\begin{thm}\label{thm: main theorem}
    Let $X = G/\Gamma$  and $F = \{g_t : t\in\R\}\subset G$, where
    \begin{itemize}
    \item $G$ is  a     Lie group with discrete center,
    \item $\Gamma$ is a non-uniform
    lattice in $G$,
\item $g_t$ is as in \eqref{gt}, where $a_0 \in \fg\nz$ is $\ad$-diagonalizable over $\Co$;
 \item the action of $F$  on $(X,m_X)$ is mixing.
\end{itemize}
    Let $y \in E(F,\infty)$. Then for any non-empty open $U\subset X$ we have
    \begin{equation}\label{eq: dim X - dim Z + 1}
        \dim \big(U\cap A(F_+,y) \cap E(F_+,\infty)\big) \geq \dim X - \dim Z + 1,
    \end{equation}
    where $Z \subset G$ is the {neutral subgroup of $G$ with respect to $F$} .
\end{thm}

\begin{thm}\label{main-lemma} \cite[Theorem 1.3]{KR1}
Let $X$ and $F$ be as above, and let $y \in E(F,\infty)$; then
\begin{equation}\label{dim1}
     \dim\big(\{ \alpha \in \R : x_\alpha \in  A(F_+,y) \cap E(F_+,\infty)\}\big) = 1.
\end{equation}
\end{thm}

\begin{defn}\label{horosph}[Expanding, contracting and neutral subgroups]\label{defn: expanding, contracting and neutral groups}
    We let $H$, $H^-$ and $Z$ be the connected Lie subgroups of $G$ associated to the algebras $\fh$, $\fh^-$ and $\fn$ respectively. We fix $m_H$, a Haar measure on $H$. We further consider the connected subgroup $\tilde{H}$ associated to the Lie algebra $\fn \oplus \fh^-$ and a left Haar measure $m_{\tilde{H}}$ on $\tilde{H}$, scaled so that $m_G$ is a direct product of $m_H$ and $m_{\tilde{H}}$, see \cite[Ch.~VII, \S 9, Proposition 13]{B} or \cite[Theorem 8.32]{Kn} for justification.
\end{defn}

\label{intro}

 Let $X$ be a   metric  space, and let $F$ be {either of the (semi) groups $\R$ or $\Z$ or $\R_{\ge 0}$ or $\Z_{\ge 0}$ represented as a family of} self-maps of $X$. We will write $F = \{g_t\}$. For $g\in F$ we will denote its action on $X$ by
$(g,x) \mapsto g
x$.
 Now fix a subset $Y\subset  X$ and define the set of points of $X$ {\sl approaching} $Y$ by $F$ as
    \begin{equation*}
        A(F,Y) := \big\{ x \in X : \exists \text{ an unbounded sequence }(g_{t_k})_{k \in \N} \subset F \text{ with }
        {\lim_{k \to \infty} g_{t_k}         x \in Y}
 \big\}.
    \end{equation*}
    Its complement,  the set of points of $X$ {\sl escaping} {$Y$} by $F$ will be denoted by $E(F,{Y})$, namely
    \begin{equation*}
        E(F,{Y}) := \bigcap_{y \in Y}\big\{ x\in X : y
        \notin\overline{(F\smallsetminus F_0) x}
         \text{ for some bounded }F_0\subset F\big\}.
    \end{equation*}
 If $X$ is not bounded, we make the convention that $x\in A(F,\infty)$ when the orbit $F
 x$ is unbounded.

 {Take $y\in Y$ and   consider $A(F,y):= A(F,\{y\})$ and $E(F,y):= E(F,\{y\})$.}  A natural question one could ask  is how large the sets $A(F,y)$   and   $E(F,y)$ can be. If the action admits an ergodic invariant probability measure, or has some hyperbolic behavior, some instances of this question can be  answered. Namely, under the assumption of existence of an ergodic measure $\mu$ of full support the set  $A(F,y)$ has full measure for any $y\in X\cup \{\infty\}$. Additionally,  for many hyperbolic systems one can prove  that the $\mu$-null sets $E(F,y)$  are \textsl{thick},  that is, their intersection with any nonempty open set has full Hausdorff dimension.  For results of this kind see e.g.\ \cite{Da3} for toral endomorphisms, \cite{U, Do} for Anosov flows, and {\cite{Da1, KM, Kl2, AGK}} for partially hyperbolic flows on \hs.
Furthermore, in many cases these sets can be proved to be winning in the sense of Schmidt games (see e.g.  \cite{AGK, BFK,Da1,Da3,Ts1,Ts2,wu1,wu2}).
Consequently in these cases  it follows that both $A(F,y)$   and   $E(F,y)$ are thick.
Moreover, both full-measure and winning conditions are stable with respect to countable intersections; thus
for any choice of countably many semigroups $F_i$ and points $y_i \in X\cup \{\infty\}$  for which the above conclusions can be established, it holds that both $\bigcap_i A(F_i,y_i)$   and   $\bigcap_i E(F_i,y_i)$  are thick.

\smallskip
This paper addresses the following question: what if one considers
a mixed case, that is, investigate the intersection \eq{mixed}{A(F,y) \cap E(F,z),} where $y,z\in X\cup \{\infty\}$? Problems of this type are amenable neither to the full-measure argument, nor to the technique based on Schmidt games. Also it is clear that some conditions on $y,z$ should be imposed; for example if $y=z$ it is clear that the intersection  \eqref{eq:mixed} is empty.

\smallskip
In what follows we restrict our attention to $F\cong \R$, also denoting $F_+ := \{g_t: t\ge 0\}$.
Assume that the $F$-action on $X$ is continuous. Then let us record {a natural}
obstruction to
the set  \eq{mixedplus}{A(F_+,y) \cap E(F_+,z)} being non-empty:

\begin{lem}\label{lem: obstruction}
    If {$y \in X$  and $z \in X \cup \{\infty\}$} are such that $y \in A(F,z)$ (that is, $z$ can be approached by the trajectory of $y$ either in the positive or in the negative time direction), then
    \begin{equation*}
        A(F_+,y) \subset A(F_+,z).
    \end{equation*}
\end{lem}

\begin{proof}
Let $x \in A(F_+,y)$, and let $(s_\ell)_{\ell \in \N}$ be an unbounded sequence in $\R_{\geq 0}$ with
\begin{equation*}
    \lim_{\ell \to \infty} g_{s_\ell}x = y.
\end{equation*}
Let $W \subset X$ be an open neighborhood of $z$. In the case, when $z = \infty$, we take $W$ to be an open set with bounded complement.
By the hypothesis, there exists a time $t \in \R$ for which $g_t y \in W$. By the continuity of the action we have
\begin{equation*}
    \lim_{\ell \to \infty} g_{s_\ell}g_tx = g_t y \in W.
\end{equation*}
Note that the sequence $(s_\ell + t)_{\ell \in \N}$ is unbounded and eventually positive. Since $W$ was arbitrary, we conclude that $x \in A(F_+, z)$.
\end{proof}
\ignore{{Suppose} we have a point $x \in A(F_+,y) \cap E(F_+,z)$. By {the} hypothesis, take an unbounded sequence $(t_k)_{k \in \N} \subset \R$ with
\begin{equation}\label{eq: y_0 to y_1}
    \lim_{k \to \infty} g_{t_k} y= z
\end{equation}
(in the case $z = \infty$ this means that $g_{t_k} y$ leaves every compact subset of $X$).
Since $x$ is an element of $E(F_+,z)$, we can take an open neighborhood $W$ containing $z$ (if $z = \infty$ we can let $W$ be the complement of a compact subset of $X$) and   $t_0>0$ such that
\begin{equation}\label{eq: tx notin W}
    g_tx \notin W \text{  for all }t> t_0.
\end{equation}
Using the fact that $x \in A(F_+,y)$, we take an unbounded sequence $(s_k)_{k \in \N} \subset \R_{\geq 0}$ with
\begin{equation}\label{eq: x to y_0}
    \lim_{k \to \infty} g_{s_k} x = y.
\end{equation}
The convergence in \eqref{eq: y_0 to y_1} guarantees us some $p \in \N$ with
\begin{equation*}
    {g_{t_{p}}     y \in W}.
\end{equation*}
The continuity of the action and the convergence in \eqref{eq: x to y_0} guarantees us that
\begin{equation*}
    \lim_{k \to \infty} g_{t_p + s_k} x = g_{t_p}    y \in W.
\end{equation*}
However, for $k$ large enough, $t_p + s_k$ will be larger than $t_0$, contradicting \eqref{eq: tx notin W}.}

{In view of the above lemma it makes sense to look for conditions guaranteeing that the set  \eqref{eq:mixedplus} is reasonably large
under the assumption that $y \in E(F,z)$.
In the present paper we will  do it in the following set-up: we let $X = G/\Gamma$, where $G$ is  a real Lie group
and $\Gamma \subset G$ is a lattice,
and    consider the $\R$-action on $X$ via the left multiplication by a one-parameter subgroup of $G$:
    \begin{equation}\label{gt}
        \big\{ g_t := \exp(ta_0) \in G: t \in \R \big\},
    \end{equation}
    where ${a_0}$ is an     element of     {$\fg:=\operatorname{Lie}(G)$, the Lie algebra of $G$}.
    We shall fix once and for all an inner product on $\fg$ to induce a right-invariant Riemannian metric $\dist_G$ on $G$.
 Note that  the invariance of the metric implies that it is geodesically complete, and so the Hopf--Rinow theorem implies that it is proper (closed and bounded sets are compact). Then one obtains a metric on $X = G/\Gamma$ by setting
\begin{equation*}
    \dist_X(g_1\Gamma, g_2\Gamma) := \inf\big\{\dist_G(g_1\gamma, g_2): \gamma \in \Gamma \big\}.
\end{equation*}
This is the set-up in which we will attempt to study sets of the form  \eqref{eq:mixedplus}. We will denote by $m_X$  the $G$-invariant Haar probability measure on $X$, and by $m_G$ the Haar measure on $G$ scaled so that the measure of any fundamental domain for $\Gamma$ is equal to $1$. Further,
   {in this paper} we will    assume that $X$ is noncompact and $z = \infty$; that is, we will seek to construct bounded $F^+$-trajectories with prescribed limit points.

  Recently the second- and third-named authors proved the following result. Take \linebreak  $G = \SL_2(\R)$, $\Gamma = \SL_2(\Z)$ and $X = G/\Gamma$. For $t,\alpha\in\R$ consider
\begin{equation*}\label{eq: gtsl2}
        g_t := \left[ {\begin{array}{cc} e^t & 0 \\
    0 & e^{-t}  \end{array}}\right]\in G,\ F = \{g_t: t\in\R\},
    \end{equation*}
    and  $${x_\alpha  := \left[ {\begin{array}{cc} 1 & \alpha \\
    0 & 1  \end{array}}\right]\Gamma\in X.}
    \ignore{the latter being identified with the lattice in $\R^2$ generated by $\left[ {\begin{array}{cc} 1 \\
    0  \end{array}}\right]$ and $ \left[ {\begin{array}{cc}  \alpha \\
    1 \end{array}}\right]$.}$$

{Take $y\in Y$ and consider $A(F,y):= A(F,\{y\})$ and $E(F,y):= E(F,\{y\})$.} A natural question one could ask is how large the sets $A(F,y)$ and $E(F,y)$ can be. If the action admits an ergodic invariant probability measure, or has some hyperbolic behavior, some instances of this question can be answered. Namely, under the assumption of existence of an ergodic measure $\mu$ of full support the set $A(F,y)$ has full measure for any $y\in X\cup \{\infty\}$. Additionally, for many hyperbolic systems one can prove that the $\mu$-null sets $E(F,y)$ are \textsl{thick}, that is, their intersection with any nonempty open set has full Hausdorff dimension. For results of this kind see e.g.\ \cite{Da3} for toral endomorphisms, \cite{U, Do} for Anosov flows, and {\cite{Da1, KM, Kl2, AGK}} for partially hyperbolic flows on \hs.
Furthermore, in many cases these sets can be proved to be winning in the sense of Schmidt games (see e.g. \cite{AGK, BFK,Da1,Da3,Ts1,Ts2,wu1,wu2}).
Consequently in these cases it follows that both $A(F,y)$ and $E(F,y)$ are thick.
Moreover, both full-measure and winning conditions are stable with respect to countable intersections; thus
for any choice of countably many semigroups $F_i$ and points $y_i \in X\cup \{\infty\}$ for which the above conclusions can be established, it holds that both $\bigcap_i A(F_i,y_i)$ and $\bigcap_i E(F_i,y_i)$ are thick.

