\begin{theorem}[The method of moment from \cite{bollobas2001random}, Theorem 1.23]\label{moment}
Let $\left\{(\Omega_i,\mathbb{P}_i)\right\}_{i\in \mathbb{N}}$ be a sequence of probability spaces.
For $m\in \mathbb{N}$, let $N_{1,i},\cdots,N_{m,i}:\Omega_i\to \mathbb{N}_0$ be random variables for all $i\in \mathbb{N}$
and suppose there exist $\lambda_1,\cdots,\lambda_m\in(0,\infty)$ such that:
$$\lim_{i\to \infty}\mathbb{E}[(N_{1,i})_{k_1}\cdots(N_{m,i})_{k_m}]=\lambda_1^{k_1}\cdots\lambda_m^{k_m}$$
for all $k_1\cdots k_m\in \mathbb{N}$. Then
$$\lim_{i\to \infty}\mathbb{P}[(N_{1,i})=n_1,\cdots,(N_{m,i})=n_m]=\prod^k_{j=1}\frac{\lambda_j^{n_j} e^{-\lambda_j}}{n_j!}.$$
That is, $(N_{1,i},\cdots,N_{m,i}):\Omega_i \to \mathbb{N}^m_0$ converges jointly in distribution to a vector which is independently Poisson distributed with means
$\lambda_1,\cdots,\lambda_m$.

\end{theorem}

\section{Some claims}\label{claim}
Before the proof of the main theorem, we have to make some claims which will simplify the situation.
We consider the stratum whose zero order is $O(1)$, and it inplies $|\kappa|=O(g)$.

\begin{clm} \label{clm1}
A closed saddle connection is non-separable.
\end{clm}
\begin{proof}
For a closed connection $\gamma$ of $(X,\omega)$, if it's separable, then it cuts $X$ into $X_1$ and $X_2$, and $\partial X_1=\partial X_2=\gamma$.
By Stokes theorem, $\int_\gamma \omega=0$, but since $\gamma$ is a saddle connection its length is $|\int_\gamma \omega|$, which is a contradiction.
\end{proof}

\begin{clm} \label{clm2}
The probability measure of the subset of $\H_g(\kappa)$ which has a closed saddle connection involving $q$ cylinders of multiplicity $p$
goes to $0$ as $g\to \infty$, where $p>1$ or $q>0$.
\end{clm}

\begin{proof}
From Theorem \ref{Siegel of closed},
the Siegel-Veech constant of configuration with closed saddle connection involving $q$ cylinders of multiplicity $p$ is $O(\frac{1}{g^{2p+q-2}})$ .
Denote by $\mathcal{C}^{p,q}_g(\kappa)[\frac{a}{\sqrt{g}},\frac{b}{\sqrt{g}}]$ the subset of $\H_g(\kappa)$ consisting of surfaces which have a closed saddle connection
of length in the interval $[\frac{a}{\sqrt{g}},\frac{b}{\sqrt{g}}]$
involving $q$ cylinders of multiplicity $p$.
Then for $p\geq 2$ or $q\geq 1$, we have
 $$\V(\mathcal{C}^{p,q}_g(\kappa)[\frac{a}{\sqrt{g}},\frac{b}{\sqrt{g}}])\leq |\kappa|O(\frac{1}{g^{q}})\frac{b^2-a^2}{g}\V(\H_g(\kappa))=O(\frac{1}{g})\V(\H_g(\kappa)$$
where the coefficient $|\kappa|$ is the selection of the zero that the closed saddle connection goes through.
So when $g\to \infty$ its probability measure goes to $0$
and it suffices to consider the closed saddle connection with multiplicity $1$ and no cylinders around it.
\end{proof}

\begin{clm} \label{clm3}
The angle of a closed saddle connection at the zero which it connects is odd multiples of $\pi$.
\end{clm}

\begin{proof}
For an Abelian differential $(X,\omega)$, if it has a closed saddle connection $\gamma$ connecting a zero $p$, if its angle at $p$ is $2k\pi$,
then the holonomy of $\gamma$ and $-\gamma$ have same direction. But
$$\int_\gamma \omega=-\int_{-\gamma} \omega.$$
It implies $\int_\gamma \omega=0$, which is a contradiction.
\end{proof}

\begin{clm} \label{clm4}
If a closed saddle connection has angle $\pi$ at one side, it must have a cylinder at the side.
\end{clm}

\begin{proof}
Let $\gamma$ be a colsed saddle connection on $(X,\omega)$ which has angle $\pi$ at one side. Since $(X,\omega)$ is oriented, its normal bundle of $\gamma$ at the side is oriented.
Consider the exponential map from the bundle to $(X,\omega)$, which is well-defined on $\gamma \times [0,s]$ for some $s$ small sufficiently and the image is a cylinder.

\end{proof}

Combine claim $2,3$ and $4$, we have that for a closed saddle connection $\gamma$ on $(X,\omega)$ in principal stratum,
since its total angle around the zero is $4\pi$ and has to be divided into odd multiples of $\pi$,
it must have angle $\pi$ at one side of $\gamma$ and $3\pi$ at another.
So there is a cyliner with $\gamma$ as a boundary. As for the other boundary, if it consists of some open saddle connections,
then these saddle connections have angles $\pi$. That means there are at least two non-homologous saddle connections have angle $\pi$,
which occurs on a set of measure zero.
So we only need to consider the cylinder bounded by curves homologous to $\gamma$.
Then from Claim \ref{clm2} the probability measure of $\mathcal{C}_g(1^{2g-2})[\frac{a}{\sqrt{g}},\frac{b}{\sqrt{g}}]$ has limit
 $$\lim_{g\to \infty}\frac{\V(\mathcal{C}_g(1^{2g-2})[\frac{a}{\sqrt{g}},\frac{b}{\sqrt{g}}])}{\V(\H_g(1^{2g-2})}=O(\frac{1}{g})\to 0.$$

So we need to consider the question on stratum with higher order zeros. Next we will consider $\H_g(2^{g-1})$ firstly.

