\label{thm:disper}
		Assume $\varepsilon_0<\varepsilon_*$, with $\varepsilon_*$ as in Lemma \ref{lem:disper-log}. Then for any given $0<\tau<\frac13$, there exists $K_1=K_1(\varepsilon_0,\tau)$ such that for any $\theta\in\mathbb{T}^d$ and any $t\in\mathbb{R}$,
\begin{equation}\label{linest}
			\norm{u\left( \cdot ,t\right)}_{\ell^\infty\left(\mathbb{Z}\right)}\leq K_1\JP{t}^{-\tau}(\norm{\varphi}_{\ell^1\left(\mathbb{Z}\right)}+\norm{\psi}_{\ell^1\left(\mathbb{Z}\right)}) \ , \quad \forall \ \varphi, \psi \in\ell^1(\mathbb{Z}) \ .
		\end{equation}

\begin{theorem}\label{thm:nonlinear}
		Let $p>7$. There exist $\varepsilon > 0$ and a constant $C$, so that whenever
		$\| \varphi \|_{\ell^2\left(\Z\right)}$ and $\| \psi\|_{\ell^2\left(\Z\right)} \leq \varepsilon
		$, there exists a unique global solution to \eqref{eq: nonlinear}, satisfying
		\begin{equation*}
		\|u\|_{L^q_t(\mathbb R;\ell^r(\mathbb Z))}\leq C\varepsilon,
		\end{equation*}
		for all admissible pairs $(q, r)$. In particular, $\| u(n,t) \|_{\ell^r(\Z)} \to 0$ as $t \to \infty$ for every $r > 2$.
	\end{theorem}

\begin{theorem}\label{thm:disper}
		Assume $\varepsilon_0<\varepsilon_*$, with $\varepsilon_*$ as in Lemma \ref{lem:disper-log}. Then for any given $0<\tau<\frac13$, there exists $K_1=K_1(\varepsilon_0,\tau)$ such that for any $\theta\in\T^d$ and any $t\in\R$,
\begin{equation}\label{linest}
			\norm{u\left( \cdot ,t\right)}_{\ell^\infty\left(\Z\right)}\leq K_1\JP{t}^{-\tau}(\norm{\varphi}_{\ell^1\left(\Z\right)}+\norm{\psi}_{\ell^1\left(\Z\right)}) \ , \quad \forall \ \varphi, \psi \in\ell^1(\Z) \ .
		\end{equation}
	\end{theorem}

We consider the Klein--Gordon equation with quasi-periodic potentials on the one-dimensional lattice $\mathbb{Z}$:
 \begin{equation}\label{eq:DKG}
 \begin{cases}
 \bigl(\partial_{tt}+G_{\theta}+m^{2}\bigr)u(n,t)=0,\qquad\qquad\quad(n,t)\in\mathbb{Z}\times\left(0,\infty\right),\\
 u(\cdot,0)=\varphi(\cdot)\in\ell^{1}\left(\mathbb{Z}\right),\quad \partial_{t}u(\cdot,0)=\psi(\cdot)\in\ell^{1}\left(\mathbb{Z}\right),\quad n\in\mathbb{Z},\\
 \end{cases}
 \end{equation}
 where $G_{\theta}$ is defined as
 \begin{equation*}
 (G_{\theta}u)(n)
 :=-\Delta_{\text{disc}}u(n)+V(\theta+n\omega)u(n),
 \qquad n\in\mathbb{Z},
 \end{equation*}
 and the discrete Laplacian is given by
 $\Delta_{\text{disc}}u(n):=u(n+1)+u(n-1)-2u(n)$.
 Here $\theta \in \mathbb{T}^d$, the potential $V\colon\mathbb{T}^{d}\to\mathbb{R}$ is a real-analytic with $d\geq 1$, and the frequency vector $\omega\in\mathbb{R}^{d}$ is \textbf{Diophantine}; i.e., there exist constants $\gamma>0$ and $\eta>d-1$ such that
 \begin{equation}\label{dio}
 \inf_{j\in\mathbb{Z}}\bigl|\langle k,\omega\rangle_{\mathbb{R}^{d}}-j\pi\bigr|
 >\frac{\gamma}{|k|^{\eta}},
 \qquad \forall\,k\in\mathbb{Z}^{d}\setminus\{0\},
 \end{equation}
 where $\langle\cdot{,}\cdot\rangle_{\mathbb{R}^{d}}$ denotes the standard scalar product on $\mathbb{R}^{d}$. We assume that $V$ admits a bounded analytic complex extension to $\left\{z\in\mathbb{C}^{d}:\left|{\rm Im}z\right|<r\right\}$. We also denote
 \begin{equation*}
 \varepsilon_{0}:=\left|V\right|_{r}=\sup_{\left\{z\in\mathbb{C}^{d}:\left|{\rm Im}z\right|<r\right\}}\left|V\left(z\right)\right|.
 \end{equation*}
 In the literature, one often sets $H_{\theta}=G_{\theta}-2$ to be the \textbf{quasi-periodic Schr\"odinger operator}. We are interested in the dispersive estimates for the equation \eqref{eq:DKG}, namely the $\ell^{1}\!\to\!\ell^{\infty}$ estimates of the form
 \begin{equation}\label{eq:dispersive}
 \norm{u(\cdot,t)}_{\ell^{\infty}(\mathbb{Z})}
 \leq C\JP{t}^{-\tau}\bigl(\norm{\varphi}_{\ell^{1}(\mathbb{Z})}+\norm{\psi}_{\ell^{1}(\mathbb{Z})}\bigr),
 \qquad \forall\,\varphi,\psi\in\ell^{1}(\mathbb{Z}),
 \end{equation}
 where $\JP{t}:=\sqrt{1+t^{2}}$ is the Japanese bracket, $C>0$ is a constant independent of $t$ and $\tau$ is so called dispersive decay rate.

\textbf{Motivation} The goal of this work is to understand to what extent dispersive properties of lattice Klein–Gordon dynamics are stable under introducing a quasi-periodic medium. On the one hand, the free discrete Klein–Gordon equation on $\mathbb Z$ is known to exhibit a precise dispersive decay rate (see \cite{SK05}), which plays a central role in both the linear and nonlinear theory. On the other hand, quasi-periodic Schr\"odinger operators $H_\theta$ are canonical models for wave propagation in quasicrystals and are known, in the small analytic and Diophantine regime, to have purely absolutely continuous spectrum of Cantor type (see for example \cite{AD08, Avi08}). From a dispersive PDE viewpoint, this spectral picture is highly singular and lies completely outside the standard setting of the fine continuous spectrum given by a finite union of intervals, so it is a priori unclear whether any robust $\ell^1\to\ell^\infty$ decay should survive.

Our main result states as follows.

\begin{equation}\label{eq:DKG}
		\begin{cases}
			\bigl(\partial_{tt}+G_{\theta}+m^{2}\bigr)u(n,t)=0,\qquad\qquad\quad(n,t)\in\mathbb{Z}\times\left(0,\infty\right),\\
            u(\cdot,0)=\varphi(\cdot)\in\ell^{1}\left(\Z\right),\quad \partial_{t}u(\cdot,0)=\psi(\cdot)\in\ell^{1}\left(\Z\right),\quad n\in\mathbb{Z},\\
		\end{cases}
	\end{equation}

\begin{lemma}\label{lem:disper-log}
		Assume that $\left|V\right|_{r}=\varepsilon_{0}\leq \varepsilon_{\ast}$ with $\varepsilon_{\ast}$ defined in Proposition \ref{propsana}. For any $k,k_{\Delta},n,n_{\Delta}$,
		\begin{equation*}
			\begin{aligned}
				\left|\mathbb{I}\left(\Sigma\right)\right|\leq\frac{C\left(m\right)\left|\ln\varepsilon_{0}\right|^{82000d\left(\ln\ln\left(2+\JP{t}\right)\right)^{2}}}{\JP{t}^{\frac{1}{3}}},
			\end{aligned}
		\end{equation*}
		for any $t\in\R$ where $m>0$ and the constant $C\left(m\right)$ only depends on $m$.
	\end{lemma}

By \eqref{infinito}, we have
 \begin{equation}\label{eq:u(t)_int}
 |u(n,t)|\leq\frac{1}{\pi}\left|\int_\Sigma \left(g_1(E,t){\cK}_n(E)+g_2(E,t){\cJ}_n(E)\right) \rho^{\prime}\d E\right|+\varepsilon_0^{\frac{\sigma^2}{10}}\|u(\cdot,t)\|_{\ell^\infty\left(\Z\right)} \ .
 \end{equation}
 To estimate $\|u(\cdot,t)\|_{\ell^\infty\left(\Z\right)}$, it suffices to control the integral term by taking $\varepsilon_{0}$ to be small. Substituting the expression of $\mathcal{S}(u(t))$ into the integral, and recalling the definitions of $g^{\varphi}$ and $g^{\psi}$, we obtain
 \begin{equation}\label{eq:appro_u}
 \begin{aligned}
 &\int_\Sigma \left(g_1(E,t){\cK}_n(E)+g_2(E,t){\cJ}_n(E)\right)\, \rho^{\prime}\d E\\
 =&\frac{1}{2} \int_\Sigma e^{{\rm i}\Omega(E)t} \sum_{k\in\Z} \left(\varphi(k) + \frac{\psi(k)}{{\rm i}\Omega(E)}\right) \left({\cK}_k(E){\cK}_n(E)+{\cJ}_k(E){\cJ}_n(E)\right)\, \rho^{\prime}\d E \\
 &+\frac{1}{2} \int_\Sigma e^{-{\rm i}\Omega(E)t} \sum_{k\in\Z} \left(\varphi(k) - \frac{\psi(k)}{{\rm i}\Omega(E)}\right) \left({\cK}_k(E){\cK}_n(E)+{\cJ}_k(E){\cJ}_n(E)\right)\, \rho^{\prime}\d E \ .
 \end{aligned}
 \end{equation}
 Note that the term $\left({\cK}_k(E){\cK}_n(E)+{\cJ}_k(E){\cJ}_n(E)\right)$ can be expanded as
 \begin{equation}\label{eq:K,J expansions}
 \sum_{k_{\Delta}, n_{\Delta}} \beta_{k,k_{\Delta}} \beta_{n,n_{\Delta}} \cos(k_{\Delta}-n_{\Delta})\rho(E) \ .
 \end{equation}
 Since $m>0$, the factor $\frac{1}{\Omega(E)}= \frac{1}{\sqrt{2+m^2+E}}$ is smooth and bounded on the spectrum of $G_{\theta}+m^2$. By the symmetry of the complex conjugate terms in \eqref{eq:appro_u}, the dispersive decay estimates for both integrals depend on the oscillatory behavior of the phase function. Thus, the problem reduces to establishing the estimates for the oscillatory integral associated with the kernel
 $e^{-{\rm i} t \sqrt{2+m^2+E}}$. For simplicity, we denote
 $$\mathbb{I}(S;k_{\Delta},n_{\Delta},\rho):=\int_{S} \beta_{k,k_{\Delta}}\beta_{n,n_{\Delta}}\cos
 \big((k_{\Delta}-n_{\Delta})\rho\left(E\right)\big)e^{-\mathrm{i}t\Omega\left(E\right)}\cdot\rho^{\prime}\d E, \quad \text{for}\ S\subset\R,$$
 and
 $$\mathbb{I}^{J}(S;k_{\Delta},n_{\Delta},\rho):=\int_{S} \beta^J_{k,k_{\Delta}} \beta^J_{n,n_{\Delta}}\cos
 \big((k_{\Delta}-n_{\Delta})\rho\left(E\right)\big)e^{-\mathrm{i}t\Omega\left(E\right)}\cdot\rho^{\prime} \d E, \quad \text{for}\ S\subset\R.$$
 Whenever no confusion can arise, we simply write $\mathbb{I}(E)$ and $\mathbb{I}^{J}(E)$ respectively.