\begin{lem}\label{lem: obstruction}
 If {$y \in X$ and $z \in X \cup \{\infty\}$} are such that $y \in A(F,z)$ (that is, $z$ can be approached by the trajectory of $y$ either in the positive or in the negative time direction), then
 \begin{equation*}
 A(F_+,y) \subset A(F_+,z).
 \end{equation*}
\end{lem}

\begin{proof}
Let $x \in A(F_+,y)$, and let $(s_\ell)_{\ell \in \N}$ be an unbounded sequence in $\R_{\geq 0}$ with
\begin{equation*}
 \lim_{\ell \to \infty} g_{s_\ell}x = y.
\end{equation*}
Let $W \subset X$ be an open neighborhood of $z$. In the case, when $z = \infty$, we take $W$ to be an open set with bounded complement.
By the hypothesis, there exists a time $t \in \R$ for which $g_t y \in W$. By the continuity of the action we have
\begin{equation*}
 \lim_{\ell \to \infty} g_{s_\ell}g_tx = g_t y \in W.
\end{equation*}
Note that the sequence $(s_\ell + t)_{\ell \in \N}$ is unbounded and eventually positive. Since $W$ was arbitrary, we conclude that $x \in A(F_+, z)$.
\end{proof}
\ignore{{Suppose} we have a point $x \in A(F_+,y) \cap E(F_+,z)$. By {the} hypothesis, take an unbounded sequence $(t_k)_{k \in \N} \subset \R$ with
\begin{equation}\label{eq: y_0 to y_1}
 \lim_{k \to \infty} g_{t_k} y= z
\end{equation}
(in the case $z = \infty$ this means that $g_{t_k} y$ leaves every compact subset of $X$).
Since $x$ is an element of $E(F_+,z)$, we can take an open neighborhood $W$ containing $z$ (if $z = \infty$ we can let $W$ be the complement of a compact subset of $X$) and $t_0>0$ such that
\begin{equation}\label{eq: tx notin W}
 g_tx \notin W \text{ for all }t> t_0.
\end{equation}
Using the fact that $x \in A(F_+,y)$, we take an unbounded sequence $(s_k)_{k \in \N} \subset \R_{\geq 0}$ with
\begin{equation}\label{eq: x to y_0}
 \lim_{k \to \infty} g_{s_k} x = y.
\end{equation}
The convergence in \eqref{eq: y_0 to y_1} guarantees us some $p \in \N$ with
\begin{equation*}
 {g_{t_{p}} y \in W}.
\end{equation*}
The continuity of the action and the convergence in \eqref{eq: x to y_0} guarantees us that
\begin{equation*}
 \lim_{k \to \infty} g_{t_p + s_k} x = g_{t_p} y \in W.
\end{equation*}
However, for $k$ large enough, $t_p + s_k$ will be larger than $t_0$, contradicting \eqref{eq: tx notin W}.}

{In view of the above lemma it makes sense to look for conditions guaranteeing that the set \eqref{eq:mixedplus} is reasonably large
under the assumption that $y \in E(F,z)$.
In the present paper we will do it in the following set-up: we let $X = G/\Gamma$, where $G$ is a real Lie group
and $\Gamma \subset G$ is a lattice,
and consider the $\R$-action on $X$ via the left multiplication by a one-parameter subgroup of $G$:
 \begin{equation}\label{gt}
 \big\{ g_t := \exp(ta_0) \in G: t \in \R \big\},
 \end{equation}
 where ${a_0}$ is an element of {$\fg:=\operatorname{Lie}(G)$, the Lie algebra of $G$}.
 We shall fix once and for all an inner product on $\fg$ to induce a right-invariant Riemannian metric $\dist_G$ on $G$.
 Note that the invariance of the metric implies that it is geodesically complete, and so the Hopf--Rinow theorem implies that it is proper (closed and bounded sets are compact). Then one obtains a metric on $X = G/\Gamma$ by setting
\begin{equation*}
 \dist_X(g_1\Gamma, g_2\Gamma) := \inf\big\{\dist_G(g_1\gamma, g_2): \gamma \in \Gamma \big\}.
\end{equation*}
This is the set-up in which we will attempt to study sets of the form \eqref{eq:mixedplus}. We will denote by $m_X$ the $G$-invariant Haar probability measure on $X$, and by $m_G$ the Haar measure on $G$ scaled so that the measure of any fundamental domain for $\Gamma$ is equal to $1$. Further,
 {in this paper} we will assume that $X$ is noncompact and $z = \infty$; that is, we will seek to construct bounded $F^+$-trajectories with prescribed limit points.

Recently the second- and third-named authors proved the following result. Take \linebreak $G = \SL_2(\R)$, $\Gamma = \SL_2(\Z)$ and $X = G/\Gamma$. For $t,\alpha\in\R$ consider
\begin{equation*}\label{eq: gtsl2}
 g_t := \left[ {\begin{array}{cc} e^t & 0 \\
 0 & e^{-t} \end{array}}\right]\in G,\ F = \{g_t: t\in\R\},
 \end{equation*}
 and $${x_\alpha := \left[ {\begin{array}{cc} 1 & \alpha \\
 0 & 1 \end{array}}\right]\Gamma\in X.}
 \ignore{the latter being identified with the lattice in $\R^2$ generated by $\left[ {\begin{array}{cc} 1 \\
 0 \end{array}}\right]$ and $ \left[ {\begin{array}{cc} \alpha \\
 1 \end{array}}\right]$.}$$

\begin{prop}\label{prop: equidistirbution for small balls}
 Let $V \subset H$ be bounded measurable with $m_H(\partial V) =0$. Let $K \subset X$ be compact and let $0<\sigma<\rho(K)$.
 Let $L \subset X$ be another compact set, and let $\eta >0$. Then there exists $t_0 := t_0(V, K, \sigma, L, \eta) >0$ such that, for all $t > t_0$, base point $x \in L$ and center $y \in K$,
 \begin{equation*}
 m_H\left\lbrace h \in V: g_thx \in B_X(y,\sigma) \right\rbrace > m_H(V) m_G\big(B_G(\sigma)\big) - \eta.
 \end{equation*}
\end{prop}
\begin{proof}
 The proof is a modification of that of \cite[Proposition 2.4]{Kl2}.
 By partitioning $V$ into finitely many smaller subsets with boundary of measure zero and noting the invariance of $m_H$, one can without loss of generality assume that $e_H\in V$, $m_H(V)<1$ and $\textup{diam}(V) < \rho(L)$. The latter condition implies that there is an open ball $U \subset G$ centered at $e_G$ and containing $V$ such that for all $x \in L$ the map
 $\pi_x$ is injective on $U$, and hence
 \begin{equation*}
 m_X(Ux) = m_G(U).
 \end{equation*}
Similarly we have, for any $0 < \alpha < 1$ and $y \in K$,
 \begin{equation*}
 m_X\big(B_X(y,\alpha \sigma)\big) = m_G\big(B_G(\alpha \sigma)\big).
 \end{equation*}
 Fix $\alpha <1$ such that
 \begin{equation}\label{eq: alpha ball close}
 m_G\big(B_G(\alpha \sigma)\big) > m_G\big(B_G(\sigma)\big) - \frac{\eta}{2}.
 \end{equation}
 Now let $\tilde{H}$ and $m_{\tilde{H}}$
 be as in Definition~\ref{horosph}.
Choose a nonempty open ball $\tilde V \subset \tilde{H}$ such that $\tilde VV \subset U$ and such that for all $\tilde v \in \tilde V$ and $t>0$ we have
 \begin{equation*}
 \dist_G\big(e_G,\Phi_t(\tilde v)\big) < (1-\alpha)\sigma. \end{equation*}
 Now for $y \in K$, $x \in L$ and $t>0$ denote
 \begin{equation*}
 W := \left\lbrace h \in V : g_t hx \in B_X(y,\sigma)\right\rbrace\ \text{ and }\ U' := \tilde V V.
 \end{equation*}
 \begin{claim}\label{claim: 99 containment}
 The set $U'x \cap g_t^{-1} B_X(y, \alpha \sigma)$ is contained in $\tilde V W x$.
 \end{claim}
 \begin{proof} Let $\tilde v \in \tilde V$, $h \in V$ be such that $g_t\tilde vhx \in B_X(y, \alpha \sigma)$. We have
 \begin{equation*}
 \begin{split}
 \dist_X(g_t h x, y) &\leq \dist_X(g_t hx, \Phi_t(\tilde v)g_thx) + \dist_X(g_t\tilde vhx, y) \\
 &< (1-\alpha)\sigma + \alpha \sigma = \sigma.
 \end{split}
 \end{equation*}
 Thus $g_thx \in B_X(y,\sigma)$.
 \end{proof}
 We now use Proposition \ref{thm: KM mixing} with \begin{equation*}
 \Xi := \left\lbrace \mathds{1}_{U'x} : x \in L\right\rbrace,\
\Psi := \{\mathds{1}_{B_X(y,\alpha \sigma}) : y \in K\},
 \end{equation*}
 and with $\eta$ replaced by $\eta m_{\tilde{H}}(\tilde V)/2$.
 This gives $t_0 >0$ such that, for all $t>t_0$, $x \in L$ and $y \in K$, we have
 \begin{equation*}
 \left|m_X\big(U'x \cap g_t^{-1}B_X(y,\alpha \sigma)\big) - m_G(U')m_G\big(B_G(\alpha \sigma)\big) \right| < \eta m_{\tilde{H}}(\tilde{V})/2.
 \end{equation*}
 In particular, using the choice of $m_{\tilde{H}}$ and Claim \ref{claim: 99 containment}, we compute
 \begin{equation*}
 \begin{split}
 m_{\tilde{H}}(\tilde{V})m_H(W) &=m_X(\tilde VWx) \\
 &\geq m_X\big(U'x \cap g_t^{-1}B_X(y,\alpha \sigma)\big) \\
 &\geq m_G(U') m_G\big(B_G(\alpha \sigma)\big) - \eta m_{\tilde{H}}(\tilde V)/2 \\
 &= m_{\tilde{H}}(\tilde V)m_H(V) m_G\big(B_G(\alpha \sigma)\big) - \eta m_{\tilde{H}}(\tilde V)/2.
 \end{split}
 \end{equation*}
 Canceling $m_{\tilde{H}}(\tilde V)$, using \eqref{eq: alpha ball close} and the fact that $m_H(V) < 1$, we get
 \begin{equation*}
 \begin{split}
 m_{H}(W) &\geq m_H(V) m_G\big(B_G(\alpha \sigma)\big) - \eta/2 \\
 &\geq m_H(V) m_G\big(B_G( \sigma)\big) - m_H(V)\eta/2 - \eta/2 \\
 &\geq m_H(V)m_G\big(B_G( \sigma)\big) - \eta
 \end{split}
 \end{equation*}
 which gives the desired result.
\end{proof}

\begin{defn}[Strongly tree-like collections and Cantor sets]\label{defn: tree like collection}\label{defn: tree-like collection}
 We say a collection of subsets $\ca{E} \subset 2^H$ is \textsl{strongly tree-like} if:
 \begin{itemize}
 \item Each $E \in \ca{E}$ is a compact set with nonempty interior.
 \item We have a partition
 \begin{equation*}
 \ca{E} = \bigcup_{k \in \Z_{\geq 0}} \ca{E}_k
 \end{equation*}
 with each $\ca{E}_k$ being finite and $\ca{E}_0$ being a singleton.
 \item If $E' \in \ca{E}_k$ with $k \in \N$, there is a unique $E \in \ca{E}_{k-1}$ with $E' \subset E$.
 \item For each $E \in \ca{E}_k$ with $k \in \Z_{\geq 0}$, there exists $E' \in \ca{E}_{k+1}$ with $E' \subset E$.
 \item If $E_1, E_2 \in \ca{E}_k$ are distinct, then $m_H(E_1 \cap E_2)=0$.
 \item If we define
 \begin{equation}\label{eq: diam}
 d_k := \sup\{\on{diam}_H(E) : E \in \ca{E}_k\}, \text{ then } \lim_{k \to \infty} d_k = 0.
 \end{equation}
 \end{itemize}
 We write
 \begin{equation*}
 \cup \ca{E}_k := \bigcup_{E \in \ca{E}_k} E
 \end{equation*}
 and define the {\sl limit set} of the collection to be $E_\infty := \bigcap_{k \in \N} \cup \ca{E}_k$.
\end{defn}
Associated to a tree-like collection $\ca{E}$ we also define, for $E \in \ca{E}_k$,
\begin{equation*}
 \on{density}(\ca{E}_{k+1},E) := \frac{m_H((\cup \ca{E}_{k+1}) \cap E)}{m_H(E)}.
\end{equation*}
Further, for each $k \in \Z_{\geq 0}$, we set
\begin{equation}\label{eq: density}
 \Delta_k := \inf\left\lbrace \operatorname{density}(\mathcal{E}_{{k+1}}, E) : E \in \mathcal{E}_k \right\rbrace
\end{equation}
\begin{thm}[Lemma 2.1 in \cite{U}]\label{thm: urbanski}
 For a tree-like collection $\ca{E}$ and the resulting limit set $E_\infty$, we have the dimension estimate
 \begin{equation*}
 \dim H - \dim E_\infty \leq \limsup_{k \to \infty} \frac{\sum_{j=0}^{k} \log \Delta_j}{\log d_k}.
 \end{equation*}
\end{thm}

Conversely, let us assume that $x_\alpha \in E(F_+, K_i)$. Then, by the compactness of $K_i$ and by Mahler's compactness criterion, there exists an $r$ with $0< r < \sqrt{r_0}$ and a time $s_0 >0$ such that for all $s > s_0$,
 \begin{equation*}
 g_s x_\alpha \notin K_i(r).
 \end{equation*}
 In particular,
 \begin{equation*}
 x_\alpha \cap \left\lbrace (z,w)\in \Co^2: |z| \leq e^{-s}r, |w| \leq e^{s}r \right\rbrace \neq \{(0,0)\}.
 \end{equation*}
 In particular, $\alpha \in \DI_i$ with constant $\frac{r^2}{r_0}$.
\end{proof}
This dynamical perspective on $\DI_i$ allows one to effortlessly establish certain properties of $\DI_i$ using existing techniques in ergodic theory and dynamics: the question of measure and dimension. (The interested reader may consult \cite[Theorem 1.5]{KW2} and \cite[Theorem 2.8]{AGK} where very general dynamical analogs of these questions are considered.)
In particular, we may use the above dynamical results to conclude that $\DI_i$ has measure zero but full Hausdorff dimension.
With respect to this current article, we are primarily concerned with the relation between Dirichlet improvability and bad approximability (cf. \cite[Theorems 1 and 2]{DS}).
Consider the set of badly approximable complex numbers with respect to $\Z[i]$; those $\alpha \in \Co$ for which there is a constant $c >0$ such that for all nonzero $q \in \Z[i]$,
\begin{equation*}
 |q| \min_{p \in \Z[i]} |q\alpha - p| >c.
\end{equation*}
We write this set as $\BA_i$.
\begin{ques}\label{ques: BA DI i}
 What is the relationship between Dirichlet improvability and bad approximabilty in the case of the Gaussian integers? Is it true that $\BA_i$ is a subset of $\DI_i$?
\end{ques}
We are able to provide a partial answer to this question as follows. However, it seems that a complete answer might require a further study of the set $K_i$ in Proposition \ref{prop: DIi dynamical}.

