot G\in \mathrm {I}\mathcal {C}_{\succcurlyeq 0}(X)$ , since in Definition 5.1, we can replace $\varphi$ by $\varphi \cdot G$ .

Remark 5.12. In Lemma 5.11, one could more generally allow $V \subset X \times k^{m'} \times \mathbb {Z}^{r'}$ (and similarly for $W$ ), and even $V \subset X \times K^{n'} \times k^{m'} \times \mathbb {Z}^{r'}$ if we use the $0$ -dimensional motivic measure on $K^{n'}$ . Indeed, one can still find $\rho \colon X \to k^{\hat m}$ as in the proof of the lemma, using orthogonality of $k$ and $\mathbb {Z}$ , and using that finite subsets of $K^{n'}$ are in definable bijection with definable subsets of $k^{m'''}$ .

Remark 6.2. For finite sets $X \subseteq K^n$ , we simply have $V_0(X) = \mu _0(X)$ . Indeed, in this case, $B_1, \dots, B_\ell$ are just the singletons of $X$ and thus $S_{0}$ is in $\emptyset$ -definable bijection with $X$ , provided that $n \ge 1$ . In the case $n = 0$ , we intentionally defined $V_0$ to make this remark hold.

Remark 6.3. For any $\emptyset$ -definable family $(X_z)_{z \in Z}$ of sets $X_z \subset K^n$ , the constructible motivic function $(V_0(X_z))_z$ is uniformly bounded by a constant function in the sense of Definition 4.10. Indeed, by Remark 3.10, the number $\ell$ of balls $B_i$ is uniformly bounded, which implies that they can be parametrized by a subset of $k^m$ for $m$ not depending on $z$ . Thus, applying Remark 4.5, the constructible motivic function on $Z$ which is constant equal to $[k^m] = \mathbb {L}^m$ is a possible bound.

Notation 6.4. We denote by $\mathrm {d} g$ the $\operatorname {GL}_n(\mathcal {O}_K)$ -invariant motivic measure on $\operatorname {GL}_n(\mathcal {O}_K)$ (see [Reference Cluckers and LoeserCL08, Section 15] for the notion of motivic measure on manifold), normalized such that the total measure of $\operatorname {GL}_n(\mathcal {O}_K)$ is equal to $[\operatorname {GL}_n(k)]\cdot \mathbb {L}^{-{\rm dim} (\operatorname {GL}_n)}$ .

Notation 6.5. Given an affine space $\bar {P} \in \bar {\operatorname {G}}_n^{n-i}$ , we write $P \in \operatorname {G}_n^{n-i}$ for the direction of $\bar P$ .

We then define a measure on $ \bar {\operatorname {G}}_n^{n-i}$ . To this end, we fix once and for all a $\emptyset$ -definable element $g_0\in \operatorname {GL}_n(\mathcal {O}_K)$ (e.g., the identity matrix) and set $P_0:=g_0\cdot (K^{n-i}\times \{0\}^{i})$ .

These choices yield a surjective map $\gamma \colon \operatorname {GL}_n(\mathcal {O}_K) \to \operatorname {G}_n^{n-i}$ sending $g$ to $g\cdot P_0$ , and a surjective map $\bar \gamma \colon \operatorname {GL}_n(\mathcal {O}_K)\times K^i \to \bar {\operatorname {G}}_n^{n-i}$ sending $(g, y)$ to $gg_0\cdot ( K^{n-i}\times \{y\})$ . Clearly, these two maps are compatible with respect to taking the direction of an affine subspace of dimension $n-i$ of $K^n$ , and note that once we fix $g \in \operatorname {GL}_n(\mathcal {O}_K)$ , $\bar \gamma$ allows us to identify $K^i$ with $K^n/(g\cdot P_0)$ . We denote by $\pi _g\colon K^n \to K^i$ the projection from $K^n$ to $K^n/(g\cdot P_0)$ using that identification. More accurately, we have for $x\in K^n$ (or for any element of $K^n$ in the same class as $x$ modulo $g\cdot P_0$ ), $\pi _g(x)=y$ , where $(gg_0)^{-1}(x)=a+y$ with $a\in K^{n-i}\times \{0\}^i$ and $y\in \{0\}^{n-i}\times K^i$ . In particular, for $y\in K^i$ and $g\in \operatorname {GL}_n(\mathcal {O}_K)$ , we have $\pi _g^{-1}(y)=\bar \gamma (g,y)$ .