\begin{proof}[Proof of Theorem \ref{thm:disper} assuming Lemma \ref{lem:disper-log}]
 By the expansions \eqref{eq:K,J expansions}, we apply Lemma \ref{lem:disper-log} to equation \eqref{eq:appro_u} and derive
 \begin{equation}\label{eq:L infty estimate}
 \begin{aligned}
 &\left|\int_\Sigma \left(g_1(E,t){\cK}_n(E)+g_2(E,t){\cJ}_n(E)\right)\, \rho^{\prime}\d E\right|\\
 \leq&C\left(m\right)\cdot\frac{\left|\ln\varepsilon_{0}\right|^{82000\left(\ln\ln\left(2+\JP{t}\right)\right)^{2}d}}{\JP{t}^{\frac{1}{3}}}\left(\sum_{k\in\Z}\left|\varphi\left(k\right)\right|+\frac{1}{\inf_{E\in\Sigma}\left|\Omega\left(E\right)\right|}\sum_{k\in\Z}\left|\psi\left(k\right)\right|\right),
 \end{aligned}
 \end{equation}
 where the absolute constant $C_{1}$ relates to the estimate of $e^{{\rm i}\Omega\left(\lambda\right)t}$ and so does $C_{2}$ for the estimate related to $e^{-{\rm i}\Omega\left(\lambda\right)t}$. Since $\epsilon_{0}<1$ is small, and by \eqref{eq:u(t)_int}, \eqref{eq:L infty estimate} we deduce
 \begin{equation*}
 \norm{u\left(t\right)}_{\ell^{\infty}\left(\Z\right)}\leq\frac{C\left(m\right)}{1-\varepsilon_0^{\frac{\sigma^2}{10}}}\frac{\left|\ln\varepsilon_{0}\right|^{82000\left(\ln\ln\left(2+\JP{t}\right)\right)^{2}d}}{\JP{t}^{\frac{1}{3}}}\left(\norm{\varphi}_{\ell^{1}\left(\Z\right)}+\frac{1}{\inf_{E\in\Sigma}\left|\Omega\left(E\right)\right|}\norm{\psi}_{\ell^{1}\left(\Z\right)}\right).
 \end{equation*}
 This completes the proof.
 \end{proof}

In the case of $|k_{\Delta}-n_{\Delta}|$ is large, we derive the following bound.
\begin{lemma}\label{lem:large}
For any $t \in \R$, the following estimate holds,
\begin{equation*}
 \left|\mathbb{I}^{J}(\Sigma)\right|\leq\frac{18}{\left|k_{\Delta}-n_{\Delta}\right|}\left|\ln \varepsilon_{0}\right|^{2J^{2}d}+\frac{3\Theta_{1}\JP{t}}{\left|k_{\Delta}-n_{\Delta}\right|}\left(\left(1+\varepsilon_{0}^{\frac{1}{4}}\right)^{2}+\varepsilon_{j}^{6\sigma}\right)\left(1+\frac{1}{\Theta_{2}}\right).\\
\end{equation*}
\end{lemma}
\begin{proof}
By Proposition~\ref{propsana1}, for each $0\leq j\leq J$ we can write
\begin{equation*}
 \Gamma_{j}^{(J)}
 =\mathcal{R}_{j}\cup\Bigl(\,\bigsqcup_{s^{J}_{j}}I_{s^{J}_{j}}\Bigr),
\end{equation*}
where $\mathcal{R}_{j}$ is a finite set of points (possibly empty) and each $I_{s^{J}_{j}}$ is an open interval in $\Gamma_{j}^{(J)}$. Recall that $\rho$ is absolutely continuous. Then
 \begin{equation}\label{eq:Ij-M large}
 \begin{aligned}
 \mathbb{I}^{J}(\Sigma)=&\int_{\inf\Sigma}^{\sup\Sigma}\beta^J_{k,k_{\Delta}} \beta^J_{n,n_{\Delta}}\cos\big((k_{\Delta}-n_{\Delta})\rho\left(E\right)\big)e^{-{\rm i}t\Omega\left(E\right)}\cdot\rho^{\prime} \d E\\
 =&\frac{1}{k_{\Delta}-n_{\Delta}}\sum_{j=0}^{J}\sum_{s_{j}^{J}}\big(\beta^J_{k,k_{\Delta}} \beta^J_{n,n_{\Delta}}\sin\big((k_{\Delta}-n_{\Delta})\rho\left(E\right)\big)e^{-{\rm i}t\Omega\left(E\right)}\big)\Big|_{\inf I_{s_{j}^{J}}}^{\sup I_{s_{j}^{J}}}\\
 &\quad-\frac{1}{k_{\Delta}-n_{\Delta}}\sum_{j=0}^{J}\sum_{s_{j}^{J}}\int_{I_{s_{j}^{J}}}\big(\beta^J_{k,k_{\Delta}} \beta^J_{n,n_{\Delta}}e^{-{\rm i}t\Omega\left(E\right)}\big)^{\prime}\sin\big((k_{\Delta}-n_{\Delta})\rho\left(E\right)\big)\d E.\\
 \end{aligned}
 \end{equation}
 Using Proposition \ref{propsana1} again, we derive the upper bound for connected components of $\bigcup \Gamma_{j}^{(J)}$:
 \begin{equation*}
 \sum_{j=0}^{J}\#\left(s_{j}^{J}\right)\leq \left|\ln \varepsilon_{0}\right|^{2J^{2}d}.
 \end{equation*}
 For the first term in equation \eqref{eq:Ij-M large}, by the $L^{\infty}$-estimate and inequality \eqref{esti_beta_0} that
 \begin{equation}\label{eq:first term}
 \begin{aligned}
 &\left|\frac{1}{k_{\Delta}-n_{\Delta}}\sum_{j=0}^{J}\sum_{s_{j}^{J}}\big(\beta^J_{k,k_{\Delta}} \beta^J_{n,n_{\Delta}}\sin\big((k_{\Delta}-n_{\Delta})\rho\left(E\right)\big)e^{-{\rm i}t\Omega\left(E\right)}\big)\Big|_{\inf I_{s_{j}^{J}}}^{\sup I_{s_{j}^{J}}}\right|\\
 \leq&\frac{2}{\left|k_{\Delta}-n_{\Delta}\right|}\left|\ln \varepsilon_{0}\right|^{2J^{2}d}\cdot\left(1+\varepsilon_{0}^{\frac{1}{4}}+\varepsilon_{0}^{3\sigma}\right)^{2}\\
 \leq&\frac{18}{\left|k_{\Delta}-n_{\Delta}\right|}\left|\ln \varepsilon_{0}\right|^{2J^{2}d}.
 \end{aligned}
 \end{equation}
For the second term of the result in equation \eqref{eq:Ij-M large}, we need the following direct estimate
\begin{equation}\label{eq:betac1}
 \left|\beta^J_{k,k_{\Delta}} \beta^J_{n,n_{\Delta}}\right|_{\mathcal{C}^{1}\left(\Gamma^{\left(J\right)}_{j}\right)}\leq
 \begin{cases}
 3\left(1+\varepsilon_{0}^{\frac{1}{4}}\right)^{2}\;{\rm when}\;j=0;\\
 3\varepsilon_{j}^{6\sigma}\qquad{\rm when}\;1\leq j\leq J.\\
 \end{cases}
\end{equation}
 Using inequality
 \eqref{eq:betac1}, we deduce

For our Theorem \ref{thm:disper}, we have the following remarks.

	\begin{itemize}
		\item[($\mathbf{R}_1$)] Stefanov and Kevrekidis~\cite{SK05} first established a sharp dispersive decay rate
		$
		\frac{1}{3}
		$
		for the free discrete Klein-Gordon equation (i.e., equation \eqref{eq:DKG} with $V=0$) on $\mathbb Z$ via Van der Corput's lemma.
		Recently, Borovyk and Goldberg \cite{BG17} derived the corresponding rate on $\mathbb Z^{2}$, namely  $\langle t\rangle^{-\frac{3}{4}}$. Later, Cuenin and Ikromov~\cite{CI21} derived the rates on~$\mathbb Z^{d}$ for $d=3,4$, namely $
		\langle t\rangle^{-\frac{7}{6}}\ (d=3)$, and
		$\langle t\rangle^{-\frac{3}{2}}\log \bigl(2+|t|\bigr)\ (d=4)$. The obstacle to extending Cuenin-Ikromov to higher dimensions is that the finite classification of stable phase singularities (Thom's catastrophes), which is essential for deriving sharp decay rates, breaks down in dimensions $d \ge 5$. Our result can be seen as a perturbation result for Stefanov and Kevrekidis, which shows that the decay rate $(\frac{1}{3})^{-}$ persists under small quasi-periodic perturbations of the potentials.

		\item[($\mathbf{R}_2$)] While several studies (e.g., \cite{CT09, GPP13, PS08, KPS09, KKK06}) have addressed dispersive equations with potentials whose continuous spectrum is a union of disjoint intervals, generic behavior of the operator $H_{\theta}$ is markedly different: its spectrum is purely absolutely continuous and forms a Cantor set; see, e.g., \cite{AD08, Avi08, E92}. Our approach builds heavily on recent advances in the spectral theory of quasi-periodic Schr\"odinger operators; see, for example, \cite{Z16}. Specifically, we employ a KAM-based perturbative construction to approximate the rotation number and decompose the spectrum into many small intervals. Using a modified  spectral transform, we represent the solution as an oscillatory integral and then apply Van der Corput's lemma on each interval to establish dispersive decay. We emphasize that a similar strategy appears in the study of ballistic motion and dispersive estimates for discrete Schr\"odinger equations with quasi-periodic potentials; see, e.g., \cite{BZ20, Z16}.

        \item[($\mathbf{R}_3$)] For discrete wave equation with quasi-periodic potentials (i.e., equation \eqref{eq:DKG} with $m=0$), our method does not work. In fact, there does not exist $\ell^1 \to \ell^{\infty}$ estimates for fundamental solution of the free wave equation. We point out the following development regarding the dispersive estimate for the free discrete wave equation. This phenomenon also holds on $\mathbb{R}^1$. Schultz \cite{Sch98} proved dispersive estimates for the wave equation on lattice graphs $\mathbb{Z}^d$ for $d=2,3$, namely  $\langle t\rangle^{-\frac{3}{4}}\ (d=2)$ and $\langle t\rangle^{-\frac{7}{6}}\ (d=3)$. Combining the singularity theory
		with results in uniform estimates of oscillatory integrals, Bi-Chen-Hua \cite{BCH1} extended this result  to $d=4$: they proved that the optimal
		time decay rate is $|t|^{-\frac{3}{2}} \log |t|$. By tools from Newton polyhedra, Bi-Chen-Hua \cite{BCH2} further extended the result to $d=5$: the sharp decay rate of the fundamental solution of the wave equation on $\mathbb{Z}^5$ is $|t|^{-\frac{6}{11}}$.
    \end{itemize}

	A typical application for dispersive estimates is the derivation of Strichartz inequalities. For the inhomogeneous discrete Klein-Gordon equation,
	\begin{equation}\label{eq: inhomogegeneous}
		\begin{cases}
			\bigl(\partial_{tt}+G_{\theta}+m^{2}\bigr)u(n,t)=F(n,t),
			&(n,t)\in\mathbb{Z}\times\left(0,\infty\right),\\[2mm]
			u(n,0)=\varphi(n),\quad \partial_{t}u(n,0)=\psi(n),
			&n\in\mathbb{Z},
		\end{cases}
	\end{equation}
	we have the following mixed space-time estimates via the standard Keel-Tao argument. For convenience, we say
	a pair of exponents $(q,r)$ with
	$
	2\le q,r\le\infty
	$
	is called \emph{$\tau$--admissible} if
	$
	\frac{2}{q}
	=\tau\Bigl(1-\frac{2}{r}\Bigr).
	$ Here $0<\tau<\tfrac13$ is fixed as in Theorem~\ref{thm:disper}.

	\begin{theorem}\label{thm:Strichartz}

        Fix
		$0<\tau<\tfrac13$ and $F\in L^{\tilde q'}(\mathbb R;\ell^{\tilde r'}(\mathbb Z))$. Let $(q,r)$ and $(\tilde q,\tilde r)$ be two $\tau$--admissible pairs with $q,\tilde q<\infty.$  Then the Cauchy
		problem \eqref{eq: inhomogegeneous} has a unique global solution
		$
		u\in C(\mathbb R;\ell^2(\mathbb Z))
		$
		and \begin{equation}\label{eq:inhom-Strichartz}
			\|u\|_{L^q_t(\mathbb R;\ell^r(\mathbb Z))}
			\;\le\; C\Bigl(
			\|\varphi\|_{\ell^2(\mathbb Z)}
			+\|\psi\|_{\ell^2(\mathbb Z)}
			+\|F\|_{L^{\tilde q'}_t(\mathbb R;\ell^{\tilde r'}(\mathbb Z))}
			\Bigr),
		\end{equation}
		where $\tilde q'$ and $\tilde r'$ denote the Hölder conjugates of
		$\tilde q$ and $\tilde r$, respectively, and the constant $C$ is
		independent of $\theta$, $\varphi$, $\psi$ and $F$.