Here and hereafter "dim" stands for \hd.
 In fact, even though it was not mentioned in \cite{KR1}, the proof actually demonstrates the thickness of the set in the left-hand side of \eqref{dim1}. Also,  by the standard "slicing" argument (see \S \ref{sec: reduction to H}), it can be easily derived from Theorem \ref{main-lemma}
that  for any $y \in E(F,\infty)$ the set $A(F_+,y) \cap E(F_+,\infty)$ is thick.
The proof of the above theorem used the technique of continued fractions, and in fact the result can be interpreted within the framework of Diophantine approximation. Namely it constructs numbers that are badly approximable but not $\nu$-Dirichlet improvable, where $\nu$ is an irreducible norm on $\R^2$. See \cite{KR, KR1, KRS, AD} for more detail, and \cite{MS} for some new results in that direction.

\smallskip
One of the main results of this paper is a generalization of Theorem \ref{main-lemma} to a wider class of flows on \hs s.

\begin{thm}\label{thm: main theorem}
    Let $X = G/\Gamma$  and $F = \{g_t : t\in\R\}\subset G$, where
    \begin{itemize}
    \item $G$ is  a     Lie group with discrete center,
    \item $\Gamma$ is a non-uniform
    lattice in $G$,
\item $g_t$ is as in \eqref{gt}, where $a_0 \in \fg\smallsetminus\{0\}$ is $\ad$-diagonalizable over $\Co$;
 \item the action of $F$  on $(X,m_X)$ is mixing.
\end{itemize}
    Let $y \in E(F,\infty)$. Then for any non-empty open $U\subset X$ we have
    \begin{equation}\label{eq: dim X - dim Z + 1}
        \dim \big(U\cap A(F_+,y) \cap E(F_+,\infty)\big) \geq \dim X - \dim Z + 1,
    \end{equation}
    where $Z \subset G$ is the {neutral subgroup of $G$ with respect to $F$} .
\end{thm}
See Definition \ref{defn: expanding, contracting and neutral groups} below for a precise definition of $Z$.
In view of Moore's Theorem \cite{Mo}, the mixing assumption is satisfied when $G$ is a connected semi-simple
    Lie group
    with finite center and with no compact factors, and $\Gamma\subset G$ is irreducible. However mixing is not restricted to the semi-simple case, see e.g.\  \cite{BM, Kl}.
Note that when $G =  \SL_2(\R)$ (that is, under the assumptions of Theorem \ref{main-lemma}) we know that {$\dim Z=\dim F = 1$}, and thus the set $A(F_+,y) \cap E(F_+,\infty)$ is thick. Moreover, in the latter case one can generalize Theorem~\ref{main-lemma}  as follows.

  \begin{thm}\label{thm: dense-orbits}
Let $G$, $\Gamma$ and $F$ be as in Theorem \ref{thm: main theorem}, and assume in addition that \linebreak {$\dim Z =
1$}.
Then for any {compact}
F-invariant
subset $B$ of $E(F,\infty)$ there exists a thick set of $x \in E(F_+,\infty)$ such that the closure of the (bounded) trajectory $F_+x$ contains $B$.
 \end{thm}

In addition to $G =  \SL_2(\R)$, the assumptions of the above theorem hold when \linebreak $G =  \SL_2(\R)\ltimes \R^2$; the choice $\Gamma =  \SL_2(\Z)\ltimes \Z^2$ yields the space of two-dimensional unimodular grids, cf.\ \cite{Kl}.
  \begin{cor}\label{cor: dim-drop}
Let $G$, $\Gamma$, $X$ and $F$ be as in Theorem \ref{thm: dense-orbits}; then for any $\varepsilon > 0$ there exists a thick set of $x\in X$ such that the orbit closure $\overline{F^+x}$ is compact and has Hausdorff dimension greater than $\dim X - \varepsilon$.
 \end{cor}
 \begin{proof}
 When $
X$ and $F$ are as in Theorem \ref{thm: main theorem}, it follows from
 \cite[Theorem 1.1]{KM} that, given $\varepsilon>0$, one can find a large enough compact set $K \subset X$ such that the compact $F$-invariant set
 \begin{equation*}
     B = \{x \in X: Fx \subset K\}
 \end{equation*}
 has Hausdorff dimension greater than $\dim X - \varepsilon$.
Under the additional hypothesis of $\dim Z = 1$, an application of Theorem \ref{thm: dense-orbits} to $B$ then gives the result.
 \end{proof}

 \noindent{\bf Acknowledgements.}
    The second-named author is grateful to ETH (Z\"urich) for its hospitality during a sabbatical stay in 2022, and to Barak Weiss for helpful discussions. The third-named author thanks N.\ Chandgotia, T.\ Mesikepp and Y.\ Peres for holding public-access office hours at TIFR-CAM, Peking University and Tsinghua University respectively.

The post-theorem discussion says that (although not stated in \cite{KR1}) “the proof actually demonstrates the thickness of the set in the left-hand side of \eqref{dim1}.” It also notes that “by the standard ‘slicing’ argument (see \S \ref{sec: reduction to H}), it can be easily derived from Theorem \ref{main-lemma} that for any $y\in E(F,\infty)$ the set $A(F_+,y)\cap E(F_+,\infty)$ is thick.” Finally, it describes the method/interpretation: “The proof of the above theorem used the technique of continued fractions,” and “can be interpreted within the framework of Diophantine approximation,” namely constructing “numbers that are badly approximable but not $\nu$-Dirichlet improvable, where $\nu$ is an irreducible norm on $\R^2$.”

The post-theorem discussion says that (although not stated in

Kenyon, R. and Ralston, J., \emph{[Theorem 1.3]} (KR1)

) “the proof actually demonstrates the thickness of the set in the left-hand side of
\begin{equation}\label{dim1}
     \dim\big(\{ \alpha \in \R : x_\alpha \in  A(F_+,y) \cap E(F_+,\infty)\}\big) = 1.
\end{equation}
.” It also notes that “by the standard ‘slicing’ argument (see \S \ref{sec: reduction to H}), it can be easily derived from the main theorem that for any $y\in E(F,\infty)$ the set $A(F_+,y)\cap E(F_+,\infty)$ is thick.” Finally, it describes the method/interpretation: “The proof of the above theorem used the technique of continued fractions,” and “can be interpreted within the framework of Diophantine approximation,” namely constructing “numbers that are badly approximable but not $\nu$-Dirichlet improvable, where $\nu$ is an irreducible norm on $\R^2$.”

\label{main-lemma} \cite[Theorem 1.3]{KR1}
Let $X$ and $F$ be as above, and let $y \in E(F,\infty)$; then
\begin{equation}\label{dim1}
     \dim\big(\{ \alpha \in \R : x_\alpha \in  A(F_+,y) \cap E(F_+,\infty)\}\big) = 1.
\end{equation},

Question:
Let
\(X=\mathrm{SL}_2(\mathbb R)/\mathrm{SL}_2(\mathbb Z)\), let
\(g_t=\begin{pmatrix}e^t&0\\0&e^{-t}\end{pmatrix}\), and let
\(F=\{g_t:t\in\mathbb R\}\) with \(F_+=\{g_t:t\ge 0\}\). For \(\alpha\in\mathbb R\), write
\(x_\alpha=u_\alpha\,\mathrm{SL}_2(\mathbb Z)\) where
\(u_\alpha=\begin{pmatrix}1&\alpha\\0&1\end{pmatrix}\). For \(y\in X\), define
\(A(F_+,y)=\{x\in X: g_{t_n}x\to y\text{ for some unbounded sequence }t_n\ge 0\}\), and let \(E(F_+,\infty)\) denote the set of points whose positive \(F_+\)-orbit does not approach the point at infinity in the one-point compactification of \(X\) (equivalently, whose positive orbit is bounded in \(X\)). If \(y\in E(F,\infty)\), i.e. the full \(F\)-orbit of \(y\) is bounded, which statement holds about the Hausdorff dimension of
\[
\{\alpha\in\mathbb R: x_\alpha\in A(F_+,y)\cap E(F_+,\infty)\}?
\]

Its Hausdorff dimension is exactly \(1\):
\[
\dim\big(\{\alpha\in\mathbb R: x_\alpha\in A(F_+,y)\cap E(F_+,\infty)\}\big)=1.
\]

Choices:
B.
Its Hausdorff dimension is exactly \(0\); in fact,
\[
\dim\big(\{\alpha\in\mathbb R: x_\alpha\in A(F_+,y)\cap E(F_+,\infty)\}\big)=0.
\]
C. Its Hausdorff dimension is at least \(0\) and at most \(1\); in particular, the set is a subset of \(\mathbb R\) and may be nonempty, but no sharper dimension conclusion is asserted here.
D.
Its Hausdorff dimension is exactly \(1\) provided \(y\in E(F_+,\infty)\); that is,
\[
\dim\big(\{\alpha\in\mathbb R: x_\alpha\in A(F_+,y)\cap E(F_+,\infty)\}\big)=1
\]
for every point whose positive \(F_+\)-orbit is bounded.
E.
The set has full Lebesgue measure in \(\mathbb R\), and hence in particular Hausdorff dimension \(1\):
\[
m\big(\{\alpha\in\mathbb R: x_\alpha\in A(F_+,y)\cap E(F_+,\infty)\}\big)=1.
\]

thickness/full-dimension conclusion replaced by zero-dimension exceptional-set heuristic

dropped the sharp equality \(=1\), keeping only the trivial ambient-dimension bound

hypothesis weakened from full-orbit boundedness \(y\in E(F,\infty)\) to only forward boundedness \(y\in E(F_+,\infty)\)

dimension/thickness conclusion strengthened to full Lebesgue measure


\begin{theorem}\label{thm:main1}
For each $n \ge 10$, the Torelli group of $X_n = 2\mathbb{CP}^2 \# n \overline{\mathbb{CP}^2}$ admits a surjective homomorphism $\Phi : T(X_n) \to \mathbb{Z}^\infty$.
\end{theorem}

Let $X$ be a compact, oriented, smooth, simply-connected $4$-manifold. The {\em mapping class group} $M(X)$ of $X$ is defined as the group of smooth isotopy classes of diffeomorphisms of $X$. The {\em Torelli group} $T(X) \subseteq M(X)$ is the subgroup of smooth isotopy classes of diffeomorphisms which are continuously isotopic to the identity. Non-trivial elements of $T(X)$ represent exotic diffeomorphisms in the sense that they are continuously isotopic to the identity but not smoothly isotopic. Thus, the Torelli group $T(X)$ can be seen as a measure of the difference between continuous and smooth isotopy.

It was first established by Ruberman that the Torelli group of a compact, simply-connected $4$-manifold can be non-trivial \cite{rub1,rub2}. Ruberman constructed versions of the Donaldson and Seiberg--Witten invariants for $1$-parameter families of $4$-manifolds and used these invariants to detect non-triviality of elements of the Torelli group. More specifically, his invariant can be used to show that for every $n \ge 2$ and $m \ge 10n+1$, the Torelli group of $2n \mathbb{CP}^2 \# m \overline{\mathbb{CP}^2}$, surjects to $\mathbb{Z}^\infty$ (where $\mathbb{Z}^\infty$ denotes a free abelian group of countably infinite rank). Baraglia--Konno proved a more general gluing formula for the families Seiberg--Witten invariants \cite{bk1}, which for example can be used to show a similar result for the Torelli groups of $2n \mathbb{CP}^2 \# 10n \overline{\mathbb{CP}^2}$ for each odd $n \ge 3$ and also for for $n(S^2 \times S^2) \# nK3$ for each $n \ge 1$. In all of these cases the parity of $b_+$ must be even due to a dimension constraint and there is a further constraint that $b_+ > 2$ so as to avoid the presence of any chamber structure for the families Seiberg--Witten invariants. Thus the smallest possible value of $b_+$ allowed by these methods is $b_+ = 4$.

The main goal of this paper is to extend the methods of \cite{rub1,rub2,bk1} to the case $b_+ = 2$, using $1$-parameter Seiberg--Witten invariants in the presence of chambers. In a series of papers \cite{tom1,tom2}, the second author has proven a general gluing formula for the families Seiberg--Witten invariants which can be applied even for low values of $b_+$ where chambers are present. The results in this paper are obtained by applying this more general gluing formula to the case of $1$-parameter families. Our first main result is that the Torelli groups of certain simply-connected $4$-manifolds with $b_+ = 2$ can be infinitely generated:

\begin{abstract}
Let $X$ be a compact, oriented, smooth, simply-connected $4$-manifold. The {\em mapping class group} of $X$ is defined as the group of smooth isotopy classes of diffeomorphisms of $X$. The {\em Torelli group} of $X$ is the subgroup of the mapping class group consisting of smooth isotopy classes of diffeomorphisms which are continuously isotopic to the identity. We prove that for each $n \ge 10$, the Torelli group of $2\mathbb{CP}^2 \# n \overline{\mathbb{CP}^2}$ surjects to $\mathbb{Z}^\infty$. We also prove that the mapping class group of $2 \mathbb{CP}^2 \# 10 \overline{\mathbb{CP}^2}$ is not finitely generated. Our proofs of these results makes use of Seiberg--Witten invariants for $1$-parameter familes of $4$-manifolds and in particular a gluing formula for connected sum families. Since the manifolds we consider have $b_+ = 2$, the chamber structure of the $1$-parameter Seiberg--Witten invariants plays an important role.
\end{abstract}

It was first established by Ruberman that the Torelli group of a compact, simply-connected $4$-manifold can be non-trivial \cite{rub1,rub2}. Ruberman constructed versions of the Donaldson and Seiberg--Witten invariants for $1$-parameter families of $4$-manifolds and used these invariants to detect non-triviality of elements of the Torelli group. More specifically, his invariant can be used to show that for every $n \ge 2$ and $m \ge 10n+1$, the Torelli group of $2n \mathbb{CP}^2 \# m \overline{\mathbb{CP}^2}$, surjects to $\mathbb{Z}^\infty$ (where $\mathbb{Z}^\infty$ denotes a free abelian group of countably infinite rank). Baraglia--Konno proved a more general gluing formula for the families Seiberg--Witten invariants \cite{bk1}, which for example can be used to show a similar result for the Torelli groups of $2n \mathbb{CP}^2 \# 10n \overline{\mathbb{CP}^2}$ for each odd $n \ge 3$ and also for for $n(S^2 \times S^2) \# nK3$ for each $n \ge 1$. In all of these cases the parity of $b_+$ must be even due to a dimension constraint and there is a further constraint that $b_+ > 2$ so as to avoid the presence of any chamber structure for the families Seiberg--Witten invariants. Thus the smallest possible value of $b_+$ allowed by these methods is $b_+ = 4$.