We use $\gamma$ to push forward the measure on $\operatorname {GL}_n(\mathcal {O}_K)$ to $ \operatorname {G}_n^{n-i}$ , i.e. for any $\varphi \in \mathcal {C}_+( \operatorname {G}_n^{n-i})$ , we say that $\varphi$ is integrable if $\varphi \circ \gamma \in \mathrm {I}\mathcal {C}_+(\operatorname {GL}_n(\mathcal {O}_K))$ with respect to the motivic measure $\mathrm {d}g$ , and we define the integral of $\varphi$ by

\begin{align*} \int _{P\in \operatorname {G}_n^{n-i} } \varphi (P) \ \mathrm {d}P & := \int _{g\in \operatorname {GL}_n(\mathcal {O}_K)} \varphi \circ \gamma (g) \ \mathrm {d}g. \end{align*}

In a similar way, we use $\bar \gamma$ to push forward the motivic product measure ${\rm d}g \otimes {\rm d}y$ on $\operatorname {GL}_n(\mathcal {O}_K)\times K^i$ to $ \bar {\operatorname {G}}_n^{n-i}$ . In particular, for $\varphi \in \mathrm {I}\mathcal {C}_+( \bar {\operatorname {G}}_n^{n-i})$ , we set

\begin{align*} \int _{\bar {P}\in \bar {\operatorname {G}}_n^{n-i}} \varphi (\bar {P}) \ \mathrm {d}\bar {P} & := \int _{g\in \operatorname {GL}_n(\mathcal {O}_K)} \int _{y\in K^i} \varphi \circ \bar {\gamma }(g,y) \ \mathrm {d}y \ \mathrm {d}g \\ & = \int _{y\in K^i} \int _{g\in \operatorname {GL}_n(\mathcal {O}_K)} \varphi \circ \bar {\gamma }(g,y) \ \mathrm {d}g \ \mathrm {d}y. \end{align*}

Remark 6.6. We note, with this definition of integration on $ \bar {\operatorname {G}}_n^{n-i}$ , that the choice of $g_0\in \operatorname {GL}_n(\mathcal {O}_K)$ and $P_0$ made to identify $K^n/P$ and $K^i$ does not matter by the change of variable theorem. So let us, for the remainder of this paper, set $g_0=I_n$ (and thus $P_0=K^{n-i}\times \{0\}^i$ ).

Remark 6.8. For a $\emptyset$ -definable bounded subset $X$ of $K^n$ and a general $\bar {P}\in \bar {\operatorname {G}}_n^{n-i}(k)$ , the set $X\cap \bar {P}$ is not $\emptyset$ -definable. Nevertheless the subset $Y:=\{(x,\bar {P}); x\in X\cap \bar {P}, \bar {P}\in \bar {\operatorname {G}}_n^{n-i}\}$ of $X\times \bar {\operatorname {G}}_n^{n-i}$ is $\emptyset$ -definable, since $\bar {\operatorname {G}}_n^{n-i}$ is itself $\emptyset$ -definable. Denoting by $\pi : Y \to \bar {\operatorname {G}}_n^{n-i}$ the projection map $\pi (x,\bar {P})=\bar {P}$ , in view of Remark 4.7, the $\emptyset$ -definable family $ (V_0(X \cap \bar P))_{\bar {P}\in \bar {\operatorname {G}}_n^{n-i}(k)}$ stands for $\pi _{!\bar {\operatorname {G}}_n^{n-i}}(\mathbf {1}_Y)\in \mathcal {C}_+( \bar {\operatorname {G}}_n^{n-i})$ , according to Remark 4.8. Consequently, $V_i(X)$ , the motivic integral of this constructible motivic function on $\bar {\operatorname {G}}_n^{n-i}$ makes sense.

Note also that since $X$ is bounded, $\bar P \mapsto V_0(X \cap \bar P)$ is non-zero only on a bounded subset of $\bar {\operatorname {G}}_n^{n-i}$ , so that the integral $V_i(X)$ is finite.

Remark 6.9. The following properties of $V_i$ follow from the nature of the measure considered on $\bar {\operatorname {G}}_n^{n-i}$ .

1. (1) For any $i=1, \ldots, n$ , for any $r \in K^\times$ ,

   \begin{equation*}V_i(r\cdot X) =|r|^i V_i(X).\end{equation*}

2. (2) For any $i=1, \ldots, n$ and any $j\le i$ ,

   \begin{equation*}V_i(X)=\int _{\bar {P}\in \bar {\operatorname {G}}_n^{n-j} } V_{i-j}(X\cap \bar {P})\ \text {d}\bar {P}.\end{equation*}

Remark 6.10. Note that in contrast to the real setting, our Vitushkin invariants are not sub-additive. In the real case the sub-additivity of the $V_i$ comes from the sub-additivity of $V_0$ itself. But in the nonarchimedean case, $V_0$ is not sub-additive. Intuitively, the reason is that it rather counts singularities than connected components. As an example, consider $X_1 := \mathcal {O}_K \times \{0\}$ , and let $X_2$ be the graph of the function $\mathcal {O}_K \to \mathcal {O}_K, x \mapsto t(x^3 - x)$ , where $t \in \mathcal {O}_K$ has valuation $1$ . Then for both, $i = 1,2$ , Theorem 3.8 yields only the ball $t^{-1}\mathcal {O}_K \times t^{-1}\mathcal {O}_K$ , so we have $V_0(X_1) = V_0(X_2) = 1$ . On the other hand, $(-1,0), (0,0),(1,0)$ are singularities of $X_1 \cup X_2$ , so that we obtain $V_0(X_1 \cup X_2) = \mu _0(\{(-1,0),(0,0),(1,0)\})=3$ .