	\end{theorem}

	Using Strichartz estimates, we establish the global well-posedness of the nonlinear Klein-Gordon equation  \begin{equation}\label{eq: nonlinear}
		\begin{cases}
			\bigl(\partial_{tt}+G_{\theta}+m^{2}\bigr)u(n,t)= \pm |u|^{p-1}u,
			&(n,t)\in\mathbb{Z}\times\mathbb{R}_{>0},\\[2mm]
			u(n,0)=\varphi(n),\quad \partial_{t}u(n,0)=\psi(n),
			&n\in\mathbb{Z},
		\end{cases}
	\end{equation}
with small initial data.
	\begin{theorem}\label{thm:nonlinear}
		Let $p>7$. There exist $\varepsilon > 0$ and a constant $C$, so that whenever
		$\| \varphi \|_{\ell^2\left(\mathbb{Z}\right)}$ and $\| \psi\|_{\ell^2\left(\mathbb{Z}\right)} \leq \varepsilon
		$, there exists a unique global solution to \eqref{eq: nonlinear}, satisfying
		\begin{equation*}
		\|u\|_{L^q_t(\mathbb R;\ell^r(\mathbb Z))}\leq C\varepsilon,
		\end{equation*}
		for all admissible pairs $(q, r)$. In particular, $\| u(n,t) \|_{\ell^r(\mathbb{Z})} \to 0$ as $t \to \infty$ for every $r > 2$.
	\end{theorem}

The remainder of the paper is organized as follows. Section \ref{sec:2} reviews the structure of the spectrum of the quasi-periodic Schr\"odinger operator and presents a spectral transform that will be used throughout. Section \ref{sec:3} establishes Theorem \ref{thm:disper} by analysing a specific oscillatory integral on the spectrum. In Section \ref{sec:4}, we prove Theorems \ref{thm:Strichartz} and \ref{thm:nonlinear} as applications of the dispersive estimate.

A proof strategy for Theorem~\ref{thm:disper} is outlined in Remark $\mathbf{R}_2$ and the paper organization: the authors "employ a KAM-based perturbative construction to approximate the rotation number and decompose the spectrum into many small intervals." Then, "[u]sing a modified spectral transform, we represent the solution as an oscillatory integral and then apply Van der Corput's lemma on each interval to establish dispersive decay." Concretely, Section~\ref{sec:3} "establishes Theorem~\ref{thm:disper} by analysing a specific oscillatory integral on the spectrum."

A proof strategy for the main theorem is outlined in Remark $\mathbf{R}_2$ and the paper organization: the authors "employ a KAM-based perturbative construction to approximate the rotation number and decompose the spectrum into many small intervals." Then, "[u]sing a modified spectral transform, we represent the solution as an oscillatory integral and then apply Van der Corput's lemma on each interval to establish dispersive decay." Concretely, next the authors establish the main theorem by analysing a specific oscillatory integral on the spectrum.

\label{thm:disper}
		Assume $\varepsilon_0<\varepsilon_*$, with $\varepsilon_*$ as in the following lemma.

\begin{lemma}\label{lem:disper-log}
		Assume that $\left|V\right|_{r}=\varepsilon_{0}\leq \varepsilon_{\ast}$ with $\varepsilon_{\ast}$ defined in Proposition \ref{propsana}. For any $k,k_{\Delta},n,n_{\Delta}$,
		\begin{equation*}
			\begin{aligned}
				\left|\mathbb{I}\left(\Sigma\right)\right|\leq\frac{C\left(m\right)\left|\ln\varepsilon_{0}\right|^{82000d\left(\ln\ln\left(2+\JP{t}\right)\right)^{2}}}{\JP{t}^{\frac{1}{3}}},
			\end{aligned}
		\end{equation*}
		for any $t\in\R$ where $m>0$ and the constant $C\left(m\right)$ only depends on $m$.
	\end{lemma}

		Then for any given $0<\tau<\frac13$, there exists $K_1=K_1(\varepsilon_0,\tau)$ such that for any $\theta\in\mathbb{T}^d$ and any $t\in\mathbb{R}$,
\begin{equation}\label{linest}
			\norm{u\left( \cdot ,t\right)}_{\ell^\infty\left(\mathbb{Z}\right)}\leq K_1\JP{t}^{-\tau}(\norm{\varphi}_{\ell^1\left(\mathbb{Z}\right)}+\norm{\psi}_{\ell^1\left(\mathbb{Z}\right)}) \ , \quad \forall \ \varphi, \psi \in\ell^1(\mathbb{Z}) \ .
		\end{equation}

Existential–Universal
Inequality or Bound

Question:
Consider the discrete Klein--Gordon equation on \(\mathbb Z\)
\[
\bigl(\partial_{tt}+G_{\theta}+m^{2}\bigr)u(n,t)=0,\qquad (n,t)\in \mathbb Z\times (0,\infty),
\]
with initial data
\[
u(\cdot,0)=\varphi\in \ell^{1}(\mathbb Z),\qquad \partial_t u(\cdot,0)=\psi\in \ell^{1}(\mathbb Z),
\]
where
\[
(G_{\theta}u)(n)=-\Delta_{\mathrm{disc}}u(n)+V(\theta+n\omega)u(n),
\qquad
\Delta_{\mathrm{disc}}u(n)=u(n+1)+u(n-1)-2u(n).
\]
Assume \(\theta\in \mathbb T^d\), \(d\ge 1\), \(m>0\), \(\omega\in\mathbb R^d\) is Diophantine, and \(V:\mathbb T^d\to\mathbb R\) extends to a bounded analytic function on \(\{z\in\mathbb C^d:|\operatorname{Im} z|<r\}\). Let
\[
\varepsilon_0:=|V|_r=
\sup_{\{z\in\mathbb C^d:|\operatorname{Im} z|<r\}} |V(z)|,
\]
and assume \(\varepsilon_0<\varepsilon_*\). Also write \(\langle t\rangle:=\sqrt{1+t^2}\). Under these assumptions, which quantitative \(\ell^1\to\ell^\infty\) dispersive estimate is valid for the solution \(u\)?

For every \(\tau\) with \(0<\tau<\tfrac13\), there exists a constant \(K_1=K_1(\varepsilon_0,\tau)\) such that, for every \(\theta\in\mathbb T^d\), every \(t\in\mathbb R\), and all \(\varphi,\psi\in \ell^1(\mathbb Z)\), the corresponding solution satisfies
\[
\|u(\cdot,t)\|_{\ell^\infty(\mathbb Z)}
\le K_1\langle t\rangle^{-\tau}
\bigl(\|\varphi\|_{\ell^1(\mathbb Z)}+\|\psi\|_{\ell^1(\mathbb Z)}\bigr).
\]

Choices:
B.
There exists a constant \(K_1=K_1(\varepsilon_0)\) such that, for every \(\theta\in\mathbb T^d\), every \(t\in\mathbb R\), and all \(\varphi,\psi\in \ell^1(\mathbb Z)\), the corresponding solution satisfies
\[
\|u(\cdot,t)\|_{\ell^\infty(\mathbb Z)}
\le K_1\langle t\rangle^{-1/3}
\bigl(\|\varphi\|_{\ell^1(\mathbb Z)}+\|\psi\|_{\ell^1(\mathbb Z)}\bigr).
\]
C.
There exists a constant \(K_1=K_1(\varepsilon_0)\) such that, for every \(\theta\in\mathbb T^d\), every \(t\in\mathbb R\), and all \(\varphi,\psi\in \ell^1(\mathbb Z)\), the corresponding solution satisfies
\[
\|u(\cdot,t)\|_{\ell^\infty(\mathbb Z)}
\le K_1
\bigl(\|\varphi\|_{\ell^1(\mathbb Z)}+\|\psi\|_{\ell^1(\mathbb Z)}\bigr).
\]
D.
For every \(\tau\) with \(0<\tau<\tfrac13\), there exists a constant \(K_1=K_1(\varepsilon_0,\tau,\theta)\) such that, for every \(t\in\mathbb R\) and all \(\varphi,\psi\in \ell^1(\mathbb Z)\), the corresponding solution satisfies
\[
\|u(\cdot,t)\|_{\ell^\infty(\mathbb Z)}
\le K_1\langle t\rangle^{-\tau}
\bigl(\|\varphi\|_{\ell^1(\mathbb Z)}+\|\psi\|_{\ell^1(\mathbb Z)}\bigr).
\]
E.
For every \(\tau\) with \(0<\tau<\tfrac12\), there exists a constant \(K_1=K_1(\varepsilon_0,\tau)\) such that, for every \(\theta\in\mathbb T^d\), every \(t\in\mathbb R\), and all \(\varphi,\psi\in \ell^1(\mathbb Z)\), the corresponding solution satisfies
\[
\|u(\cdot,t)\|_{\ell^\infty(\mathbb Z)}
\le K_1\langle t\rangle^{-\tau}
\bigl(\|\varphi\|_{\ell^1(\mathbb Z)}+\|\psi\|_{\ell^1(\mathbb Z)}\bigr).
\]

endpoint decay exponent from oscillatory-integral bound

dropped time-decay factor \(\langle t\rangle^{-\tau}\)

uniformity in phase parameter \(\theta\)

allowed range of admissible decay exponents

The stem does not state the conclusion or explicitly reveal the correct option. It gives hypotheses and asks for the resulting dispersive estimate, so there is no direct answer leakage.

This is largely a theorem-recall item: the hypotheses are presented and the learner is asked to identify the theorem's conclusion. The nearby variants prevent it from being a completely trivial restatement, but it is still close to direct recall rather than a genuinely independent inference task.

There is some reasoning pressure because the options differ in subtle but important ways: open vs. closed endpoint, uniformity in θ, exact endpoint decay, and a weaker existential claim. However, the problem mainly tests memory of the exact theorem statement, not substantial generative reasoning.

Most distractors are mathematically plausible and target common failure modes, especially endpoint inclusion and loss of uniformity in θ. However, option C is a weaker true statement if A is true, so the item is not cleanly single-correct unless the stem asks for the strongest statement. That weakens distractor quality.

A reasonably well-constructed theorem-discrimination MCQ with no answer leakage and mostly strong distractors, but it is close to theorem restatement and is weakened by the inclusion of a weaker true option, making the single-best-answer format somewhat ambiguous.


Question:
Consider the discrete Klein--Gordon equation on \(\mathbb Z\)
\[
\bigl(\partial_{tt}+G_{\theta}+m^{2}\bigr)u(n,t)=0,\qquad (n,t)\in \mathbb Z\times (0,\infty),
\]
with initial data
\[
u(\cdot,0)=\varphi\in \ell^{1}(\mathbb Z),\qquad \partial_t u(\cdot,0)=\psi\in \ell^{1}(\mathbb Z),
\]
where
\[
(G_{\theta}u)(n)=-\Delta_{\mathrm{disc}}u(n)+V(\theta+n\omega)u(n),
\qquad
\Delta_{\mathrm{disc}}u(n)=u(n+1)+u(n-1)-2u(n).
\]
Assume \(\theta\in \mathbb T^d\), \(d\ge 1\), \(m>0\), \(\omega\in\mathbb R^d\) is Diophantine, and \(V:\mathbb T^d\to\mathbb R\) extends to a bounded analytic function on \(\{z\in\mathbb C^d:|\operatorname{Im} z|<r\}\). Let
\[
\varepsilon_0:=|V|_r=
\sup_{\{z\in\mathbb C^d:|\operatorname{Im} z|<r\}} |V(z)|,
\]
and assume \(\varepsilon_0<\varepsilon_*\). Also write \(\langle t\rangle:=\sqrt{1+t^2}\). Under these assumptions, which quantitative \(\ell^1\to\ell^\infty\) dispersive estimate is valid for the solution \(u\)?