The main goal of this paper is to extend the methods of \cite{rub1,rub2,bk1} to the case $b_+ = 2$, using $1$-parameter Seiberg--Witten invariants in the presence of chambers. In a series of papers \cite{tom1,tom2}, the second author has proven a general gluing formula for the families Seiberg--Witten invariants which can be applied even for low values of $b_+$ where chambers are present. The results in this paper are obtained by applying this more general gluing formula to the case of $1$-parameter families. Our first main result is that the Torelli groups of certain simply-connected $4$-manifolds with $b_+ = 2$ can be infinitely generated:

Prior to this result it had been shown by Konno--Mallick--Taniguchi \cite{kmt} that $2\mathbb{CP}^2 \# m \overline{\mathbb{CP}^2}$ has an exotic diffeomorphism for each $m \ge 11$. Qiu proves that $2\mathbb{CP}^2 \# 10 \overline{\mathbb{CP}^2}$ admits an exotic diffeomorphism with infinite order in the Torelli group \cite{qiu}. Qiu's method is similar to the one used in this paper, although the conclusion is not as strong.

\begin{theorem}\label{thm:main2}
The mapping class group of $X = 2\mathbb{CP}^2 \# 10 \overline{\mathbb{CP}^2}$ is not finitely generated. More precisely, there is an index $2$-subgroup $M_+(X) \subseteq M(X)$ and a surjective homomorphism $\Phi : M_+(X) \to \mathbb{Z}^\infty$.
\end{theorem}

\begin{theorem}
For each $n \ge 10$, the Torelli group of $X_n = 2\mathbb{CP}^2 \# n \overline{\mathbb{CP}^2}$ admits a surjective homomorphism $\Phi : T(X_n) \to \mathbb{Z}^\infty$ to a free abelian group of countably infinite rank.
\end{theorem}
\begin{proof}
Let $\mathbb{Z}[\mathcal{S}(X_n)] = \bigoplus_{\mathfrak{s} \in \mathcal{S}(X_n)} \mathbb{Z}$. Elements of $\mathbb{Z}[\mathcal{S}(X_n)]$ can be regarded as functions $\mathcal{S}(X_n) \to \mathbb{Z}$ with finite support. Define a homomorphism $\Phi' : T(X_n) \to \bigoplus_{\mathfrak{s} \in \mathcal{S}(X_n) } \mathbb{Z}$ by taking $(\Phi'(f))(\mathfrak{s}) = SW^c_{X_n , \mathfrak{s} , \mathbb{Z}}(f)$. The function $\mathfrak{s} \mapsto SW^c_{X_n , \mathfrak{s} , \mathbb{Z}}(f)$ has finite support because of the compactness properties of the Seiberg--Witten equations.

The image $im(\Phi')$ of $\Phi'$ is a subgroup of the free abelian group $\mathbb{Z}[\mathcal{S}(X_n)]$ and hence is free abelian. Let $\Phi : T(X_n) \to im(\Phi')$ be the homomorphism obtained by factoring $\Phi'$ through its image. The theorem will follow if we can show that $im(\Phi')$ has infinite rank (since $\mathbb{Z}[\mathcal{S}(X_n)]$ has countably infinite rank, the rank of $im(\Phi')$ can be at most countably infinite). For $n=10$, Lemma \ref{lem:torelli} immediately implies that $im(\Phi')$ has countably infinite rank. For $n > 10$ we use induction on $n$ and the blow-up formula. Write $X_n = X_{n-1} \# \overline{\mathbb{CP}^2}$. Let $f \in T(X_{n-1})$. We can isotopy $f$ so that it is the identity in a neighbourhood of a point. Then we can form the connected sum diffeomorphism $f \# id_{\overline{\mathbb{CP}^2}} \in T(X_n)$. Let $\kappa$ be a spin$^c$-structure on $\overline{\mathbb{CP}^2}$ with $c(\kappa)^2 = -1$. Let $\mathfrak{s} \in \mathcal{S}(X_{n-1})$. Then $\mathfrak{s} \# \kappa \in \mathcal{S}(X_n)$. The blowup formula gives
\[
SW^c_{X_n , \mathfrak{s} \# \kappa , \mathbb{Z}}(f \# id_{\overline{\mathbb{CP}^2}}) = SW^c_{X_{n-1} , \mathfrak{s} , \mathbb{Z}}(f).
\]
Hence there are infinitely many spin$^c$-structures on $X_n$ for which $SW^c_{X_n , \mathfrak{s} , \mathbb{Z}} : T(X_n) \to \mathbb{Z}$ is non-zero and hence $\Phi'( T(X_n) )$ has infinite rank.
\end{proof}

\begin{theorem}
The mapping class group of $X = 2\mathbb{CP}^2 \# 10 \overline{\mathbb{CP}^2}$ is not finitely generated. More precisely there is a surjective homomorphism $\Phi : M_+(X) \to \mathbb{Z}^\infty$.
\end{theorem}
\begin{proof}
Define $\Phi' : M_+(X) \to \mathbb{Z}^\infty$ to be given by
\[
\Phi' = \bigoplus_{n=1}^{\infty} SW_{X , \mathcal{O}_{2n-1}} : M_+(X) \to \bigoplus_{n=1}^{\infty} \mathbb{Z}.
\]
To prove the result, it suffices to show that the image of $\Phi'$ is not finitely generated. Then we let $\Phi$ be the homomorphism obtained by replacing the codomain of $\Phi'$ with the image of $\Phi'$. The argument is almost identical to the proof of \cite[Theorem 3.1]{bar}, except that now we use $E(1)_{2,2n+1}$ in place of $E(n)_q$.
\end{proof}

Prior to this result it had been shown by Konno--Mallick--Taniguchi \cite{kmt} that $2\mathbb{CP}^2 \# m \overline{\mathbb{CP}^2}$ has an exotic diffeomorphism for each $m \ge 11$. Qiu proves that $2\mathbb{CP}^2 \# 10 \overline{\mathbb{CP}^2}$ admits an exotic diffeomorphism with infinite order in the Torelli group \cite{qiu}. Qiu's method is similar to the one used in this paper, although the conclusion is not as strong.

Our second main result concerns infinite generation of the mapping class group $M(X)$ (by infinite generation, we mean to say that $M(X)$ is not finitely generated). Note that infinite generation of $M(X)$ does not follow from infinite generation of $T(X)$, because a finitely generated group can have infinitely generated subgroups.

\begin{theorem}\label{thm:main2}
The mapping class group of $X = 2\mathbb{CP}^2 \# 10 \overline{\mathbb{CP}^2}$ is not finitely generated. More precisely, there is an index $2$-subgroup $M_+(X) \subseteq M(X)$ and a surjective homomorphism $\Phi : M_+(X) \to \mathbb{Z}^\infty$.
\end{theorem}

Note that by Schreier's lemma \cite{ser} every finite index subgroup of a finitely generated group is finitely generated. So the infinite generation of $M_+(X)$ implies the infinite generation of $M(X)$.

The first examples of compact, simply-connected $4$-manifolds whose mapping class groups are not finitely generated were given by Baraglia \cite{bar} and Konno \cite{kon}, namely $M(X)$ is not finitely generated for $X = 2n \mathbb{CP}^2 \# 10n \overline{\mathbb{CP}^2}$ where $n \ge 3$ is odd and also for $X = n (S^2 \times S^2) \# nK3$, $n \ge 1$. Note that these are precisely the manifolds of the form $E(m) \# (S^2 \times S^2)$, $m \ge 2$. Theorem \ref{thm:main2} says the same result is true for the case $m=1$, that is, for $X = E(1) \# (S^2 \times S^2) = 2\mathbb{CP}^2 \# 10 \overline{\mathbb{CP}^2}$.

\subsection{Outline of the proofs of the main results}
To each diffeomorphism $f \in M(X)$, one can form the mapping cylinder $E(f)$. This is the $1$-parameter family of $4$-manifolds obtained from $[0,1] \times X$ by identifying the ends via $f$. If $\mathfrak{s}$ is a spin$^c$-structure which is preserved by $f$ and for which the expected dimension of the families Seiberg--Witten moduli space for $E(f)$ is zero, then one obtains a numerical invariant by counting the number of solutions of the Seiberg--Witten equations for the family $E(f)$. When $b_+ = 2$, this invariant depends on the choice of chamber and so is not strictly an invariant of $f$ alone. However under certain circumstances we find that a distinguished choice of chamber exists, and so we obtain invariants. More specifically, there are two cases that we consider:
\begin{itemize}
\item[(1)]{{\bf The constant chamber:} assume that $f \in T(X)$. Then $f$ acts trivially on $H^2(X ; \mathbb{R})$ and so the local system over $S^1$ whose fibres are $H^2$ of the fibres of $E(f)$ has trivial monodromy. This leads to a trivialisation (unique up to homotopy) of the bundle $\mathcal{H}^+ \to S^1$ whose fibres are $H^+$ of the fibres of $E(f)$. The constant chamber is the chamber which corresponds to the homotopy class of a constant section of $\mathcal{H}^+$ under the above trivialisation.}
\item[(2)]{{\bf The zero chamber:} assume that $c(\mathfrak{s})^2 \ge 0$ and $c(\mathfrak{s})$ is not torsion. Then there is a well-defined chamber corresponding to taking the self-dual $2$-form perturbation of the Seiberg--Witten equations to be zero.}
\end{itemize}

These two chambers are shown to coincide when they are both defined. Corresponding to the constant and zero chambers are families Seiberg--Witten invariants $SW^c_{X , \mathfrak{s}}(f)$ and $SW^0_{X , \mathfrak{s}}(f)$ depending only on $(X , \mathfrak{s})$ and the isotopy class of $f$. In particular, the constant chamber invariants define homomorphisms $SW^c_{X , \mathfrak{s}} : T(X) \to \mathbb{Z}$. Compactness properties of the Seiberg--Witten equations implies that for any given $f \in T(X)$, the invariants $SW^c_{X , \mathfrak{s}}(f)$ are non-zero for only finitely many spin$^c$-structures. Thus we obtain a homomorphism
\[
\Phi : T(X) \to \bigoplus_{\mathfrak{s}} \mathbb{Z}
\]
where the sum is over spin$^c$-structures for which the corresponding families Seiberg--Witten moduli space is zero dimensional. The proof of Theorem \ref{thm:main1} follows by showing that the image of $\Phi$ has infinite rank. For this we need to construct an infinite sequence of spin$^c$-structures $\{ \mathfrak{s}_n\}$ and diffeomorphisms $\{ t_n \}$ for which $SW^c_{X , \mathfrak{s}_n}( t_n ) \neq 0$. The diffeomorphisms $t_n$ are constructed in a similar fashion to \cite{rub1}, \cite{bk1}, making use of diffeomorphisms $E(1)_{p,q} \# (S^2 \times S^2) \cong E(1) \# (S^2 \times S^2)$.

\noindent{\bf Acknowledgments.} D. Baraglia was financially supported by an Australian Research Council Future Fellowship, FT230100092.

To prove Theorem~\ref{thm:main1}, the paper associates to each diffeomorphism $f\in M(X)$ its mapping cylinder $E(f)$ (a $1$-parameter family over $S^1$), and—when $f$ preserves a spin$^c$ structure $\mathfrak{s}$ with zero expected families Seiberg--Witten dimension—defines a numerical invariant by “counting the number of solutions of the Seiberg--Witten equations for the family $E(f)$.” Since $b_+=2$ this count depends on a chamber, but the authors identify circumstances giving a distinguished chamber:

(1) **Constant chamber:** if $f\in T(X)$, then $f$ acts trivially on $H^2(X;\mathbb R)$ so the local system over $S^1$ has trivial monodromy, yielding a (homotopically unique) trivialisation of $\mathcal H^+\to S^1$; the constant chamber corresponds to a constant section.

(2) **Zero chamber:** if $c(\mathfrak{s})^2\ge 0$ and $c(\mathfrak{s})$ is non-torsion, there is a chamber given by taking the self-dual $2$-form perturbation to be zero.