Choices:
B.
There exists a constant \(K_1=K_1(\varepsilon_0)\) such that, for every \(\theta\in\mathbb T^d\), every \(t\in\mathbb R\), and all \(\varphi,\psi\in \ell^1(\mathbb Z)\), the corresponding solution satisfies
\[
\|u(\cdot,t)\|_{\ell^\infty(\mathbb Z)}
\le K_1\langle t\rangle^{-1/3}
\bigl(\|\varphi\|_{\ell^1(\mathbb Z)}+\|\psi\|_{\ell^1(\mathbb Z)}\bigr).
\]
C.
There exists a constant \(K_1=K_1(\varepsilon_0)\) such that, for every \(\theta\in\mathbb T^d\), every \(t\in\mathbb R\), and all \(\varphi,\psi\in \ell^1(\mathbb Z)\), the corresponding solution satisfies
\[
\|u(\cdot,t)\|_{\ell^\infty(\mathbb Z)}
\le K_1
\bigl(\|\varphi\|_{\ell^1(\mathbb Z)}+\|\psi\|_{\ell^1(\mathbb Z)}\bigr).
\]
D.
For every \(\tau\) with \(0<\tau<\tfrac13\), there exists a constant \(K_1=K_1(\varepsilon_0,\tau,\theta)\) such that, for every \(t\in\mathbb R\) and all \(\varphi,\psi\in \ell^1(\mathbb Z)\), the corresponding solution satisfies
\[
\|u(\cdot,t)\|_{\ell^\infty(\mathbb Z)}
\le K_1\langle t\rangle^{-\tau}
\bigl(\|\varphi\|_{\ell^1(\mathbb Z)}+\|\psi\|_{\ell^1(\mathbb Z)}\bigr).
\]
E.
For every \(\tau\) with \(0<\tau<\tfrac12\), there exists a constant \(K_1=K_1(\varepsilon_0,\tau)\) such that, for every \(\theta\in\mathbb T^d\), every \(t\in\mathbb R\), and all \(\varphi,\psi\in \ell^1(\mathbb Z)\), the corresponding solution satisfies
\[
\|u(\cdot,t)\|_{\ell^\infty(\mathbb Z)}
\le K_1\langle t\rangle^{-\tau}
\bigl(\|\varphi\|_{\ell^1(\mathbb Z)}+\|\psi\|_{\ell^1(\mathbb Z)}\bigr).
\]

The stem gives hypotheses and asks for the valid dispersive estimate, but it does not state or strongly hint at the precise decay rate, admissible range of exponents, or uniformity in b8. The correct option is not leaked.

This is essentially a theorem-recall item: the question presents the assumptions and asks for the exact conclusion of the dispersive estimate. The options are small perturbations of the theorem statement rather than genuinely different mathematical outcomes.

There is some pressure to distinguish subtle variants (endpoint \(1/3\) vs. \(\tau<1/3\), uniform vs. \(\theta\)-dependent constants, stronger vs. weaker decay), but the task mainly rewards recalling the precise theorem statement rather than generating a conclusion from reasoning.

The distractors are strong: they reflect common failure modes such as claiming the endpoint decay, dropping decay entirely, weakening uniformity in \(\theta\), or overstating the admissible exponent range. They are plausible and mathematically distinct.

A solid multiple-choice item with no answer leakage and high-quality distractors, but it is mostly a direct theorem-restatement question and only moderately tests generative reasoning.


\label{thm} If $C$ is stably embedded in $\mathbf{M}$ and $\mu$ an $A$-valued Fubini measure on $\operatorname{Def}(C)$,
then $\mu$ extends uniquely to an $A$-valued Fubini measure on $\operatorname{Def}(C)^{\operatorname{f}}$.

\begin{proof} For every $p\in M^k$ the set $\cal{S}(p)$ has by Lemma~\ref{l5} an $(r+1)$-step fibering over $C$. As in the proof of Lemma~\ref{l5}, this yields a finite set $E$ of tuples
$$\big( f_1,\dots, f_{r+1}, f_{r+2}; m_1,\dots,m_{r+1}, n_1,\dots, n_{r+1},n_{r+2}\big)$$
where for $j=1,\dots, r+2$: $f_j$ is a definable map $ \Sigma_j\to M^{n_i}$ with (definable) domain
$\Sigma_j\subseteq M^l\times M^{m_1}\times\cdots\times M^{m_{j-1}}\times M^{m+n_1+\cdots + n_{j-1}}$, and for each $p\in M^k$ there is
a tuple in $E$ as displayed and $q\in M^{l}$
such that $\cal{S}(p)=\Sigma_1(q)$ and the tuple
$$ \big(f_1(q,-),\dots, f_{r+1}(q,-), f_{r+2}(q,-); m_1,\dots, m_{r+1}, n_1,\dots, n_{r+1}, n_{r+2}\big)$$
is an $(r+1)$-step fibering of the domain $\Sigma_1(q)\subseteq M^m$ of $f_1(q,-)$ over $C$.

Consider one such tuple $\tau\in E$ as displayed above. Let $Q$ be the definable
set of $q\in M^{l}$ such that
for some $p\in M^k$ we have $\cal{S}(p)=\Sigma_1(q)$ and
$$ \big(f_1(q,-),\dots, f_{r+1}(q,-), f_{r+2}(q,-); m_1,\dots, m_{r+1}, n_1,\dots, n_{r+1},n_{r+2}\big)$$
is an $(r+1)$-step fibering of the domain $\Sigma_1(q)\subseteq M^m$ of $f_1(q,-)$ over $C$.
Each $q\in Q$ yields a map
$f_1(q,-): \Sigma_1(q)\to M^{n_1}$ such that its preimages of points $y\in M^{n_1}$ are in $\cC_r$. The maps $f_1(q,-)$ with $q\in Q$ make up a definable family $\cal{F}_\tau$ of maps from subsets of $M^m$ into $M^{n_1}$ as described just before Lemma~\ref{l2+}. That lemma then tells us that $\{\mu\big(\Sigma_1(q)\big): q\in Q\}$ is finite, and that
$\{q\in Q:\ \mu\big(\Sigma_1(q)\big)=a\}$ is definable for every $a$. This holds for all $\tau\in E$, which yields the
desired result.
\end{proof}

\noindent
This finishes the inductive proof of Theorem~\ref{thm} in the case of $\omega$-saturated
$\cM$.

\subsection*{Reduction to the $\omega$-saturated case}  Here $C$ is stably embedded in $\cM$, but we do not assume $\cM$ is $\omega$-saturated.

Let $\cM^*$ be an elementary extension of $\cM$. We define an $A$-valued Fubini measure $\mu^*$ on  the full subcategory
$\Def(C^*)$ of $\Def(\cM^*)$: Let $Y\subseteq (C^*)^n$ be definable in $\cM^*$; take an $\cal{L}$-formula $\phi(x,y)$ with $x=(x_1,\dots, x_m)$, $y=(y_1,\dots, y_n)$, and a point $p_Y\in (C^*)^m$ such that $Y=\{q\in (C^*)^n: \cM^*\models \phi(p_Y,q)\}$. To simplify notation, set $\phi(p, C^n):=\{q\in C^n:\ \cM\models \phi(p,q)\}$ for $p\in C^m$, and likewise define
$\phi\big(p, (C^*)^n\big)\subseteq (C^*)^n$ for $p\in (C^*)^m$; in particular, $Y=\phi\big(p_Y, (C^*)^n\big)$.
Now $C^m$ is the disjoint union of the definable sets $X_a:=\{p\in C^m:\ \mu\big(\phi(p,C^m)\big)=a\}$ (which is empty for all but finitely many $a$).
Set $\mu^*(Y):=a$, where $a$ is unique such that $p_Y\in (X_a)^*$. It is routine to verify that this particular $a$ does not depend on the choice of $\phi$ and that it yields indeed a Fubini measure on $\Def(C^*)$.

Now take such an elementary extension $\cM^*$ to be $\omega$-saturated. Then $\mu^*$ extends uniquely to an $A$-valued Fubini measure $\mu^*$ on the full subcategory $\Def(C^*)^{\f}$ of $\Def(\cM^*)$.
For $X\in \Def(C)^{\f}$ we have $X^*\in \Def(C^*)^{\f}$; set $\mu(X):= \mu^*(X^*)$.
This extends the original $A$-valued Fubini measure $\mu$ on $\Def(C)$ to an $A$-valued Fubini measure $\mu$ on $\Def(C)^{\f}$.  As to uniqueness, suppose $X$ is fiberable over $C$, and take an $r$-step fibering $(f_1,\dots, f_r, f_{r+1}; m_1,\dots, m_{r}, n_1,\dots, n_{r+1})$
of $X$ over $C$.
For $r=0$ the definable map $f_1: X \to C^{n_1}$ is injective, and so by Fubini
$\mu(X)$ is uniquely determined by $\mu: \Def(C)\to A$. For $r\ge 1$ the definable map $f_1: X\to M^{n_1}$ is such that
$f_1(X)$ and every
$f_1^{-1}(y)$ with $y\in f_1(X)$ is $(r-1)$-step fiberable over $C$, and we can assume inductively that the values of
$\mu\big(f(X)\big)$ and $\mu\big(f_1^{-1}(y)\big)$ for
$y\in f_1(X)$ are uniquely determined by $\mu: \Def(C)\to A$, which again by Fubini uniquely deternines $\mu(X)$.
This concludes the proof of Theorem~\ref{thm}.

\section{Fiberability and Co-analyzability}\label{co}

\noindent
Fiberability is a somewhat ad hoc notion, but agrees under natural model-theoretic conditions with the more intrinsic notion of co-analyzability from \cite{HHM}, as we shall see. {\it Throughout this section $X\subseteq M^m$ is definable}.

\subsection*{Fiberability implies co-analyzability}
We recall here the recursion on $r$ that defines ``$X$ is co-analyzable in $r$ steps" (tacitly: relative to $C$ in $\cM$), where $\cM$ is assumed to be $\omega$-saturated: \begin{enumerate}
\item  $X$ is co-analyzable in $0$ steps iff $X$ is finite;
\item  $X$ is co-analyzable in $r+1$ steps iff for some definable set $R \subseteq M^m\times C$ we have: $\breve{R}(y)$ is co-analyzable in $r$ steps for every $y\in C$, and
$X:= \bigcup_{y\in C}\breve{R}(y)$.
\end{enumerate}
 See \cite[p. 35]{ADH} for how this extends to not necessarily $\omega$-saturated $\cM$ in such a way that
 for any elementary extension $\cM^*$ of $\cM$: $X$ is co-analyzable in $r$ steps relative to $C$ in $\cM$ iff $X^*$ is co-analyzable in $r$ steps relative to $C^*$ in $\cM^*$.  Call $X$ {\em co-analyzable\/} if $X$ is co-analyzable in $r$ steps for some $r$.

 By \cite{HHM} this notion can be characterized as follows:

 \begin{prop}\label{hhm}
Let $T$ be a complete $L$-theory, where $L$ is a countable one-sorted language. Let $y$ be a single variable and $D(y)$ an $L$-formula that defines in each model $\cal{N}=(N;\cdots)$ of $T$ a set $D(N)$ with $|D(N)|\ge 2$.
Then the following conditions on an $L$-formula $\varphi(x)$ with $x=(x_1,\dots,x_m)$ are equivalent:
\begin{enumerate}
\item[\textup{(i)}] for some  $\cal{N}\models T$, $\varphi(N^m)$ is co-analyzable relative to $D(N)$;
\item[\textup{(ii)}] for all $\cal{N}\models T$, $\varphi(N^m)$ is co-analyzable relative to $D(N)$;
\item[\textup{(iii)}] for all  $\cal{N}\models T$, if
$D(N)$ is countable, then so is $\varphi(N^m)$;
\item[\textup{(iv)}] for all $\cal{N}\preceq\cal{N}^*\models T$, if
$D(N)=D(N^*)$, then $\varphi(N^m)=\varphi((N^*)^m)$.
\end{enumerate}
\end{prop}

\noindent
The equivalences (i)$\Leftrightarrow$(ii)$\Leftrightarrow$(iv) hold without assuming that $L$ is countable:
For (i)$\Leftrightarrow$(ii)$\Rightarrow$(iv), reduce to countable $L$ by taking suitable reducts. For (iv)$\Rightarrow$(ii), use also that (iv) amounts to $T_2\models \sigma$ where $T_2$ is the theory of elementary pairs of models of $T$ and $\sigma$ is a certain sentence, depending on $\varphi$, in the language of $T_2$. Note that in addition we can restrict $\cal{N}$ and $\cal{N}^*$ in (iv) to be  $\omega$-saturated.

\begin{lemma}\label{fico} If $X$ is fiberable over $C$, then  $X$ is co-analyzable relative to $C$.
\end{lemma}
\begin{proof}
We arrange that $\cM$ is $\omega$-saturated.
Below we prove by induction on $r$:  if $X$ is $r$-step fiberable over $C$, then $X$ is co-analyzable relative to $C$.
For $r=0$, if $f: X\to C^n$ is definable and injective, then $X$ is co-analyzable in $n$ steps relative to $C$, by an easy induction on $n$. Thus the claimed implication holds for $r=0$. Assume this implication holds for a certain $r$, and let
$X$ be $(r+1)$-step fiberable over $C$.
To show $X$ is co-analyzable relative to $C$ in $\cM$ we
augment $\cal{L}$ by names for finitely many elements of $M$ to arrange that $X$ is $0$-definable, so $X=\varphi(M^m)$ with $\varphi(x)$ an $\cal{L}$-formula, $x=(x_1,\dots, x_m)$. We now apply (iv)$\Rightarrow$(ii) of Proposition~\ref{hhm} without assuming $L=\cal{L}$ is countable, with $T:=\Th(\cM)$ and $D(y)$ an $\cal{L}$-formula such that $D(M)=C$.
So let $\cal{N}\preceq \cal{N}^*\models T$ with $D(N)=D(N^*)$; it suffices to show that then $\varphi(N^m)=\varphi\big((N^*)^m\big)$.
We also assume $\cal{N}$ is $\omega$-saturated, as we may by a remark following Proposition~\ref{hhm}. Since $\cal{M}\equiv \cal{N}$, the set $\varphi(N^m)$ is $(r+1)$-step fiberable over $C^{\cal{N}}:= D(N)$ in $\cal{N}$. Take $f: \varphi(N^m)\to Y$ where $Y\subseteq N^n$ and $f$ are definable in $\cal{N}$, such that $Y$ is $r$-step fiberable over $C^{\cal{N}}$ and for all
$y\in Y$ the fiber $f^{-1}(y)$ is $r$-step fiberable over $C^{\cal{N}}$. Thus by our inductive assumption and Proposition~\ref{hhm}, $Y$ and the fibers
$f^{-1}(y)$ for $y\in Y$ do not change in passing from $\cal{N}$ to $\cal{N}^*$. Hence the same is true for $\varphi(N^m)$, that is,
$\varphi(N^m)=\varphi\big((N^*)^m\big)$.
\end{proof}

\label{ms}

\noindent
So far we assumed $\cM$ to be one-sorted. But many structures are more naturally viewed as many-sorted; for example, valued fields as $3$-sorted structures with residue field a

\begin{proof} Take definable $f: Y\to M^m$ with $Y\subseteq C^\nu$, $\nu\in \N$, such that $X=f(Y)$. Since $(C;\cM)$ has EI, we have a definable map $g:Y\to C^n $ such that for all $y,z\in Y$ we have $f(y)=f(z)\Leftrightarrow g(y)=g(z)$. This yields a definable bijection $$\qquad\qquad \qquad \qquad f(y)\mapsto g(y)\ :\  X\to g(Y),\qquad (y\in Y). \qquad \qquad \qquad \qquad \qedhere$$
\end{proof}

\subsection*{Analyzability}
The concept of {\em analyzability\/} is more popular in model theory than co-analyzability, but is usually defined for types
in the environment of a monster model, instead of definable sets. I have been told that for definable sets and $\omega$-saturated $\cM$ its recursive definition should be as follows:

{\em $X$ is $0$-step analyzable in $C$ iff $X$ is internal to $C$, and $X$ is $(r+1)$-step analyzable in $C$ iff there is a $\Def(\cM)$-morphism $f: X \to Y$ such that $Y$ is $r$-step analyzable over $C$ and all fibers $f^{-1}(y)$, $y\in Y$, are internal to $C$. Finally, $X$ is analyzable in $C$ iff $X$ is $r$-step analyzable in $C$ for some $r$}.

This extends to not necessarily $\omega$-saturated $\cM$ by requiring in the $(r+1)$-clause the existence of a definable
family of maps from subsets of $C^n$ (for a fixed $n$) into $M^m$ such that for each $y\in Y$ some member of the family has image $f^{-1}(y)$.
With this extended notion and any elementary extension $\cM^*$ of $\cM$,
$$X \text{ is $r$-step analyzable in }C\ \Longleftrightarrow\  X^* \text{ is $r$-step analyzable in }C^*.$$
Using this it follows from Proposition~\ref{hhm} that if $X$ is analyzable in $C$, then $X$ is co-analyzable relative to $C$. Ben Castle and Rahim Moosa have assured me that the converse should also hold under some natural model-theoretic conditions to be satisfied in particular by differentially closed fields (and probably by $\T$) with $C$ its constant field. I have not pursued this, but it would be good to have this confirmed. One motivation for this: it might help in showing that for $\T$ and for differentially closed fields the notions of  fiberability over $C$ and co-analyzability relative to $C$ do not collapse for any $r$ to $r$-step fiberability over $C$ and co-analyzability in $r$-steps relative to $C$, respectively. This is plausible in view of \cite{J} where Jin constructs in a ``big" differentially closed field for any $r\ge 1$ a $1$-type that is $r$-step analyzable but not $(r-1)$-step analyzable (in the constant field).

\section{Application to $\T$ and to differentially closed fields} \label{special}

\subsection*{The case of $\T$} We construe here $\T$ as an ordered differential field. Its subfield $\R$ is its constant field, and
a set $Y\subseteq \R^n$ is definable in $\T$ iff $X$ is semialgebraic in the sense of $\R$. In particular, $\R$ is stably embedded in
$\T$. Thus by Theorem~\ref{thm}, the $\N_{\trop}$-valued Fubini measure $Y\mapsto \dim Y$ on $\Def(\R)$ extends uniquely to an $\N_{\trop}$-valued Fubini measure on the full subcategory $\Def(\R)^{\f}$ of $\Def(\T)$. Likewise for the $\Z$-valued Fubini measure $Y\mapsto E(Y)$ on $\Def(\R)$.

All this holds also for any $\upo$-free, newtonian, Liouville closed $H$-field $K$ and its constant field instead of
$\T$ and its constant field $\R$; this includes all ordered differential fields $K \equiv\T$.  The extended Euler characteristic now yields an answer to a test question raised in \cite[p. 34, end of Section 5]{ADH}:

\begin{cor} With $K$ as above, let $X\subseteq K$ be definable in $K$. Then $X$ with the ordering induced by $K$
is not elementarily equivalent to the ordered set $\omega$.
\end{cor}
\begin{proof} Suppose $X\equiv \omega$. Then $X$ is discrete as a subspace of $K$ with its order topology. Hence $X$ is fiberable
over $C$ by \cite{ADH}. Now $X\equiv \omega$ also yields  a definable bijection between $X$ and $X\setminus \{\text{the least element of $X$}\}$, but the extended Euler characteristic then gives $E(X)=E(X)-1$, a contradiction.
\end{proof}

\label{int}

\noindent
Semialgebraic sets $Y\subseteq  {\bbold R} ^n$ have a dimension $\dim(Y)\in  {\bbold N} \cup\{-\infty\}$ and an Euler characteristic $E(Y)\in  {\bbold Z} $ with good properties; see \cite[Ch. 4]{D}. In \cite{ADH} we asked for a natural extension of this dimension and Euler characteristic to the definable sets $X\subseteq  {\bbold T} ^m$
that are ``fiberable by $ {\bbold R} $" as defined there; here $ {\bbold T} $ is the real closed differential field of transseries, which has its subfield $ {\bbold R} $ as constant field.

There are many similar situations, and so it makes sense to do the above as a special case of a general extension result for what we call {\em Fubini measures}, with ``Fubini'' signaling good behavior in families. For example, at the end of Section~\ref{special} we shall also apply this to differentially closed fields (models of $\operatorname{DCF}_0$).

\medskip\noindent
Let $\mathbf{M}=(M;\cdots)$ be a one-sorted $\mathcal{L}$-structure;
``definable" will mean ``definable in $\mathbf{M}$ with parameters" unless we indicate otherwise.  For definable $X\subseteq M^m$ and an elementary extension $\mathbf{M}^*=(M^*;\cdots)$ of $\mathbf{M}$ we let $X^*\subseteq (M^*)^m$ be defined in $\mathbf{M}^*$ by any $\mathcal{L}_M$-formula that defines $X$ in $\mathbf{M}$.
The (small) category $\operatorname{Def}(\mathbf{M})$ has as its objects the definable sets $X\subseteq M^m$; the number $m\in  {\bbold N} $ is part of  specifying $X$ as an object of $\operatorname{Def}(\mathbf{M})$.  For objects $X,Y$ of $\operatorname{Def}(\mathbf{M})$, the morphisms $X\to Y$ are the definable maps $X\to Y$,
and composition of morphisms is composition of maps, so an isomorphism $X\to Y$ is the same as a definable bijection $X\to Y$. We shall abuse language by
indicating the class of objects of a category $\mathcal{C}$ also by $\mathcal{C}$, so $X\in \mathcal{C}$ means that $X$ is an object of $\mathcal{C}$.

\medskip\noindent
Let $\cal{C}$ be a full subcategory of $\operatorname{Def}(\mathbf{M})$ such that for all $X,Y\in \cal{C}$: \begin{itemize}
\item[(a)] if $X, Y\subseteq M^m$, then $X\cup Y\subseteq M^m$ is an object of $\cal{C}$;
\item[(b)] every definable subset of $X$ is an object of $\cal{C}$;
\item[(c)]  $X\times Y\in \cal{C}$ (where $X\times Y\subseteq  M^{m+n}$ for $X\subseteq M^m, Y\subseteq M^n$).
\end{itemize}
Let $A$ be a commutative semiring, that is, a set with binary operations of addition and multiplication and distinguished elements
$0$ and $1$ such that $A$ is a commutative monoid with respect to both addition and multiplication, with $0$ and $1$ as neutral elements for addition and multiplication, respectively, such that multiplication is distributive with respect to addition, and such that also $0\cdot a=0$ for all $a\in A$.

Let  $\mu$ an $A$-valued Fubini measure on $\cal{C}$, that is, $\mu: \cal{C}\to A$, and for all $X,Y\in  \cal{C}$:\begin{enumerate}
\item[(d)] $\mu(X)=0$ whenever $X$ is empty; $\mu(X)=1$ whenever $X$ is a singleton;
\item[(e)] $\mu(X\cup Y)=\mu(X)+\mu(Y)$ whenever $X,Y\subseteq M^m$ are disjoint;
\item[(f)] every $\cal{C}$-morphism $f: X \to Y$ satisfies ``Fubini": \begin{enumerate}
\item[(F1)] the subset $\{\mu\big(f^{-1}(y)\big):\ y\in Y\}$ of $A$ is finite;
\item[(F2)] for all $a\in A$, the set
$Y_a := \{y\in Y:\ \mu\big(f^{-1}(y)\big)=a\}$ is definable;
\item[(F3)] if $\mu\big(f^{-1}(y)\big)$ takes the constant value $a$ on $Y$, then
$\mu(X)= a\mu(Y)$.
\end{enumerate}
\end{enumerate}
Thus for $X,Y\in  \cal{C}$:  $\mu(X\times Y)=\mu(X)\cdot \mu(Y)$; if $X$ and $Y$ are $\cal{C}$-isomorphic, then $\mu(X)=\mu(Y)$ (a ``motivic'' property);  for $X,Y\subseteq M^m$ we can represent $X\cup Y$ as the disjoint union of $X\setminus Y, X\cap Y$, and $Y\setminus X$ to obtain
$$\mu(X\cup Y)+\mu(X\cap Y)\ =\ \mu(X)+\mu(Y).$$
If $f: X\to Y$ is a $\cal{C}$-morphism, then $\mu(X)=\sum_a a\mu(Y_a)$, using  notation of (F2).

\medskip\noindent
{\bf Example.} Let $\mathbf{M}= {\bbold T} $ as in the beginning of the introduction. Then the subsets of $ {\bbold R} ^n$ definable in $\mathbf{M}$ are  semialgebraic in the sense of $ {\bbold R} $, so the full subcategory $\cal{C}$ of $\operatorname{Def}(\mathbf{M})$
whose objects are these semialgebraic sets satisfies conditions (a), (b), (c). The above Euler characteristic $Y\mapsto E(Y)$ is then a Fubini measure on $\cal{C}$ with values in the ring $ {\bbold Z} $ of integers.  Also $Y\mapsto \dim(Y)$ is a Fubini measure on $\cal{C}$, taking values in the tropical semiring $\N_{\operatorname{trop}}$, which has underlying set
$ {\bbold N} \cup\{-\infty\}$ and addition and multiplication given respectively by
$(p,q)\mapsto \max(p,q)$ and $(p,q)\mapsto p+q$; this semiring has $-\infty$ as its zero element for addition and $0$ as its multiplicative identity.