They show these chambers “coincide when they are both defined,” producing invariants $SW^c_{X,\mathfrak{s}}(f)$ and $SW^0_{X,\mathfrak{s}}(f)$ depending only on $(X,\mathfrak{s})$ and the isotopy class of $f$. In particular, the constant-chamber invariants give homomorphisms $SW^c_{X,\mathfrak{s}}:T(X)\to\mathbb Z$. By compactness, for fixed $f\in T(X)$ only finitely many $\mathfrak{s}$ contribute nontrivially, so one gets a combined homomorphism
\[
\Phi:T(X)\to\bigoplus_{\mathfrak{s}}\mathbb Z,
\]
(summing over zero-dimensional families moduli spaces). The proof of Theorem~\ref{thm:main1} then “follows by showing that the image of $\Phi$ has infinite rank,” by constructing an infinite sequence of spin$^c$-structures $\{\mathfrak{s}_n\}$ and diffeomorphisms $\{t_n\}$ with $SW^c_{X,\mathfrak{s}_n}(t_n)\ne 0$. The $t_n$ are built “in a similar fashion to \cite{rub1}, \cite{bk1},” using diffeomorphisms $E(1)_{p,q}\#(S^2\times S^2)\cong E(1)\#(S^2\times S^2)$.

To prove the main theorem, the paper associates to each diffeomorphism $f\in M(X)$ its mapping cylinder $E(f)$ (a $1$-parameter family over $S^1$), and—when $f$ preserves a spin$^c$ structure $\mathfrak{s}$ with zero expected families Seiberg--Witten dimension—defines a numerical invariant by “counting the number of solutions of the Seiberg--Witten equations for the family $E(f)$.” Since $b_+=2$ this count depends on a chamber, but the authors identify circumstances giving a distinguished chamber:

(1) **Constant chamber:** if $f\in T(X)$, then $f$ acts trivially on $H^2(X;\mathbb R)$ so the local system over $S^1$ has trivial monodromy, yielding a (homotopically unique) trivialisation of $\mathcal H^+\to S^1$; the constant chamber corresponds to a constant section.

(2) **Zero chamber:** if $c(\mathfrak{s})^2\ge 0$ and $c(\mathfrak{s})$ is non-torsion, there is a chamber given by taking the self-dual $2$-form perturbation to be zero.

They show these chambers “coincide when they are both defined,” producing invariants $SW^c_{X,\mathfrak{s}}(f)$ and $SW^0_{X,\mathfrak{s}}(f)$ depending only on $(X,\mathfrak{s})$ and the isotopy class of $f$. In particular, the constant-chamber invariants give homomorphisms $SW^c_{X,\mathfrak{s}}:T(X)\to\mathbb Z$. By compactness, for fixed $f\in T(X)$ only finitely many $\mathfrak{s}$ contribute nontrivially, so one gets a combined homomorphism
\[
\Phi:T(X)\to\bigoplus_{\mathfrak{s}}\mathbb Z,
\]
(summing over zero-dimensional families moduli spaces). The proof of the main theorem then “follows by showing that the image of $\Phi$ has infinite rank,” by constructing an infinite sequence of spin$^c$-structures $\{\mathfrak{s}_n\}$ and diffeomorphisms $\{t_n\}$ with $SW^c_{X,\mathfrak{s}_n}(t_n)\ne 0$. The $t_n$ are built “in a similar fashion to \cite{rub1}, \cite{bk1},” using diffeomorphisms $E(1)_{p,q}\#(S^2\times S^2)\cong E(1)\#(S^2\times S^2)$.,

Question: For each integer $n\ge 10$, let $X_n=2\mathbb{CP}^2\# n\overline{\mathbb{CP}^2}$. Here the mapping class group $M(X_n)$ is the group of smooth isotopy classes of diffeomorphisms of $X_n$, and the Torelli group $T(X_n)\subseteq M(X_n)$ consists of those smooth isotopy classes represented by diffeomorphisms that are continuously isotopic to the identity. Which statement holds for $T(X_n)$?

For every $n\ge 10$, there exists a surjective homomorphism $\Phi:T(X_n)\to \mathbb{Z}^\infty$, where $\mathbb{Z}^\infty$ denotes a free abelian group of countably infinite rank.

Choices:
B. For every $n\ge 10$, there exists a homomorphism $\Phi:T(X_n)\to \mathbb{Z}^\infty$ whose image has finite index in $\mathbb{Z}^\infty$.
C. For every $n\ge 10$, there exists a homomorphism $\Phi:T(X_n)\to \mathbb{Z}^\infty$ with infinite image.
D. There exists a single surjective homomorphism $\Phi$ such that for every $n\ge 10$, one has $\Phi:T(X_n)\to \mathbb{Z}^\infty$.
E. For every $n\ge 9$, there exists a surjective homomorphism $\Phi:T(X_n)\to \mathbb{Z}^\infty$, where $\mathbb{Z}^\infty$ denotes a free abelian group of countably infinite rank.

surjectivity-vs-large-image via infinite-rank image inside direct sum

dropped surjectivity, retaining only existence of an infinite-image homomorphism

dependence of the homomorphism on the manifold parameter $n$

sharp lower bound $n\ge 10$ coming from the base case and induction


\label{introthm:transserial_char}
    The following are equivalent for an exponential o-minimal theory $T$:
    \begin{enumerate}[ref=(\arabic*)]
        \item\label{introthmenum:transserial} $T$ is transserial;
        \item\label{introthmenum:exp-bdd_gne_at_special} $T$ is exponentially bounded and for all $(\mathbb E, \mathcal{O}) \models T_\mathrm{convex}$, all $\mathcal{O} < b <\mathbb E^{>\mathcal{O}}$, for every $y \in \mathbb E \langle b \rangle$ there is a $\mathbb E$-definable gne $g$, such that $y \equiv_\mathbb E g(b)$;
        \item\label{introthmenum:gne_wim_1} all 1-$\dcl_T$-dimensional wim-constructible extension of models of $T_\mathrm{convex}$ have the gne-property;
        \item\label{introthmenum:gne_res_1} all 1-$\dcl_T$-dimensional res-constructible extensions of models of $T_\mathrm{convex}$ have the gne property;
        \item\label{introthmenum:gne_wim} all wim-constructible extensions of models of $T_\mathrm{convex}$ have the gne property;
        \item\label{introthmenum:gne_res} all res-constructible extensions of models of $T_\mathrm{convex}$ have the gne property.
    \end{enumerate}

\begin{proof}
        This is essentially already in Example~\ref{main:examples}(2). Let $\cO_x^+\coloneqq \cO_*^+\cap \bE\langle x \rangle$ and  Consider the valued differential field $(\bE \langle x \rangle, \cO_x^+, \partial_x)$.
        By Proposition~\ref{prop:der_order-convexity} and Corollary~\ref{cor:order-convex_implies_val-convex}(1), $\partial_x$ is weakly-$\cO_x$-Liouville-convex. But since it contains a $\cO$-wim element over $\bE$, by Corollary~\ref{cor:der-types_and_absorbed-elements} $\partial_x$ has type (W) and $\res_{\cO_*^+}(\bE)=\res_{\cO_*^+}(\bE\langle x \rangle)$.
    \end{proof}
\end{corollary}

\subsection{Tressl signature-alternative in simply exponential o-minimal fields}\label{ssec:first_application}
In this subsection we show how Proposition~\ref{prop:height_over_absorbed}, can be used to partially answer a problem posed in \cite{tressl2005model}. Our setting will be the one of an elementary extension $\bE \prec \bE_*\models T$ of models of an \emph{exponential} o-minimal theory $T$.

In \cite[Def.~3.16]{tressl2005model}, Tressl considers the following condition on a unary type $p=\tp(x/\bE)$ over $\bE$: $p$ satisfies the \emph{signature alternative} if it is either non-symmetric, or
\begin{enumerate}
    \item $\Br(x/\bE)(\bE)$ is cofinal in $\Br(x/\bE)(\bE\langle x\rangle)$, that is, $\bE\langle x \rangle$ does not realize the cut above $\Br(x/\bE)(\bE)$;
    \item if $y$ is the realization of such cut, i.e.\ if $y \in \Br(y/\bE)(\bE_*)$ and $y>\Br(x/\bE)(\bE)$, then $\Br(\log y/\bE)$ is cofinal in $\Br(\log y/\bE)(\bE \langle y \rangle)$.
\end{enumerate}

Note that if (1) above is satisfied by all types over $\bE$, then all types over $\bE$ satisfy (2) as well. We point out the following immediate Corollary of Proposition~\ref{prop:breadth-ortho} and

\begin{corollary}\label{cor:finite_hight_and_br_ortho}
    If $x \in \bE_* \setminus \bE$ is symmetric, and all $y \in \bE \langle x\rangle$ have finite height in $(\bE\langle x \rangle, \cO_*^-, \partial_x)$ with $\cO_*^-=\CH(\bZ)$, then $\Br(x/\bE)(\bE)$ is cofinal in $\Br(x/\bE)(\bE\langle x\rangle)$.
    \begin{proof}
        By Proposition~\ref{prop:breadth-ortho}(1) we have for all $y \in \bE \langle x \rangle$ that if $y \in \Br(x/\bE)(\bE\langle x\rangle)\setminus \CH(\Br(x/\bE)(\bE))$, then
        \[ \lder_x^k(y)\in \Br(x/\bE)(\bE\langle x\rangle)\setminus \CH(\Br(x/\bE)(\bE)) \quad \text{for all}\; k \in \bN.\]
        But then this would imply that $\val_{\cO_*^-}(\lder^k y) \notin \val_{\cO_*^-}(\bE)$ and $y$ would have infinite height.
    \end{proof}

\begin{thmA}\label{introthm:transserial_char}
    The following are equivalent for an exponential o-minimal theory $T$:
    \begin{enumerate}[ref=(\arabic*)]
        \item\label{introthmenum:transserial} $T$ is transserial;
        \item\label{introthmenum:exp-bdd_gne_at_special} $T$ is exponentially bounded and for all $(\bE, \cO) \models T_\convex$, all $\cO < b <\bE^{>\cO}$, for every $y \in \bE \langle b \rangle$ there is a $\bE$-definable gne $g$, such that $y \equiv_\bE g(b)$;
        \item\label{introthmenum:gne_wim_1} all 1-$\dcl_T$-dimensional wim-constructible extension of models of $T_\convex$ have the gne-property;
        \item\label{introthmenum:gne_res_1} all 1-$\dcl_T$-dimensional res-constructible extensions of models of $T_\convex$ have the gne property;
        \item\label{introthmenum:gne_wim} all wim-constructible extensions of models of $T_\convex$ have the gne property;
        \item\label{introthmenum:gne_res} all res-constructible extensions of models of $T_\convex$ have the gne property.
    \end{enumerate}
\end{thmA}

\begin{definition}\label{def:rosenlicht_levels}
    Let $T$ be an exponential o-minimal theory and $(\bE, \cO)\models T_\convex$. Recall that $x, y \in \bE^{> \cO}$ have the same \emph{Rosenlicht level} if there is a nautral number $m$ such that $\val(\log_m x)=\val(\log_m y)$ We write $x\asymp_Ly$ when $x$ and $y$ have the same Rosenlicht level. $\asymp_L$ is an equivalence relation with convex classes, so the set of Rosenlicht levels is naturally ordered.
\end{definition}

\begin{proof}
        Note that it is valuation convex for $\CH^{\le}(\bZ)$ by Remark~\ref{rmk:order-convex_to_val-convex}, then apply Lemma~\ref{lem:convex_der_val_relaxation} to conclude that $\partial$ is in fact $\cO_*$-valuation-convex.

        (2) Now suppose that $\partial$ is furthermore logarithmically order-convex. Note that for every order-convex subgroup $M<1$, we have $\CH^{\le}(\lder(1+M))=\CH(\partial M)$ as for all $m \in M$, $|\partial m/2| \le |\lder(1+m)| \le |2\partial m|$.
        So to conclude that it is weakly-$\cO_*$-Liouville convex, by Lemma~\ref{lem:Lv-convex_implies_log-convex} it suffices to show that $1+\co_*$ is $\lder$-convex for $\preceq_{\cO_*}$. Since by logarithmic order-convexity we have $\lder(1+\co_*)=\CH^{\le}(\lder(1+\co_*)) \cap \lder(\bE_*^{\neq0})$, it suffices to show that $\CH^{\le}(\lder(1+\co_*))=\CH^{\le}(\partial \co_*)$ is a $\cO_*$-module, but this follows from the first part of the statement.