\bigskip\noindent
To state the main result of this paper, we consider a $0$-definable set $C\subseteq M$ with more than one element, and let $\operatorname{Def}(C)$ be the full subcategory of $\operatorname{Def}(\mathbf{M})$ whose objects are the definable subsets of the powers $C^n$.  Below we define the notion, for definable $X\subseteq M^m$, of {\em $X$ being fiberable over $C$}. These $X$ are the objects of a full subcategory
$\operatorname{Def}(C)^{\operatorname{f}}\supseteq  \operatorname{Def}(C)$ of $\operatorname{Def}(\mathbf{M})$. The categories $\operatorname{Def}(C)$ and $\operatorname{Def}(C)^{\operatorname{f}}$ satisfy conditions (a), (b), (c)  imposed earlier on $\cal{C}$.

There are many similar situations, and so it makes sense to do the above as a special case of a general extension result for what we call {\em Fubini measures}, with ``Fubini'' signaling good behavior in families. For example, at the end of Section~\ref{special} we shall also apply this to differentially closed fields (models of $\operatorname{DCF}_0$).

\medskip\noindent
Let $\mathbf{M}=(M;\cdots)$ be a one-sorted $\mathcal{L}$-structure;
``definable" will mean ``definable in $\mathbf{M}$ with parameters" unless we indicate otherwise. For definable $X\subseteq M^m$ and an elementary extension $\mathbf{M}^*=(M^*;\cdots)$ of $\mathbf{M}$ we let $X^*\subseteq (M^*)^m$ be defined in $\mathbf{M}^*$ by any $\mathcal{L}_M$-formula that defines $X$ in $\mathbf{M}$.
The (small) category $\operatorname{Def}(\mathbf{M})$ has as its objects the definable sets $X\subseteq M^m$; the number $m\in {\bbold N} $ is part of specifying $X$ as an object of $\operatorname{Def}(\mathbf{M})$. For objects $X,Y$ of $\operatorname{Def}(\mathbf{M})$, the morphisms $X\to Y$ are the definable maps $X\to Y$,
and composition of morphisms is composition of maps, so an isomorphism $X\to Y$ is the same as a definable bijection $X\to Y$. We shall abuse language by
indicating the class of objects of a category $\mathcal{C}$ also by $\mathcal{C}$, so $X\in \mathcal{C}$ means that $X$ is an object of $\mathcal{C}$.

\medskip\noindent
Let $\cal{C}$ be a full subcategory of $\operatorname{Def}(\mathbf{M})$ such that for all $X,Y\in \cal{C}$: \begin{itemize}
\item[(a)] if $X, Y\subseteq M^m$, then $X\cup Y\subseteq M^m$ is an object of $\cal{C}$;
\item[(b)] every definable subset of $X$ is an object of $\cal{C}$;
\item[(c)] $X\times Y\in \cal{C}$ (where $X\times Y\subseteq M^{m+n}$ for $X\subseteq M^m, Y\subseteq M^n$).
\end{itemize}
Let $A$ be a commutative semiring, that is, a set with binary operations of addition and multiplication and distinguished elements
$0$ and $1$ such that $A$ is a commutative monoid with respect to both addition and multiplication, with $0$ and $1$ as neutral elements for addition and multiplication, respectively, such that multiplication is distributive with respect to addition, and such that also $0\cdot a=0$ for all $a\in A$.

Let $\mu$ an $A$-valued Fubini measure on $\cal{C}$, that is, $\mu: \cal{C}\to A$, and for all $X,Y\in \cal{C}$:\begin{enumerate}
\item[(d)] $\mu(X)=0$ whenever $X$ is empty; $\mu(X)=1$ whenever $X$ is a singleton;
\item[(e)] $\mu(X\cup Y)=\mu(X)+\mu(Y)$ whenever $X,Y\subseteq M^m$ are disjoint;
\item[(f)] every $\cal{C}$-morphism $f: X \to Y$ satisfies ``Fubini": \begin{enumerate}
\item[(F1)] the subset $\{\mu\big(f^{-1}(y)\big):\ y\in Y\}$ of $A$ is finite;
\item[(F2)] for all $a\in A$, the set
$Y_a := \{y\in Y:\ \mu\big(f^{-1}(y)\big)=a\}$ is definable;
\item[(F3)] if $\mu\big(f^{-1}(y)\big)$ takes the constant value $a$ on $Y$, then
$\mu(X)= a\mu(Y)$.
\end{enumerate}
\end{enumerate}
Thus for $X,Y\in \cal{C}$: $\mu(X\times Y)=\mu(X)\cdot \mu(Y)$; if $X$ and $Y$ are $\cal{C}$-isomorphic, then $\mu(X)=\mu(Y)$ (a ``motivic'' property); for $X,Y\subseteq M^m$ we can represent $X\cup Y$ as the disjoint union of $X\setminus Y, X\cap Y$, and $Y\setminus X$ to obtain
$$\mu(X\cup Y)+\mu(X\cap Y)\ =\ \mu(X)+\mu(Y).$$
If $f: X\to Y$ is a $\cal{C}$-morphism, then $\mu(X)=\sum_a a\mu(Y_a)$, using notation of (F2).

\medskip\noindent
{\bf Example.} Let $\mathbf{M}= {\bbold T} $ as in the beginning of the introduction. Then the subsets of $ {\bbold R} ^n$ definable in $\mathbf{M}$ are semialgebraic in the sense of $ {\bbold R} $, so the full subcategory $\cal{C}$ of $\operatorname{Def}(\mathbf{M})$
whose objects are these semialgebraic sets satisfies conditions (a), (b), (c). The above Euler characteristic $Y\mapsto E(Y)$ is then a Fubini measure on $\cal{C}$ with values in the ring $ {\bbold Z} $ of integers. Also $Y\mapsto \dim(Y)$ is a Fubini measure on $\cal{C}$, taking values in the tropical semiring $\N_{\operatorname{trop}}$, which has underlying set
$ {\bbold N} \cup\{-\infty\}$ and addition and multiplication given respectively by
$(p,q)\mapsto \max(p,q)$ and $(p,q)\mapsto p+q$; this semiring has $-\infty$ as its zero element for addition and $0$ as its multiplicative identity.

\bigskip\noindent
To state the main result of this paper, we consider a $0$-definable set $C\subseteq M$ with more than one element, and let $\operatorname{Def}(C)$ be the full subcategory of $\operatorname{Def}(\mathbf{M})$ whose objects are the definable subsets of the powers $C^n$. Below we define the notion, for definable $X\subseteq M^m$, of {\em $X$ being fiberable over $C$}. These $X$ are the objects of a full subcategory
$\operatorname{Def}(C)^{\operatorname{f}}\supseteq \operatorname{Def}(C)$ of $\operatorname{Def}(\mathbf{M})$. The categories $\operatorname{Def}(C)$ and $\operatorname{Def}(C)^{\operatorname{f}}$ satisfy conditions (a), (b), (c) imposed earlier on $\cal{C}$.

\bigskip\noindent
To state the main result of this paper, we consider a $0$-definable set $C\subseteq M$ with more than one element, and let $\Def(C)$ be the full subcategory of $\Def(\cM)$ whose objects are the definable subsets of the powers $C^n$. Below we define the notion, for definable $X\subseteq M^m$, of {\em $X$ being fiberable over $C$}. These $X$ are the objects of a full subcategory
$\Def(C)^{\f}\supseteq \Def(C)$ of $\Def(\cM)$. The categories $\Def(C)$ and $\Def(C)^{\f}$ satisfy conditions (a), (b), (c) imposed earlier on $\cC$.

\noindent
Suppose $\cM$ is $\omega$-saturated, and let $X\subseteq M^m$ be definable. By recursion on $r\in \N$ we define ``$X$ is $r$-step fiberable over $C$": this means the existence of a definable $f: X \to M^n$ such that
if $r=0$, then $f$ is injective and $f(X)\subseteq C^n$, and if $r\ge 1$, then $f(X)$ and every fiber $f^{-1}(y)$, $y\in M^n$,
is $(r-1)$-step fiberable over $C$.

\begin{lemma}\label{smfub} If $\cM$ is strongly minimal $($which includes $M$ being infinite$)$, then $X\mapsto \operatorname{MR}(X)$ is an $\N_{\trop}$-valued Fubini measure on $\Def(\cM)$.
\end{lemma}

\subsection*{The case of $\T$} We construe here $\T$ as an ordered differential field. Its subfield $\R$ is its constant field, and
a set $Y\subseteq \R^n$ is definable in $\T$ iff $X$ is semialgebraic in the sense of $\R$. In particular, $\R$ is stably embedded in
$\T$. Thus by Theorem~\ref{thm}, the $\N_{\trop}$-valued Fubini measure $Y\mapsto \dim Y$ on $\Def(\R)$ extends uniquely to an $\N_{\trop}$-valued Fubini measure on the full subcategory $\Def(\R)^{\f}$ of $\Def(\T)$. Likewise for the $\Z$-valued Fubini measure $Y\mapsto E(Y)$ on $\Def(\R)$.

\begin{prop}\label{uniq} If $\operatorname{char}(\k)>0$, then there is no $\Z$-valued Fubini measure on $\Def(\k)$. If
$\operatorname{char}(\k)=0$, then there is a unique $\Z$-valued Fubini measure on $\Def(\k)$.
\end{prop}
\begin{proof} Suppose $\mu$ is a $\Z$-valued Fubini measure on $\Def(\k)$. Then $\mu(\k)=1$:
 if $\operatorname{char}(\k)\ne 2$, the fibers of
 $y\mapsto y^2: \k\to \k$
have size $2$ except at $0$, so
$$\mu(\k)\ =\ 1+2\big(\mu(\k)-1\big),$$
 and thus $\mu(\k)=1$; if $\operatorname{char}(\k)=2$, the map $y\mapsto y^3: \k\to \k$ gives likewise $\mu(\k)=1+3\big(\mu(\k)-1\big)$, again resulting in $\mu(\k)=1$.

\noindent
Let now $\k$ be a proper subfield of the algebraically closed field $K$, and $\cM:=(K, \k)$. Then
$\cM$ is $\omega$-stable with $\text{MR}(\k)=1$, $\text{MR}(K)=\omega$, and a set $Y\subseteq \k^n$ is definable in $\cM$ iff $Y$ is constructible in the sense of the algebraically closed field $\k$. Thus $\k$ is stably embedded in $\cM$.
Let $X\subseteq K^m$ be definable. Then
$$X\text{ is almost internal to }\k\
\Leftrightarrow\ X \text{ is co-analyzable relative to }\k\
 \Leftrightarrow\ \operatorname{MR}(X)<\omega.$$
These equivalences are in \cite{AD}, which also has an example of a definable $X\subseteq K$ that is almost internal to $\k$ but not internal to $\k$. Another proof of the first equivalence using \cite{P} was pointed out to me by Pillay. Lemmas~\ref{fico} and ~\ref{cofi} also give:
$$X \text{ is fiberable over }\k\ \Longleftrightarrow\ X \text{ is co-analyzable relative to }\k.$$
 By Lemma~\ref{smfub} the function $Y\mapsto \text{MR}(Y)$
on $\Def(\k)$ is a Fubini measure with values in $\N_{\trop}$. By Theorem~\ref{thm} it extends uniquely to
an $\N_{\trop}$-valued Fubini measures on $\Def(\k)^{\f}$; this extension is necessarily
$X\mapsto \text{MR}(X)$: the above relation to almost internality implies that it is a Fubini measure on $\Def(\k)^{\f}$.

\medskip\noindent
Now assume $\operatorname{char}(\k)=0$. Then we have the unique $\Z$-valued Fubini measure $Y\mapsto E(Y)$ on $\Def(\k)$ from the proof of Proposition~\ref{uniq}. By Theorem~\ref{thm} it extends uniquely to a $\Z$-valued Fubini measure on
$\Def(\k)^{\f}$.
Although "fiberable over $\k$" collapses here to ``almost internal to $\k$", we still need all of Section~\ref{prelim} and much of Section~\ref{proof} to obtain this extension result.

\medskip
{\em If $\cC$ is stably embedded in $\cM$ and $\mu$ is an $A$-valued Fubini measure on $\cC$,
then $\mu$ extends uniquely to an $A$-valued Fubini measure on $\cC^{\f}$}.

\label{prelim}

\noindent
It will be convenient to consider measures that do not necessarily satisfy (F3). We call  them {\em semi-Fubini measures\/} and they are defined in the first subsection below

\begin{lemma}  If $A, B$ are commutative semirings and $\mu: \cC\to A$ and $\nu: \cC\to B$ are Fubini measures, then $(\mu,\nu): \cC\to A\times B$ is a Fubini measure.
\end{lemma}

\section{Proof of the Theorem}\label{proof}

\noindent
{\it In this section $\mu$ is an $A$-valued Fubini measure on $\Def(C)$}. Till further notice we also assume $\cM$ is $\omega$-saturated, indicating at the end of the section how to reduce to that case.

\medskip\noindent
By recursion on $r$ we now introduce a sequence $(\cC_r)$ of full subcategories of $\Def(\cM)$, such that $ \cC_r\subseteq \cC_{r+1}$ for all $r$ and $\bigcup_r \cC_r=\Def(C)^{\f}$:

\medskip\noindent
 $\cC_0:= \Def(C)^{\iso}$, and the objects of $\cC_{r+1}$ are the definable $X\subseteq M^m$ for which there exists a definable
$f: X\to Y$ with $Y\in \cC_r$, such that $f^{-1}(y)\in  \cC_r$ for all $y\in Y$; such $f$ is called a $\cC_{r+1}$-witness. Thus the objects of $\cC_r$ are the definable $X\subseteq M^m$ that are $r$-step fiberable over $C$ as defined in the Introduction.
In particular, the objects of $\cC_0$ are the definable $X\subseteq M^m$ that admit a definable injective map $X\to C^n$. Also, if the definable set $X\subseteq M^m$ admits a definable map $X\to C^n$ all whose fibers are finite, then $X\in \cC_1$.

\medskip\noindent
 Induction on $r$ shows that $\cC_r$ satisfies conditions (a), (b), (c) for all $r$, and that $\cC_r^{\iso}=\cC_r$ for all $r$.  In the inductive step we note that if $f: X\to Y$ is a $\cC_{r+1}$-witness and $X'\subseteq X$ and $Y'\subseteq Y$ are definable with
$f(X')\subseteq Y'$, then $f|_{X'}: X'\to Y'$ is also a $\cC_{r+1}$-witness. Induction on $r$ also shows that there is at most one
$A$-valued Fubini measure on $\cC_r$ extending the given $A$-valued Fubini measure $\mu$ on $\Def(C)$. In the beginning of the previous section we have extended $\mu$ to an $A$-valued Fubini measure on $\cC_0$, also to be denoted by $\mu$.

\subsection*{More about $\cC_0$} {\it In this subsection we assume that $C$ is stably embedded in $\cM$}.
Let  $\cal{S}\subseteq M^k\times C^n$ be definable. Think of $\cal{S}$ as the definable family of sets
$\cal{S}(p)\subseteq C^n$ parametrized by the points $p\in M^k$.

\begin{lemma}\label{l1}  The set $\{\mu\big(\cal{S}(p)\big):\ p\in M^k\}$ is finite, and for every $a$ the set $\{p\in M^k:\ \mu\big(\cal{S}(p)\big)=a\}$ is definable.
\end{lemma}

\begin{theorem}\label{thm} If $C$ is stably embedded in $\cM$ and $\mu$ an $A$-valued Fubini measure on $\Def(C)$,
then $\mu$ extends uniquely to an $A$-valued Fubini measure on $\Def(C)^{\f}$.
\end{theorem}

\noindent
Suppose $\mathbf{M}$ is $\omega$-saturated, and let $X\subseteq M^m$ be definable. By recursion on $r\in  {\bbold N} $ we define ``$X$ is $r$-step fiberable over $C$":  this means the existence of a definable $f: X \to M^n$ such that
if $r=0$, then $f$ is injective and $f(X)\subseteq C^n$, and if $r\ge 1$, then $f(X)$ and every fiber $f^{-1}(y)$, $y\in M^n$,
is $(r-1)$-step fiberable over $C$.

Now drop the assumption that $\mathbf{M}$ is $\omega$-saturated, and let $X\subseteq M^m$ be definable.
Then the definition of ``$X$ is $r$-step fiberable
over $C$'' (with $\mathbf{M}$ as ambient structure) is more complicated, see end of Section~\ref{proof}, but will be such that it is equivalent to ``$X^*$ is $r$-step fiberable  over $C^*$" with $\mathbf{M}^*$ as ambient structure, where $\mathbf{M}^*=(M^*,\dots)$
is any $\omega$-saturated elementary extension of $\mathbf{M}$. Define $X$ to be {\em fiberable over $C$\/} if  it is $r$-step fiberable over $C$ for some $r$.

Using this equivalence the proof of Theorem~\ref{thm} reduces to the case where $\mathbf{M}$ is $\omega$-saturated, which is very convenient. The idea is to show by induction on $r$ that $\mu$ extends to an $A$-valued Fubini measure on the full subcategory of $\operatorname{Def}(\mathbf{M})$ whose objects are the definable $X\subseteq M^m$ that are $r$-step fiberable over $C$. In the step from $r$ to $r+1$ we consider an $(r+1)$-step fiberable $X\subseteq M^m$ over $C$. This is witnessed by a definable $f: X\to M^n$ such that $f(X)$ and all $f^{-1}(y)$ are $r$-step fiberable over $C$. A carefully chosen inductive assumption will tell us that $\mu\big(f(X)\big)$ is defined and that $\mu\big(f^{-1}(y)\big)$ is defined and takes only finitely many values for $y\in f(X)$. Then we define
$\mu_f(X)$ as a suitably weighted sum of these values and show that $\mu_f(X)$ does not depend on $f$, but only on $X$, and that the resulting $\mu(X)=\mu_f(X)$ defined for such $X$ preserves the Fubini property. We also have to show that the inductive assumption is preserved in this step from $r$ to $r+1$. All this is carried out in detail in Section~\ref{proof}.

Section~\ref{prelim} contains preliminary results. In section~\ref{co} we relate fiberability over $C$ to the more intrinsic notion of {\em co-analyzability relative to $C$\/}  from \cite{HHM}, and show their equivalence under various model theoretic conditions. These conditions are satisfied by $ {\bbold T} $ and by differentially closed fields, with $C$ their constant field. This is discussed in Section~\ref{special}, where it is used to answer questions left open in \cite{ADH}. In the short last section~\ref{ms} we state for the record a many-sorted version of Theorem~\ref{thm}.

\medskip\noindent
The  measures from  \cite{MS, AMSW} also take values in semirings and satisfy {\em Fubini}, but seem to be of a rather different nature. Nevertheless, some variant of Theorem~\ref{thm} might hold in that setting. One could also take a look at motivic measures as in ~\cite{HK}, and consider their ``Fubini" nature, or lack of it.

\medskip\noindent
I thank Ben Castle, Artem Chernikov, Rahim Moosa, Anand Pillay, and Charles Steinhorn for useful feedback after a talk I gave on an earlier version of this paper.

\subsection*{Notations and Terminology} We let $d,e, k, l, m,n,r$, sometimes subscripted, range over $ {\bbold N} =\{0,1,2,\dots\}$.  For sets $P, Q$, and $\mathcal{S}\subseteq P\times Q$ and $p\in P$ we set $$\mathcal{S}(p)\ :=\ \{q\in Q:\ (p,q)\in \mathcal{S}\}. $$  (Think of $\mathcal{S}$ as describing the family $\big(\mathcal{S}(p)\big)_{p\in P}$ of subsets of $Q$.) Throughout, $\mathbf{M}=(M;\dots)$ is a one-sorted $\mathcal{L}$-structure. When $C$ gets mentioned, we assume $C\subseteq M$ is $0$-definable and $|C|\ge 2$. Recall, for example from \cite{HHM}, that $C$ is said to be {\em stably embedded in $\mathbf{M}$\/} if for every $0$-definable $\mathcal{S}\subseteq M^{m}\times M^n$ there is a $0$-definable $\mathcal{T}\subseteq M^{\mu}\times M^n$, $\mu\in  {\bbold N} $, such that for all $p\in M^m$ there exists $c\in C^{\mu}$ for which $\mathcal{S}(p)\cap C^n = \mathcal{T}(c)\cap C^n$.
  For $\omega$-saturated $\mathbf{M}$, this is equivalent to all definable $X\subseteq C^n$ being $C$-definable.

Suppose $X\subseteq M^m$ and $Y\subseteq M^n$ are definable. Then a {\em definable family of maps from subsets of $X$ into $Y$\/} is a definable set
$\mathcal{F}\subseteq P\times X \times Y$, with definable $P\subseteq M^k$ such that for each ``parameter" $p\in P$ the set
$\mathcal{F}(p)\subseteq X\times Y$ is the graph of a map $f_p$ with domain a subset of $X$ and codomain $Y$.  We think of such $\mathcal{F}$ as the family $(f_p)_{p\in P}$. Abusing language we write $f\in \mathcal{F}$ to indicate that $f=f_p$ for some $p\in P$, and to state that the set of $f\in \mathcal{F}$ with a certain property is definable means that the set of $p\in P$ such that $f_p$ has that property is definable. Given $\mathcal{S}\subseteq P\times Q$, $p\in P$, and a map $f: \mathcal{S} \to Z$, we let $f(p,-)$ denote the map $q\mapsto f(p,q): \mathcal{S}(p)\to Z$.

Our definition of ``$r$-step fiberable over $C$'' differs from that of ``fiberable by $C$ in $r$ steps''
in \cite{ADH} for a Liouville closed $H$-field $K$ and its constant field $C$. But, in that case,  ``fiberable over $C$'' as defined here is  equivalent to ``fiberable by $C$ in $r$ steps for some $r$" as defined in \cite{ADH}.

Using the equivalence (for an $omega$-saturated elementary extension $\mathbf{M}^*$) between “$X$ is $r$-step fiberable over $C$” and “$X^*$ is $r$-step fiberable over $C^*$”, the proof of Theorem~\ref{thm} “reduces to the case where $\mathbf{M}$ is $\omega$-saturated”. One then proves the extension by “induction on $r$”, showing that $\mu$ extends to an $A$-valued Fubini measure on the full subcategory of $\operatorname{Def}(\mathbf{M})$ consisting of definable sets that are $r$-step fiberable over $C$.

For the step $r\to r+1$, take an $(r+1)$-step fiberable $X\subseteq M^m$, witnessed by a definable $f:X\to M^n$ with $f(X)$ and all fibers $f^{-1}(y)$ ($y\in M^n$) $r$-step fiberable over $C$. A “carefully chosen inductive assumption” gives that $\mu\big(f(X)\big)$ is defined and that $\mu\big(f^{-1}(y)\big)$ is defined and “takes only finitely many values for $y\in f(X)$”. Define $\mu_f(X)$ as a “suitably weighted sum of these values”, and show $\mu_f(X)$ “does not depend on $f$, but only on $X$”, so one may set $\mu(X)=\mu_f(X)$. Finally, verify that this $\mu$ “preserves the Fubini property” and that the inductive assumption is preserved from $r$ to $r+1$. (Details are said to be in Section~\ref{proof}.)

Using the equivalence (for an $\omega$-saturated elementary extension $\mathbf{M}^*$) between “$X$ is $r$-step fiberable over $C$” and “$X^*$ is $r$-step fiberable over $C^*$”, the proof of the main theorem reduces to the case where $\mathbf{M}$ is $\omega$-saturated. One then proves the extension by induction on $r$, showing that $\mu$ extends to an $A$-valued Fubini measure on the full subcategory of $\operatorname{Def}(\mathbf{M})$ consisting of definable sets that are $r$-step fiberable over $C$.

More precisely, one introduces by recursion on $r$ a sequence $(\cC_r)$ of full subcategories of $\Def(\cM)$, such that $ \cC_r\subseteq \cC_{r+1}$ for all $r$ and $\bigcup_r \cC_r=\Def(C)^{\f}$:

$\cC_0:= \Def(C)^{\iso}$, and the objects of $\cC_{r+1}$ are the definable $X\subseteq M^m$ for which there exists a definable
$f: X\to Y$ with $Y\in \cC_r$, such that $f^{-1}(y)\in  \cC_r$ for all $y\in Y$; such $f$ is called a $\cC_{r+1}$-witness. Thus the objects of $\cC_r$ are the definable $X\subseteq M^m$ that are $r$-step fiberable over $C$ as defined in the Introduction.
In particular, the objects of $\cC_0$ are the definable $X\subseteq M^m$ that admit a definable injective map $X\to C^n$. Also, if the definable set $X\subseteq M^m$ admits a definable map $X\to C^n$ all whose fibers are finite, then $X\in \cC_1$.