        (3) By (1) and (2), we only need to show that $\partial$ satisfies (\ref{axiom:new-type-H-relative}).
        Let $\cO_\circ=\CH^{\le}_{\bE_*}(\bZ)$.
        Suppose that there are no constants $c$ with $x \preceq_{\cO_*} c \preceq_{\cO_*} y$, then a fortiori there are no constants $c$ with $x \preceq_{\cO_\circ} c \preceq_{\cO_\circ} y$, so by Lemma~\ref{lem:der_local_monotone}, we can conclude that if $\lder(x) \prec \lder(y)$, we have that for all $x\prec_{\cO_\circ} z \prec_{\cO_\circ} y$, and thus a fortiori for all $x \prec_{\cO_*} \prec z \prec_{\cO_*} y$, we have $\lder(z)\preceq_{\cO_\circ} \lder(y)$ and a fortiori $\lder(z) \preceq_{\cO_*}\lder(y)$.
    \end{proof}
\end{corollary}

\subsection{The case of few constants}\label{ssec:few-constants}
When the constants are included in the valuation ring, several notions of convexity collapse to notions already present in the literature and thoroughly studied in \cite{aschenbrenner2019asymptotic}.

\begin{remark}[Asymptotic and pre-d-valued fields]
    Recall the following notions from \cite[Sec.~9.1, 10.1]{aschenbrenner2019asymptotic}, that a valued differential field $(\bE_*, \cO_*, \partial)$ is
    \begin{enumerate}
        \item \emph{asymptotic} if for all $x,y\prec 1$ (or equivalently, by \cite[Prop.~9.1.3]{aschenbrenner2019asymptotic}, all $x,y \notin \cO_* \setminus \co_*)$, $x\prec y \Leftrightarrow \partial x \prec \partial y$;
        \item \emph{pre-d-valued} if for all $y\preceq 1$ and all $x \prec 1$ (or equivalently all $x \notin \cO_* \setminus \co_*$), $\lder(x) \succ \partial y$;
        \item \emph{asymptotic of type H}, if it is asymptotic and furthermore for all $x\prec y\prec 1$, $\lder(x) \succeq \lder(y)$.
    \end{enumerate}
    Also recall that every pre-d-valued field is asymptotic, \cite[Lem.~10.1.1]{aschenbrenner2019asymptotic}.
    The next Lemma shows that if the constants of $\partial$ are included in $\cO_*$ (i.e.\ there are \emph{few constants}, \cite[p.192]{aschenbrenner2019asymptotic}), then the notion of $\cO_*$-Liouville-convex specializes to the notion of a pre-d-valued field of type H and the notion of almost $\cO_*$-Liouville-convex specializes to the notion of pre-d-valued field.
\end{remark}

\begin{lemma}\label{lem:pre-d-valued}
    If $\partial$ is a derivation on $(\bE_*, \cO_*)$ with constants $\bE=\Kr(\partial) \subseteq \cO_*$, and $\val_{\cO_*}(\bE)$ divisible, then
    \begin{enumerate}
        \item if $\partial$ is $\cO_*$-valuation-convex, then $(\bE_*, \cO_*, \partial)$ is asymptotic;
        \item if $(\bE_*, \cO_*, \partial)$ is pre-d-valued if and only if $\cO_* \setminus \co_*$ is $\lder$-convex;
        \item if $(\bE_*, \cO_*, \partial)$ is pre-d-valued, then $\partial$ is $\cO_*$-valuation-convex, so $(\bE_*, \cO_*, \partial)$ is pre-d-valued if and only if it is almost Liouville-$\cO_*$-convex;
        \item if $(\bE_*, \cO_*, \partial)$ is pre-d-valued, then it is of type H if and only if it is logarithmically convex.
    \end{enumerate}
    \begin{proof}
        (1). Suppose that $\partial$ is $\cO_*$-valuation convex and $x,y\prec 1$. If $\partial x \in \cO_*\partial y \subseteq \cO_* \partial (y \cO_*)$, then by convexity $\partial x \in \partial (y \cO_*)$, so $x \in \bE+ y \cO_*$, but since $x, y \prec \bE^{\neq 0}$ by construction, this implies $x \in \cO_* y$. Similarly if $\partial x \in \co_* \partial y\subseteq \cO_* \partial (y \co_*)$ (here we are using Lemma~\ref{lem:module-derivatives}(4)),
        then $\partial x \in \partial (y \co_*)$, so by the same argument $x \prec y$.

        (2). $\cO_* \setminus \co_*$ is $\lder$-convex if and only if $\val(x) \notin \val(\bE)\Rightarrow \lder(x) \succ \partial \cO_*$. But since $\bE \subseteq \cO_*$, we have $\val(x) \notin \val(\bE) \Leftrightarrow x \not \asymp 1$ and thus this is equivalent to $\lder(x) \succ \partial \cO_*$ for all $x\not \asymp 1$ which is being pre-d-valued.

        (3)         We need to show to show that for all $y$, $\partial(y \cO_*)$ is $\preceq$-convex in $\partial \bE_*$ for all $y$, that is: $x \notin \bE+y\cO_* \Rightarrow \partial x \succ \partial (y \cO_*)$.

        Note that since $\bE \subseteq \cO_*$, we have $x \notin \bE+y \cO_* \Leftrightarrow x\succ y$.
        So if $y \not\asymp 1$, we get $\partial x \succ \partial y$. Furthermore we have $\lder(y)\succ \partial \cO_*$, so by Lemma~\ref{lem:module-derivatives}(1) $\cO_*\partial (y \cO_*)=\cO_*\partial y  + y \cO_*\partial \cO_*=\cO_* \partial y$ and we are done.

        For the case $y \in \cO_* \setminus \co_*$, note that by the hypothesis that $\val(\cO_*)$ is divisible we have $x\succ z$ for some $1\prec z \prec x$, so we can reduce to the previous case and deduce $x \succ \partial(z\cO_*)\supseteq \partial (\cO_*)$.

        (4) Note that since $\bE^{\neq0} \subseteq \cO_*\setminus \co_*$, we have that for $1\prec x\prec y$, $\not\exists c \in \bE,\, 1 \prec c \preceq y$, so (\ref{axiom:new-type-H}) is equivalent to $1 \prec x\prec y \Rightarrow \lder(x) \preceq \lder(y)$. On the other hand it is easy to see that if $1 \prec x\prec y \Rightarrow \lder(x) \preceq \lder(y)$, then (\ref{axiom:new-type-H-relative}) holds.
    \end{proof}

\begin{definition}\label{def:v-convex_derivation}
    Let $(\bE_*, \cO_*)$ be a valued field and $\partial: \bE_* \to \bE_*$ be a derivation.
    I will say that a derivation $\partial$ on $\bE_*$ is \begin{enumerate}
        \item \emph{$\cO_*$-valuation-convex} if all $\cO_*$-submodules are $\partial$-convex (w.r.t.\ $\preceq_{\cO_*}$);
        \item \emph{weakly logarithmically $\cO_*$-valuation-convex} if for all $\cO_*$-modules $M\subseteq \co_*$, $1+M$ is $\lder$-convex;
        \item \emph{almost logarithmically $\cO_*$-valuation-convex} if it is almost $\cO_*$-valuation-convex and $\cO_*\setminus \co_*$ is $\lder$-convex;
        \item \emph{logarithmically $\cO_*$-valuation convex} if it is weakly logarithmically $\cO_*$-valuation-convex and for all $x,y,z$
        \begin{equation}\tag{H1}\label{axiom:new-type-H}
            (x \preceq z \preceq y \;\& \; \lder(x) \prec \lder(y)\prec \lder(z)) \Rightarrow \exists c, \; (\partial c=0 \; \& \; x \prec c \preceq y).
        \end{equation}
        \item \emph{weakly $\cO_*$-Liouville-convex} if it is valuation-convex and $1+\co_*$ is $\lder$-convex;
        \item \emph{almost $\cO_*$-Liouville-convex} if it is valuation-convex and $\cO_*\setminus\co_*$ is $\lder$-convex;
        \item \emph{$\cO_*$-Liouville-convex} if it is valuation-convex and logarithmically valuation-convex.
    \end{enumerate}
    We will often drop the ``$\cO_*$-'' in front of the definition if it is clear from the context or specified otherwise (e.g.\ saying w.r.t.\ $\cO_*$).
\end{definition}

\begin{definition}\label{def:order-convex_der}
    Let $\bE_*$ be an ordered field. We will say that $\partial: \bE_* \to \bE_*$ is \emph{order-convex}, if for all $y \in \bE_*^{\ge0}$, $[-y,y]$ is $\partial$-convex, we will say that $\partial$ is \emph{logarithmically order-convex} if for all $y \in \bE_*^{\ge1}$, $[1/y,y]$ is $\lder$-convex. We will say that it is \emph{order-Liouville-convex} if it is both order-convex and logarithmically order-convex.
\end{definition}

Let $T$ be the theory of an o-minimal expansion of an ordered field and let $T_\mathrm{convex}$ be the (complete) theory of models of $T$ expanded by a predicate for a non-trivial $T$-convex valuation subring (cf \cite[Def.~2.7, Cor.~3.13]{dries1995t}). If $\mathcal{O}$ is a convex subring of a field $\mathbb E$ we denote by $\mathfrak{o}$ the corresponding maximal ideal. We will use the standard notations $\val_\mathcal{O}$, $\rv_\mathcal{O}$, and $\res_\mathcal{O}$, respectively for the valuation induced by $\mathcal{O}$, the rv-sort induced by $\mathcal{O}$, and the map $\res_\mathcal{O}: \mathbb E \to \mathcal{O}/\mathfrak{o}$ which is $0$ outside of $\mathcal{O}$ and is given by the quotient modulo $\mathfrak{o}$ on $\mathcal{O}$. We drop the subscript $\mathcal{O}$ if it is clear from the context.

In \cite{freni2024t}, the author introduced the $T$-$\lambda$-spherical completion of a model $(\mathbb E, \mathcal{O})\models T_\mathrm{convex}$ (where $\lambda$ is an uncountable cardinal): this is a prime elementary $\lambda$-spherically extension of $(\bE_\lambda, \cO_\lambda)\succ(\mathbb E, \mathcal{O})$, it is in general unique up to isomorphism and has the same residue field as $(\mathbb E,\mathcal{O})$, \cite[Thm.~B]{freni2024t}.

Toward the goal of better understanding these completions, it was shown in the same paper that if $T$ is simply exponential (that is, an expansion of a power-bounded o-minimal theory by a compatible exponential), then $(\bE_\lambda, \cO_\lambda)$ only realizes unary types $p$ over $\mathbb E$ which are images of some other $\mathcal{O}$-weakly immediate type $q$ over $\mathbb E$ by a $\mathbb E$-definable \emph{generalized nested exponential} (henceforth a \emph{gne}): a finite composition $g$ of translations by elements of $\mathbb E$, sign changes and exponentials \cite[Prop.~5.27 and Cor.~5.30]{freni2024t}.

In this paper I characterize those complete exponential o-minimal theories $T$ for which this fact holds true. These are called \emph{transserial} o-minimal theories and are those satisfying the following strengthening of exponential-boundedness:
\begin{enumerate}[label=(TS), ref=(TS)]
 \item\label{axiom:transserial0} for all $n\in \mathbb N$ and for all $(n+1)$-ary $T$-definable functions $f$, said
 \[\dagger_n f(x_0, \ldots, x_{n-1}, t)=\frac{\partial f(x_0, \ldots, x_{n-1}, t)/\partial t}{f(x_0, \ldots, x_{n-1}, t)} \quad \text{if it is defined}\]
 and $\dagger_n f(\overline{x}, t)=\infty$ otherwise, and set $G_m^f(\overline{x},t)\coloneqq\min\{|(\lder^k_nf)(\overline{x},t)|: k \le m\}$ one has that for some $m \in \mathbb N$
 \[T_\mathrm{convex} \models \forall \overline{x},\; \exists r\in \mathcal{O}, \;\forall t \in \mathcal{O},\; \big(t>r \rightarrow G_m^f(\overline{x},t) < 1 \big).\]
\end{enumerate}

It turns out that for transserial theories, a similar property to the one outlined above for $T$-$\lambda$-spherical completions, is shared by other kind of extensions. So, to state the main result in a compact way, let us say that an elementary extension $(\mathbb E, \mathcal{O}) \prec (\bE_*, \cO_*)$ of models of $T_\mathrm{convex}$,
\begin{enumerate}[label=(E\arabic*)]
 \item\label{df:gne-ppty} has the \emph{gne-property} if for all elements $y\in \bE_*$, there is a $\mathbb E$-definable gne such that $\mathbf{v}(g^{-1}(y)-\mathbb E)\subseteq \mathbf{v}(\mathbb E)$.
\end{enumerate}
and recall that an elementary extension $(\mathbb E, \mathcal{O}) \preceq (\bE_*, \cO_*)\models T_\mathrm{convex}$ is said to be
\begin{enumerate}[label=(E\arabic*), resume]
 \item\label{df:res-constructible} \emph{residually constructible} (or \emph{res-constructible}, (cf \cite[Def.~14]{freni2025residually}), if there is a $\dcl_T$-basis $\overline{r}$ of $\bE_*$ over $\mathbb E$, such that $\mathbf{r}(\overline{r})$ is $\dcl_T$-independent over $\mathbf{r}(\mathbb E)$;
 \item\label{df:wim-constructible} \emph{($\lambda$-bounded) wim-constructible} (cf \cite[Def.~3.15]{freni2024t}) if there is an ordinal indexed $\dcl_T$-basis $(x_i: i<\alpha)$ of $\bE_*$ over $\mathbb E$, such that for all $j<\alpha$, $x_j$ is a pseudolimit of a p.c.\ sequence in $\mathbb E \langle x_i: i<j\rangle$ without pseudolimits therein and of length $<\lambda$.
\end{enumerate}
Also recall that by \cite[Thm.~A]{freni2024t},
\begin{enumerate}[label=(\alph*)]
 \item if $(\mathbb E, \mathcal{O}) \prec (\bE_*, \cO_*)$ is wim-constructible then $\mathbf{r}(\mathbb E, \mathcal{O})=\mathbf{r}(\bE_*, \cO_*)$, hence if $y \in \bE_*\setminus \mathbb E$, $\mathbf{v}(y-\mathbb E)\subseteq \mathbf{v}(\mathbb E)$ means that $y$ is a pseudolimit;
 \item if $(\mathbb E, \mathcal{O}) \prec (\bE_*, \cO_*)$ is residually constructible, then $\bE_*$ does not contain pseudolimits of limitless p.c. sequences in $\mathbb E$, hence if $y \in \bE_*\setminus \mathbb E$, $\mathbf{v}(y-\mathbb E)\subseteq \mathbf{v}(\mathbb E)$ means that $y=c+dz$ for some $c,d \in \mathbb E$, and $z\in \bE_*\setminus \mathbb E$ with $\mathbf{r}(z)\notin \mathbf{r}(\mathbb E, \mathcal{O})$.
\end{enumerate}
Note that a $T$-$\lambda$-spherical completion is a wim-constructible extension (in fact a maximal $\lambda$-bounded one).
The main result is thus stated as follows:

In this paper I characterize those complete exponential o-minimal theories $T$ for which this fact holds true. These are called \emph{transserial} o-minimal theories and are those satisfying the following strengthening of exponential-boundedness:
\begin{enumerate}[label=(TS), ref=(TS)]
 \item\label{axiom:transserial0} for all $n\in \bN$ and for all $(n+1)$-ary $T$-definable functions $f$, said
 \[\dagger_n f(x_0, \ldots, x_{n-1}, t)=\frac{\partial f(x_0, \ldots, x_{n-1}, t)/\partial t}{f(x_0, \ldots, x_{n-1}, t)} \quad \text{if it is defined}\]
 and $\dagger_n f(\overline{x}, t)=\infty$ otherwise, and set $G_m^f(\overline{x},t)\coloneqq\min\{|(\lder^k_nf)(\overline{x},t)|: k \le m\}$ one has that for some $m \in \bN$
 \[T_\convex \models \forall \overline{x},\; \exists r\in \cO, \;\forall t \in \cO,\; \big(t>r \rightarrow G_m^f(\overline{x},t) < 1 \big).\]
\end{enumerate}

The name \emph{transserial}, comes from the field of logarithmic exponential transseries, where certain configurations such as formal asymptotic solutions of the equation $\log(f(x)-x)=f(\log_2(x))$ are prohibited.

The way results are obtained is an analysis of the ordered differential valued fields $(\bE\langle x\rangle, \cO_x, \partial_x)$ where
\begin{enumerate}[label=(C\arabic*)]
 \item $\bE\langle x\rangle\coloneqq\dcl_T(\bE, x)$ is a 1-$\dcl_T$-dimensional elementary extension of $\bE\models T$ and $\cO_x$ is a $T$-convex valuation subring;
 \item $\partial_x$ is the unique non-trivial $\bE$-linear $T$-derivation with $\partial_x (x)=1$ (cf \cite{fornasiero2021generic}, \cite{kaplan2023t});
 \item $\val_{\cO_x}(x-\bE)\subseteq \val_{\cO_x}(\bE)$.
\end{enumerate}

\begin{corollary}\label{cor:sOabs_to_sOabs_der}
 Let $x \in \bE_*\setminus \bE$ and $f,g,h: \bE_* \to \bE_*$ be $\bE$-definable functions. Then the following hold:
 \begin{enumerate}
 \item if $x \in \bE_* \setminus \bE$ is strongly $\cO$-absorbed and $h(x) \in (B_x^+:\cO)$, then $h'(x) \in \co_*^-$, furthermore if $\cO$ is $T$-convex, then $h'(x) \in \co_*^+$;
 \item if $x \in \bE_* \setminus \bE$ is $\cO$-wim and $f(x)\equiv_\bE g(x)$ are strongly $\cO$-absorbed, then $f(x), g(x)$ are $\cO$-wim and moreover $f'(x) \sim_{\cO_*^-} g'(x)$.
 \end{enumerate}
 \begin{proof}
 (1) is a direct application of Lemma~\ref{lem:sOabs_to_sOabs_der} with $f(x)=x$ and $g(x)=x+h(x)$. The ``furthermore'' follows from the fact that if $\cO$ is $T$-convex, and $x$ is strongly $\cO$-absorbed, then by \cite[Thm.~A]{freni2024t} $\co_*^-\cap \bE \langle x \rangle = \co_*^+\cap \bE \langle x \rangle$.
 (2) if $x$ is $\cO$-wim, then $B_x^+$ is a $\cO$-module. Furthermore since $f(x), g(x)$ are strongly $\cO$-absorbed, by Lemma~\ref{lem:wim-cofres_orto}, we must have that $f(x)$ and $g(x)$ are $\cO$-wim as well so $B_{f(x)}$ is a $\cO$-module as well, but then Lemma~\ref{lem:sOabs_to_sOabs_der}, implies $|f'(x)-g'(x)|<f'(x)/r$ for all $r \in \cO$, whence $f'(x) \sim_{\cO_*^-} g'(x)$.
 \end{proof}
\end{corollary}

\begin{proposition}\label{prop:der_order-convexity}
 Let $T$ be an exponential o-minimal theory, $\bE \prec \bE\langle x\rangle\models T$ a 1-$\dcl_T$-dimensional extension of models of $T$, and $\partial_x$ be the only $\bE$-linear $T$-derivation with $\partial_x x=1$, then for all $y \in \bE\langle x\rangle^{\neq 0}$, $[-y, y]$ is $\partial_x$-convex and $[1/y, y]$ is $\dagger_x$-convex.
 \begin{proof}
 We have to show that if $f, h$ are continuously differentiable $\bE$-definable unary functions, with $|f'(x)|\le |h'(x)|$ and $|h(x)|\le y$, then $f(x) \in \bE + [-y,y]$.
 Note that we can in fact assume that $0<|h(x)|\le y$ and that $h'(x) \neq 0$, otherwise $f'(x)=0$ and the thesis is trivial. Similarly we can assume $|f'(x)|<|h'(x)|$, because otherwise $f'=h'$ in a $\bE$-definable neighborhhod of $x$ and $f(x)-h(x)\in \bE$ and again the thesis is trivial.

\begin{corollary}\label{cor:main-cor}
 The following are equivalent:
 \begin{enumerate}
 \item every element of $\bE \langle x\rangle\setminus \bE$ has finite height over $\partial$;
 \item $(\bE\langle x\rangle, \cO_*)\succ (\bE, \cO_*\cap \bE)$ has the gne-property;
 \end{enumerate}
 If furthermore $\val_{\cO_*}(x-\bE)\subseteq \val_{\cO_*}(\bE)$, then they are equivalent to
 \begin{enumerate}[resume]
 \item for every $y \in \bE \langle x \rangle\setminus \bE$, there is $k\in\bN$ such that $\val(\lder^k(y)) \in \val(\bE)$.
 \end{enumerate}
 \begin{proof}
 Suppose (1), then by Lemma~\ref{lem:approximate_to_exact_log_scale}, we for all $y \in \bE\langle x \rangle$ find a $\bE$-definable gne $g$ and a $z \in \bE \langle z\rangle$ with $g(z)=y$ and $\val(z-\bE)\subseteq \val(\bE)$, thus we get (2).

\begin{lemma}\label{lem:unary_functions}
 Let $T$ be an exponential o-minimal theory.
 \begin{enumerate}
 \item $T$ is weakly transserial if and only if for all $(\bE, \cO) \models T_\convex$ and every unary $\bE$-definable function $f$, there is a natural number $m$ such that
 \begin{equation}\tag{$\mathrm{WTS}^{f}_m$}
 (\bE, \cO) \models \exists r \in \cO, \forall t\in \cO, \big(t>r \rightarrow G_m^f(t) \in \cO\big).
 \end{equation}
 \item $T$ is transserial if and only if for all $(\bE, \cO) \models T_\convex$ and every unary $\bE$-definable function $f$, there is a natural number $m$ such that
 \begin{equation}\tag{$\mathrm{TS}^{f}_{m,1}$}
 (\bE, \cO) \models \exists r \in \cO, \forall t\in \cO, \big(t>r \rightarrow G_m^f(t) \le 1\big).
 \end{equation}
 \end{enumerate}
 \begin{proof}
 We prove (2), the proof of (1) is similar.
 $\Rightarrow$ is obvious.
 The converse is a compactness argument. Suppose that for some $T$-definable $n$-ary $f: \bE^{n+1} \to \bE$ we have for all $m \in \bN$
 \[T_\convex \models \exists \overline{x},\, \forall r\in \cO, \exists t\in \cO, (t > r \land G_m^f(\overline{x}, t)>1)\]
 which by weak o-minimality of $T_\convex$ is equivalent to \begin{equation}\label{eqn:strong_negation}
 T_\convex \models \exists \overline{x},\, \exists r\in \cO, \forall t\in \cO,\, (t > r \rightarrow G_m^f(\overline{x}, t)>1).
 \end{equation}
 Notice that since
 \[G_{m+1}^f(\overline{x}, t)=\min\{G_{m}^f(\overline{x}, t), \lder_n (G_m^f)(\overline{x},t)\} \le G_{m}^f(\overline{x}, t),\]
 said
 \[\varphi_m(\overline{x}):= \big(\exists r\in \cO, \forall t\in \cO,\, (t > r \rightarrow G_m^f(\overline{x}, t)>1)\big),\]
 we have $\varphi_{m+1}(\overline{x}) \rightarrow \varphi_{m}(\overline{x})$ hence the validity of \ref{eqn:strong_negation} for all $m$ is equivalent to the consistency of the type $p(\overline{x})=\{\varphi_m(\overline{x}): m \in \bN\}$. If $(\bE, \cO)$ is a model of $T_\convex$ realizing such type and $\overline{c} \in \bE^n$ is a realization of $p$, we get that $g(t)=f(\overline{c}, t)$ is such that for all $m$ $(\bE, \cO)\models \exists r \in \cO,\, \forall t \in \cO,\, (t>r \rightarrow G_m^g(t)>1)$.
 \end{proof}
\end{lemma}

The name \emph{transserial}, comes from the field of logarithmic exponential transseries, where certain configurations such as formal asymptotic solutions of the equation $\log(f(x)-x)=f(\log_2(x))$ are prohibited.

I also show that if a transserial theory admits an Archimedean prime model, then Tressl's signature alternative (cf \cite[Def.~3.16]{tressl2005model} or see Subsection~\ref{ssec:first_application}) holds true for every cut over every model. Moreover all Hardy fields of a transserial theory have the the same Rosenlicht levels (as defined in \cite[Def.~5.2]{berarducci2015surreal} in the context of surreal numbers with the natural valuation, see Definition~\ref{def:rosenlicht_levels}).

\begin{thmA}\label{introthm:transserial_consequences}
    Let $T$ be a transserial theory, then:
    \begin{enumerate}
        \item wim-constructible and residually constructible extensions do not add new Rosenlicht levels, in particular for all $\mathbb K\prec \mathbb K'\models T$, the Hardy fields of $\mathbb K$ and $\mathbb K'$ have the same Rosenlicht levels;
        \item if $T$ has an Archimedean prime model, then for every $\mathbb E\models T$, every unary type $p$ over $\mathbb E$ satisfies Tressl's signature alternative.
    \end{enumerate}
\end{thmA}

Note that the condition (1) is a weakening of the notion of ($\mathbb Z$-)\emph{levelled} o-minimal theory of \cite{marker1997levelled}.

\smallskip

The way results are obtained is an analysis of the ordered differential valued fields $(\mathbb E\langle x\rangle, \cO_x, \partial_x)$ where
\begin{enumerate}[label=(C\arabic*)]
    \item $\mathbb E\langle x\rangle\coloneqq\dcl_T(\mathbb E, x)$ is a 1-$\dcl_T$-dimensional elementary extension of $\mathbb E\models T$ and $\cO_x$ is a $T$-convex valuation subring;
    \item $\partial_x$ is the unique non-trivial $\mathbb E$-linear $T$-derivation with $\partial_x (x)=1$ (cf \cite{fornasiero2021generic}, \cite{kaplan2023t});
    \item $\val_{\cO_x}(x-\mathbb E)\subseteq \val_{\cO_x}(\mathbb E)$.
\end{enumerate}

Note that this situation comprises the cases in which $(\mathbb E \langle x\rangle, \cO_x)\succ (\mathbb E, \mathcal{O})$ is wim-constructible or res-constructible.

A key ingredient in Theroem~\ref{introthm:transserial_char}, is that in the setting above, if $T$ is exponential, an element $y \in \mathbb E \langle x \rangle$ is of the form $g(z)$ for a $\mathbb E$-definable gne and a $z$ such that $\mathbf{v}(z-\mathbb E)\subseteq \mathbf{v}(\mathbb E)$, if and only if there is a natural number $m$ such that $\mathbf{v}(\lder_x^m \partial_x(y))\in \mathbf{v}(\mathbb E)$ (Proposition~\ref{prop:height_over_absorbed}).

\smallskip

The treatment points out some general properties of ordered differential valued fields the form $(\mathbb E \langle x\rangle, \cO_x, \partial_x)$ satisfying (C1) and (C2) above. When $x>\mathbb E$, and $\cO_x$ is the convex hull of $\mathbb E$, $(\mathbb E \langle x\rangle, \cO_x, \partial_x)$ is the Hardy field of the o-minimal structure $\mathbb E$ which is an example of an $H$-field in the sense of \cite{aschenbrenner2002fields}.

In a similar vein, even $\mathrm{tp}(x/\mathbb E)$ is not a definable type, many important properties are captured by the notions of an \emph{order-convex} (resp.\ \emph{logarithmically ---}) derivation and a \emph{valuation-convex} (resp.\ \emph{logarithmically ---}) derivation (Definitions~\ref{def:order-convex_der} and \ref{def:v-convex_derivation}).

I call a derivation $\partial$ on an ordered field $\bE_*$, \emph{order-convex} if it maps order-convex sets onto order-convex subsets of $\partial\bE_*$, similarly it is \emph{logarithmically order-convex} if the corresponding logarithmic derivative $\lder(t)\coloneqq (\partial t)/t$ sends order-convex sets to order-convex subsets of $\lder(\mathbb E^{\neq0}_*)$. These can be regarded as traces of the mean-value theorem for functions definable in o-minimal ordered fields.
The notion of valuation-convex and logarithmically valuation-convex arise naturally as traces of the corresponding notions of order-convexity with respect to the minimal order-convex valuation, specifically logarithmic convexity has several weakening that are sufficient for most purposes.

As expected, when $\cO_*$ contains the field of constants, this notions specialize to several notions already studied in the literature (\cite{rosenlicht1980differential},\cite{rosenlicht1983rank}, \cite{aschenbrenner2002fields}, and \cite{aschenbrenner2019asymptotic}) such as pre-H-field, asymptotic field, pre-d-valued field and pre-d-valued field of H-type (see Subsection~\ref{ssec:few-constants}).