For the step $r\to r+1$, take an $(r+1)$-step fiberable $X\subseteq M^m$, witnessed by a definable $f:X\to Y$ with $Y\in \cC_r$ and all fibers $f^{-1}(y)$ ($y\in Y$) lying in $\cC_r$. The inductive setup yields that $\mu$ is already defined on $Y$ and on each fiber $f^{-1}(y)$. To ensure the finiteness/definability needed to form a Fubini-type sum, one uses the following lemma.

\begin{lemma}\label{l1}  The set $\{\mu\big(\cal{S}(p)\big):\ p\in M^k\}$ is finite, and for every $a$ the set $\{p\in M^k:\ \mu\big(\cal{S}(p)\big)=a\}$ is definable.
\end{lemma}

Using this, one obtains that $\mu\big(f^{-1}(y)\big)$ takes only finitely many values as $y$ ranges over $Y=f(X)$. One then defines $\mu_f(X)$ as the corresponding suitably weighted sum of these values and shows that $\mu_f(X)$ does not depend on the choice of witness $f$ but only on $X$, allowing one to set $\mu(X)=\mu_f(X)$. Finally, one checks that this definition preserves the Fubini property and that the inductive hypotheses propagate from $r$ to $r+1$.

The details are carried out later.

\begin{lemma}  If $A, B$ are commutative semirings and $\mu: \cC\to A$ and $\nu: \cC\to B$ are Fubini measures, then $(\mu,\nu): \cC\to A\times B$ is a Fubini measure.
\end{lemma}

\section{Proof of the Theorem}\label{proof}

\noindent
{\it In this section $\mu$ is an $A$-valued Fubini measure on $\Def(C)$}. Till further notice we also assume $\cM$ is $\omega$-saturated, indicating at the end of the section how to reduce to that case.

\medskip\noindent
By recursion on $r$ we now introduce a sequence $(\cC_r)$ of full subcategories of $\Def(\cM)$, such that $ \cC_r\subseteq \cC_{r+1}$ for all $r$ and $\bigcup_r \cC_r=\Def(C)^{\f}$:

\medskip\noindent
 $\cC_0:= \Def(C)^{\iso}$, and the objects of $\cC_{r+1}$ are the definable $X\subseteq M^m$ for which there exists a definable
$f: X\to Y$ with $Y\in \cC_r$, such that $f^{-1}(y)\in  \cC_r$ for all $y\in Y$; such $f$ is called a $\cC_{r+1}$-witness. Thus the objects of $\cC_r$ are the definable $X\subseteq M^m$ that are $r$-step fiberable over $C$ as defined in the Introduction.
In particular, the objects of $\cC_0$ are the definable $X\subseteq M^m$ that admit a definable injective map $X\to C^n$. Also, if the definable set $X\subseteq M^m$ admits a definable map $X\to C^n$ all whose fibers are finite, then $X\in \cC_1$.

\medskip\noindent
 Induction on $r$ shows that $\cC_r$ satisfies conditions (a), (b), (c) for all $r$, and that $\cC_r^{\iso}=\cC_r$ for all $r$.  In the inductive step we note that if $f: X\to Y$ is a $\cC_{r+1}$-witness and $X'\subseteq X$ and $Y'\subseteq Y$ are definable with
$f(X')\subseteq Y'$, then $f|_{X'}: X'\to Y'$ is also a $\cC_{r+1}$-witness. Induction on $r$ also shows that there is at most one
$A$-valued Fubini measure on $\cC_r$ extending the given $A$-valued Fubini measure $\mu$ on $\Def(C)$. In the beginning of the previous section we have extended $\mu$ to an $A$-valued Fubini measure on $\cC_0$, also to be denoted by $\mu$.

\subsection*{More about $\cC_0$} {\it In this subsection we assume that $C$ is stably embedded in $\cM$}.
Let  $\cal{S}\subseteq M^k\times C^n$ be definable. Think of $\cal{S}$ as the definable family of sets
$\cal{S}(p)\subseteq C^n$ parametrized by the points $p\in M^k$.

\begin{lemma}\label{l1}  The set $\{\mu\big(\cal{S}(p)\big):\ p\in M^k\}$ is finite, and for every $a$ the set $\{p\in M^k:\ \mu\big(\cal{S}(p)\big)=a\}$ is definable.
\end{lemma}

Question: Let \(\mathbf M\) be a one-sorted \(\mathcal L\)-structure, let \(C\subseteq M\) be a \(0\)-definable set with more than one element, and let \(A\) be a commutative semiring. Write \(\operatorname{Def}(C)\) for the full subcategory of definable sets whose objects are the definable subsets of powers \(C^n\). An \(A\)-valued Fubini measure on a full subcategory assigns to each object an element of \(A\), sends the empty set to \(0\) and singletons to \(1\), is additive on disjoint definable unions, and satisfies the stated Fubini fiber conditions for definable maps. Let \(\operatorname{Def}(C)^{\operatorname f}\) be the full subcategory of definable sets fiberable over \(C\), equivalently the union of the classes \(\mathcal C_r\) defined by \(\mathcal C_0=\operatorname{Def}(C)^{\operatorname{iso}}\) and \(X\in \mathcal C_{r+1}\) iff there is a definable map \(f:X\to Y\) with \(Y\in \mathcal C_r\) and every fiber \(f^{-1}(y)\in \mathcal C_r\). Assume that \(C\) is stably embedded in \(\mathbf M\), and let \(\mu\) be an \(A\)-valued Fubini measure on \(\operatorname{Def}(C)\). Which statement holds about extending \(\mu\)?

There exists exactly one \(A\)-valued Fubini measure \(\widetilde\mu\) on \(\operatorname{Def}(C)^{\operatorname f}\) such that \(\widetilde\mu(X)=\mu(X)\) for every object \(X\in \operatorname{Def}(C)\); equivalently, \(\mu\) extends uniquely to an \(A\)-valued Fubini measure on \(\operatorname{Def}(C)^{\operatorname f}\).

Choices:
B. There exists exactly one \(A\)-valued semi-Fubini measure \(\widetilde\mu\) on \(\operatorname{Def}(C)^{\operatorname f}\) such that \(\widetilde\mu(X)=\mu(X)\) for every object \(X\in \operatorname{Def}(C)\); in general one cannot require \(\widetilde\mu\) to satisfy the full Fubini axiom on all of \(\operatorname{Def}(C)^{\operatorname f}\).
C. There exists an \(A\)-valued Fubini measure \(\widetilde\mu\) on \(\operatorname{Def}(C)^{\operatorname f}\) such that \(\widetilde\mu(X)=\mu(X)\) for every object \(X\in \operatorname{Def}(C)\).
D. For every \(r\in \mathbb N\), there exists exactly one \(A\)-valued Fubini measure \(\widetilde\mu_r\) on \(\mathcal C_r\) extending \(\mu\), but in general these extensions need not glue to a single \(A\)-valued Fubini measure on the whole union \(\operatorname{Def}(C)^{\operatorname f}=\bigcup_r \mathcal C_r\).
E. There exists exactly one \(A\)-valued Fubini measure \(\widetilde\mu\) on the full subcategory of all definable sets \(X\subseteq M^m\) for which there is a definable map \(X\to C^n\) with all fibers finite, such that \(\widetilde\mu(X)=\mu(X)\) for every object \(X\in \operatorname{Def}(C)\); equivalently, \(\mu\) extends uniquely to \(\mathcal C_1\), and this already exhausts \(\operatorname{Def}(C)^{\operatorname f}\).

promotion from semi-Fubini to full Fubini in the inductive step

witness-independence and compatibility across the chain \(\mathcal C_r\)

all fiberable sets are already in \(\mathcal C_1\)

The stem does not explicitly reveal the correct option or uniquely encode choice A. It gives the hypotheses in full, but the conclusion still has to be selected from several closely related possibilities.

This is essentially a direct theorem-recall item: the stem presents the assumptions of an extension theorem and asks for the valid conclusion, with one option matching the theorem statement almost verbatim.

Some reasoning is needed to distinguish exact existence-and-uniqueness from weaker, stronger, or hypothesis-dropping variants, but the item mainly tests recognition/recall of the theorem rather than generating a conclusion from first principles.

The distractors are mathematically plausible and target realistic failure modes: weakening existence to uniqueness only, dropping stable embeddedness, confusing semi-Fubini with full Fubini, and asserting non-uniqueness from witness-based construction.

A mathematically careful MCQ with strong distractors, but it is largely a theorem restatement rather than a genuinely generative reasoning question.


Question: Let \(\mathbf M\) be a one-sorted \(\mathcal L\)-structure, let \(C\subseteq M\) be a \(0\)-definable set with more than one element, and let \(A\) be a commutative semiring. Write \(\operatorname{Def}(C)\) for the full subcategory of definable sets whose objects are the definable subsets of powers \(C^n\). An \(A\)-valued Fubini measure on a full subcategory assigns to each object an element of \(A\), sends the empty set to \(0\) and singletons to \(1\), is additive on disjoint definable unions, and satisfies the stated Fubini fiber conditions for definable maps. Let \(\operatorname{Def}(C)^{\operatorname f}\) be the full subcategory of definable sets fiberable over \(C\), equivalently the union of the classes \(\mathcal C_r\) defined by \(\mathcal C_0=\operatorname{Def}(C)^{\operatorname{iso}}\) and \(X\in \mathcal C_{r+1}\) iff there is a definable map \(f:X\to Y\) with \(Y\in \mathcal C_r\) and every fiber \(f^{-1}(y)\in \mathcal C_r\). Assume that \(C\) is stably embedded in \(\mathbf M\), and let \(\mu\) be an \(A\)-valued Fubini measure on \(\operatorname{Def}(C)\). Which statement holds about extending \(\mu\)?

Choices:
B. There exists exactly one \(A\)-valued semi-Fubini measure \(\widetilde\mu\) on \(\operatorname{Def}(C)^{\operatorname f}\) such that \(\widetilde\mu(X)=\mu(X)\) for every object \(X\in \operatorname{Def}(C)\); in general one cannot require \(\widetilde\mu\) to satisfy the full Fubini axiom on all of \(\operatorname{Def}(C)^{\operatorname f}\).
C. There exists an \(A\)-valued Fubini measure \(\widetilde\mu\) on \(\operatorname{Def}(C)^{\operatorname f}\) such that \(\widetilde\mu(X)=\mu(X)\) for every object \(X\in \operatorname{Def}(C)\).
D. For every \(r\in \mathbb N\), there exists exactly one \(A\)-valued Fubini measure \(\widetilde\mu_r\) on \(\mathcal C_r\) extending \(\mu\), but in general these extensions need not glue to a single \(A\)-valued Fubini measure on the whole union \(\operatorname{Def}(C)^{\operatorname f}=\bigcup_r \mathcal C_r\).
E. There exists exactly one \(A\)-valued Fubini measure \(\widetilde\mu\) on the full subcategory of all definable sets \(X\subseteq M^m\) for which there is a definable map \(X\to C^n\) with all fibers finite, such that \(\widetilde\mu(X)=\mu(X)\) for every object \(X\in \operatorname{Def}(C)\); equivalently, \(\mu\) extends uniquely to \(\mathcal C_1\), and this already exhausts \(\operatorname{Def}(C)^{\operatorname f}\).

The stem does not explicitly reveal the correct option and does not contain a giveaway phrase matching choice A. It sets up definitions and hypotheses, but the conclusion still has to be identified.

This is very close to a theorem-recall item: the correct option is essentially the theorem’s conclusion stated verbatim. However, it is not pure tautology because the student must distinguish it from nearby variants involving existence vs. uniqueness and weaker extension claims.

Some reasoning is needed to select the strongest valid extension statement and reject weaker or subtly false alternatives. Still, the task is mainly recognition of the exact theorem rather than substantial generative mathematical reasoning.

The distractors are strong: B confuses full Fubini with semi-Fubini, C is a weaker true statement, D challenges compatibility/gluing across the inductive classes, and E incorrectly truncates the hierarchy at C1. These are plausible and mathematically meaningful failure modes.

A solid theorem-discrimination MCQ with good distractors and little answer leakage, but it remains fairly close to direct theorem recall rather than testing deep generative reasoning.