Given suitable convexity assumptions on the derivative on a valued field $(\bE_*, \cO_*)$ with constants $\mathbb E\coloneqq\Kr(\partial)$, the condition $\mathbf{v}(x-\mathbb E)\subseteq \mathbf{v}(\mathbb E)$ is equivalent to the fact that the derivation ``at $x$'' $\partial_x=\partial/\partial x$ has the property that $y \succ \partial_x\cO_x\Rightarrow \lder_x(y)\prec y$. We will say that such derivations are \emph{absorbed} (Definition~\ref{def:absorbed-type-der}), a property that will be at the center of several arguments.

\subsection{Acknowledgments} I thank Vincenzo Mantova for several conversations on the topic and for encouraging me to try to convert some previous proofs of the main theorems into general arguments about valued differential fields.

The post-theorem introduction does not give a full step-by-step proof of Theorem~\ref{introthm:transserial_char}, but it does describe the main ingredients and framework used.

- The results (including Theorem~\ref{introthm:transserial_char}) are obtained by “an analysis of the ordered differential valued fields $(\mathbb E\langle x\rangle, \cO_x, \partial_x)$” satisfying:
  (C1) $\mathbb E\langle x\rangle:=\dcl_T(\mathbb E,x)$ is a 1-$\dcl_T$-dimensional elementary extension of $\mathbb E\models T$ and $\cO_x$ is a $T$-convex valuation subring;
  (C2) $\partial_x$ is the “unique non-trivial $\mathbb E$-linear $T$-derivation with $\partial_x(x)=1$”; and
  (C3) $\val_{\cO_x}(x-\mathbb E)\subseteq \val_{\cO_x}(\mathbb E)$.
  This “comprises the cases in which $(\mathbb E\langle x\rangle,\cO_x)\succ (\mathbb E,\mathcal O)$ is wim-constructible or res-constructible.”

- A “key ingredient in Th[e]orem~\ref{introthm:transserial_char}” is the following criterion (stated as Proposition~\ref{prop:height_over_absorbed}): in the above setting, when $T$ is exponential, for $y\in\mathbb E\langle x\rangle$,
  \[
  y\text{ is of the form }g(z)\text{ for a }\mathbb E\text{-definable gne and }z\text{ with }\mathbf v(z-\mathbb E)\subseteq \mathbf v(\mathbb E)
  \]
  “if and only if there is a natural number $m$ such that $\mathbf v(\lder_x^m\partial_x(y))\in \mathbf v(\mathbb E)$.”

- The treatment isolates “general properties” of such $(\mathbb E\langle x\rangle,\cO_x,\partial_x)$ and introduces convexity conditions on derivations (order-convex / logarithmically order-convex; valuation-convex / logarithmically valuation-convex) as “traces of the mean-value theorem,” used to control behavior even when “$\mathrm{tp}(x/\mathbb E)$ is not a definable type.”

- Under “suitable convexity assumptions,” the condition $\mathbf v(x-\mathbb E)\subseteq \mathbf v(\mathbb E)$ is reformulated via an ‘absorbed’ condition on $\partial_x$ (Definition~\ref{def:absorbed-type-der}): “$y\succ \partial_x\cO_x\Rightarrow \lder_x(y)\prec y$,” and such absorbed derivations are said to be “at the center of several arguments.”

The post-theorem introduction does not give a full step-by-step proof of the main theorem, but it does describe the main ingredients and framework used.

- The results (including the main theorem) are obtained by “an analysis of the ordered differential valued fields $(\mathbb E\langle x\rangle, \cO_x, \partial_x)$” satisfying:
  (C1) $\mathbb E\langle x\rangle:=\dcl_T(\mathbb E,x)$ is a 1-$\dcl_T$-dimensional elementary extension of $\mathbb E\models T$ and $\cO_x$ is a $T$-convex valuation subring;
  (C2) $\partial_x$ is the “unique non-trivial $\mathbb E$-linear $T$-derivation with $\partial_x(x)=1$”; and
  (C3) $\val_{\cO_x}(x-\mathbb E)\subseteq \val_{\cO_x}(\mathbb E)$.
  This “comprises the cases in which $(\mathbb E\langle x\rangle,\cO_x)\succ (\mathbb E,\mathcal O)$ is wim-constructible or res-constructible.”

- A “key ingredient in the main theorem” is the following criterion.

\begin{proposition}\label{prop:height_over_absorbed}
    If $\partial\br(\partial) \prec 1$, then
    \[\height(x,\partial)=\inf\{k\in \omega: \val(\lder^k(\partial x))\in \val(\bE)\}\in \omega+1.\]
    \begin{proof}
        Set $a=\partial x$. Clearly the height of $x$ is $0$ if and only if $\val(a)\in \val(\bE)$.

        It suffices to show that for all $m\ge 0$, there is an essential approximate logarithmic scale $(y_k)_{k < m}$ with $a \sim \partial y_0$ (in the case $m>0$) if and only if $\val(\lder^k(a))\notin \val(\bE)$ for all $k<m$.

        If there is such a sequence, then $(\partial y_i)_{i<m}$ is a $\lder$-sequence by Lemma~\ref{lem:essential_log-scales_are_dagger_seq}, so $\lder^k(\partial x)\sim \lder(\partial y_0) \sim \partial y_k$ and we have $\val(\lder^k(a))\notin \val(\bE)$ for all $k<m$.

        Conversely if by $\val(\lder^k(a))\notin \val(\bE)$ for all $k<m$, then by Lemma~\ref{lem:essential_log-scales_are_dagger_seq}(2), there is an essential approximate $\partial$-logarithmic scale $(y_{k})_{k<m}$ with $\partial y_{0}\sim a=\partial x$.
    \end{proof}
\end{proposition}

  In the above setting, when $T$ is exponential, for $y\in\mathbb E\langle x\rangle$,
  \[
  y\text{ is of the form }g(z)\text{ for a }\mathbb E\text{-definable gne and }z\text{ with }\mathbf v(z-\mathbb E)\subseteq \mathbf v(\mathbb E)
  \]
  “if and only if there is a natural number $m$ such that $\mathbf v(\lder_x^m\partial_x(y))\in \mathbf v(\mathbb E)$.”

- The treatment isolates “general properties” of such $(\mathbb E\langle x\rangle,\cO_x,\partial_x)$ and introduces convexity conditions on derivations (order-convex / logarithmically order-convex; valuation-convex / logarithmically valuation-convex) as “traces of the mean-value theorem,” used to control behavior even when “$\mathrm{tp}(x/\mathbb E)$ is not a definable type.”

- Under “suitable convexity assumptions,” the condition $\mathbf v(x-\mathbb E)\subseteq \mathbf v(\mathbb E)$ is reformulated via an ‘absorbed’ condition on $\partial_x$.

\begin{definition}\label{def:absorbed-type-der}
    Let $(\bE_*, \cO_*)$ be a valued field, $\partial$ be an almost $\cO_*$-Lioville-convex derivation on $\bE_*$. Let $\bE\coloneqq\Kr(\partial)$.
    We will say that $\partial$ is \emph{weakly $\cO_*$-absorbed} if $\partial(\br_{\cO_*}(\partial))\prec 1$. We will say that it is \emph{$\cO_*$-absorbed} if furthermore there is $x\in \bE_* \setminus \bE$ such that $\val(x-\bE)\subseteq \val(\bE)$.
\end{definition}

  In particular, such absorbed derivations are said to be “at the center of several arguments.”

Biconditional or Equivalence
Universal–Existential

Question:
Let $T$ be an exponential o-minimal theory, and let $T_{\mathrm{convex}}$ be the complete theory of models $(\mathbb E,\mathcal O)$ of $T$ expanded by a predicate for a nontrivial $T$-convex valuation subring $\mathcal O$. For a $T$-definable function $f(\bar x,t)$, write
\[
\dagger_n f(\bar x,t)=\frac{\partial f(\bar x,t)/\partial t}{f(\bar x,t)}
\]
when defined and $\infty$ otherwise, and let
\[
G_m^f(\bar x,t):=\min\{\,|(\dagger_n^k f)(\bar x,t)|:k\le m\,\}.
\]
Call $T$ transserial if for every $n$ and every $(n+1)$-ary $T$-definable $f$ there exists $m\in\mathbb N$ such that
\[
T_{\mathrm{convex}}\models \forall \bar x\,\exists r\in\mathcal O\,\forall t\in\mathcal O\,(t>r\to G_m^f(\bar x,t)<1).
\]
A generalized nested exponential (gne) over $\mathbb E$ is a finite composition of translations by elements of $\mathbb E$, sign changes, and exponentials. An elementary extension $(\mathbb E,\mathcal O)\preceq(\mathbb E_*,\mathcal O_*)$ has the gne-property if for every $y\in\mathbb E_*$ there exist an $\mathbb E$-definable gne $g$ and some $z\in\mathbb E_*$ with $g(z)=y$ and $\mathbf v(z-\mathbb E)\subseteq \mathbf v(\mathbb E)$. A res-constructible extension is one admitting a $\operatorname{dcl}_T$-basis over $\mathbb E$ whose residues are $\operatorname{dcl}_T$-independent over the base residue field; a wim-constructible extension is one admitting an ordinal-indexed $\operatorname{dcl}_T$-basis such that each new element is a pseudolimit of a pseudo-Cauchy sequence over the previously generated structure with no pseudolimit there; and 1-$\operatorname{dcl}_T$-dimensional means $\operatorname{dcl}_T$-dimension $1$ over the base. If $b$ lies in an elementary extension and realizes the cut $\mathcal O<b<\mathbb E^{>\mathcal O}$, let $\mathbb E\langle b\rangle=\operatorname{dcl}_T(\mathbb E b)$, and write $y\equiv_{\mathbb E} g(b)$ to mean that $y$ and $g(b)$ have the same type over $\mathbb E$. Which statement is valid?

The following are equivalent for $T$: (1) $T$ is transserial; (2) $T$ is exponentially bounded and for every $(\mathbb E,\mathcal O)\models T_{\mathrm{convex}}$, every $b$ realizing $\mathcal O<b<\mathbb E^{>\mathcal O}$, and every $y\in\mathbb E\langle b\rangle$, there exists an $\mathbb E$-definable gne $g$ such that $y\equiv_{\mathbb E} g(b)$; (3) every 1-$\operatorname{dcl}_T$-dimensional wim-constructible elementary extension of models of $T_{\mathrm{convex}}$ has the gne-property; (4) every 1-$\operatorname{dcl}_T$-dimensional res-constructible elementary extension of models of $T_{\mathrm{convex}}$ has the gne-property; (5) every wim-constructible elementary extension of models of $T_{\mathrm{convex}}$ has the gne-property; and (6) every res-constructible elementary extension of models of $T_{\mathrm{convex}}$ has the gne-property.

Choices:
B. The following are equivalent for $T$: (1) $T$ is transserial; (2) for every $(\mathbb E,\mathcal O)\models T_{\mathrm{convex}}$, every $b$ realizing $\mathcal O<b<\mathbb E^{>\mathcal O}$, and every $y\in\mathbb E\langle b\rangle$, there exists an $\mathbb E$-definable gne $g$ such that $y\equiv_{\mathbb E} g(b)$; (3) every 1-$\operatorname{dcl}_T$-dimensional wim-constructible elementary extension of models of $T_{\mathrm{convex}}$ has the gne-property; (4) every 1-$\operatorname{dcl}_T$-dimensional res-constructible elementary extension of models of $T_{\mathrm{convex}}$ has the gne-property; (5) every wim-constructible elementary extension of models of $T_{\mathrm{convex}}$ has the gne-property; and (6) every res-constructible elementary extension of models of $T_{\mathrm{convex}}$ has the gne-property.
C. If $T$ is transserial, then every wim-constructible elementary extension of models of $T_{\mathrm{convex}}$ has the gne-property, and every res-constructible elementary extension of models of $T_{\mathrm{convex}}$ has the gne-property.
D. The following are equivalent for $T$: (1) $T$ is transserial; (2) $T$ is exponentially bounded and for every $(\mathbb E,\mathcal O)\models T_{\mathrm{convex}}$, every $b$ realizing $\mathcal O<b<\mathbb E^{>\mathcal O}$, and every $y\in\mathbb E\langle b\rangle$, there exists a single $\mathbb E$-definable gne $g$ depending only on $b$ such that $y\equiv_{\mathbb E} g(b)$ for all such $y$; (3) every 1-$\operatorname{dcl}_T$-dimensional wim-constructible elementary extension of models of $T_{\mathrm{convex}}$ has the gne-property; (4) every 1-$\operatorname{dcl}_T$-dimensional res-constructible elementary extension of models of $T_{\mathrm{convex}}$ has the gne-property; (5) every wim-constructible elementary extension of models of $T_{\mathrm{convex}}$ has the gne-property; and (6) every res-constructible elementary extension of models of $T_{\mathrm{convex}}$ has the gne-property.
E. The following are equivalent for $T$: (1) $T$ is transserial; (2) $T$ is exponentially bounded and for every $(\mathbb E,\mathcal O)\models T_{\mathrm{convex}}$, every $b$ realizing $\mathcal O<b<\mathbb E^{>\mathcal O}$, and every $y\in\mathbb E\langle b\rangle$, there exists an $\mathbb E$-definable gne $g$ such that $y=g(b)$; (3) every 1-$\operatorname{dcl}_T$-dimensional wim-constructible elementary extension of models of $T_{\mathrm{convex}}$ has the gne-property; (4) every 1-$\operatorname{dcl}_T$-dimensional res-constructible elementary extension of models of $T_{\mathrm{convex}}$ has the gne-property; (5) every wim-constructible elementary extension of models of $T_{\mathrm{convex}}$ has the gne-property; and (6) every res-constructible elementary extension of models of $T_{\mathrm{convex}}$ has the gne-property.