\section{Surgery}\label{surgery}
This section we will introduce some surgeries that, when combined, can collapse a closed saddle connection.
\subsection{Open up a higher-order zero}
Masur, Rafi and Randecker introduce a surgery in \cite{masur2024lengths} which is a variation of a surgery from \cite{eskin2003moduli}.
This surgery can collapse a saddle connection which is not closed and has multiplicity $1$.

Let $(X,\omega)$ be an Abelian differential, $\sigma$ be an open saddle connection with endpoints $v_1$ and $v_2$,
whose orders are $n_1$ and $n_2$ respectively.
Since the total angle at $v_1$ is $2(n_1+1)\pi$,
one can extend $\sigma$ from $v_1$ along its direction to $v_3$ and denote by $\sigma'$ the geodesic segment $v_1v_3$
such that $\ell_\omega(\sigma)=\ell_\omega(\sigma')$
and the angle between $\sigma$ and $\sigma'$ is
$2k_1\pi$ and $2k_2\pi$, where $k_1+k_2=n_1+1$.
If $\sigma'$ does not go through other zeros of $(X,\omega)$,
which means $\sigma'$ is a ray from $v_1$, the following surgery can be carried out.

Cut along $\sigma+\sigma'$ and denote the two copies of $\sigma+\sigma'$ and $v_1$ by $\sigma^{\pm}+(\sigma')^{\pm}$ and $v_1^{\pm}$.
Then glue $\sigma^+$ and $(\sigma')^+$, $\sigma^-$ and $(\sigma')^-$.
This surgery reduces the order of $v_1$ and constructs a new Abelian differential $(X',\omega')$.
On $(X',\omega')$, the total angles at $v_1^+$, $v_1^-$ and $v_2$ are $2k_1\pi$, $2k_2\pi$, and $2(n_2+2)\pi$ respectively,
that is, the orders of $v_1^+$, $v_1^-$ and $v_2$ are $k_1-1$, $k_2-1$, and $n_2+1$.
Particularly, if $v_1$ is a simple zero, the surgery collapses the saddle connection. We call this surgery \emph{collapsing surgery}.
\begin{figure}[h]
    \centering
    \includegraphics[width=0.6\linewidth]{collapsing}
    \caption{The collapsing surgery}
    \label{collapsing}
\end{figure}

In our situation, we need to use the inverse of collapsing surgery which we call \emph{opening surgery}.
From Claim \ref{clm2}, it suffices to consider the subset $\mathcal{C}^{1,0}_g(2^{g-1})[\frac{a}{\sqrt{g}},\frac{b}{\sqrt{g}}]$.
For an Abelian differential $(X,\omega)$ in $\mathcal{C}^{1,0}_g(2^{g-1})[\frac{a}{\sqrt{g}},\frac{b}{\sqrt{g}}]$,
choose a closed saddle connection $\gamma$ with length in $[\frac{a}{\sqrt{g}},\frac{b}{\sqrt{g}}]$ at a double zero $p$.
We can reverse the surgery introduced above as follows.

Since the total angle at $p$ is $6\pi$, from Claim \ref{clm3} and Claim \ref{clm4}, the angles on both side of $\gamma$ at $p$ must be $3\pi$.
Fix the orientation of $\gamma$, choose two rays $\sigma^+$ and $\sigma^-$ from $p$ such that
$\ell_\omega(\sigma^+)=\ell_\omega(\sigma^-)=\ell_\omega(\gamma)$,
and the angle between $\gamma$ and $\sigma^+$
is $\pi$, the angle between $\gamma$ and $\sigma^-$ is $-\pi$, where the sign is consistent with the orientation of the surface.
Denote by $q^+$ and $q^-$ the other endpoints of $\sigma^+$ and $\sigma^-$.
Then cut along $\sigma^+ + \sigma^-$ and glue the two copies of $\sigma^+$ and $\sigma^-$.
After the surgery, the two copies of $p$ become a regular and a simple zero, $q^+$ and $q^-$ are glued to become a new simple zero $q$.

\begin{figure}[h]
   \centering
   \includegraphics[width=0.6\linewidth]{openinglocally}
  \caption{The opening surgery}
  \label{opening}
\end{figure}

This surgery is the inverse of the collapsing surgery above: the double zero is replaced by two simple zeros,
and it can be carried out except for the surgery locus goes through some zero,
that is, it has two non-homologous saddle connections with angle $2\pi$.
From period mapping, such subset has measure zero. We call the subset on which the surgery can be carried out \emph{permissible set},
and denote it by $\mathcal{C}^{1,0}_{g,Per}(2^{g-1})[\frac{a}{\sqrt{g}},\frac{b}{\sqrt{g}}]$, from the discussion above we have
\begin{proposition}
$\mathcal{C}^{1,0}_{g,Per}(2^{g-1})[\frac{a}{\sqrt{g}},\frac{b}{\sqrt{g}}]$ is a full measure subset of $\mathcal{C}^{1,0}_g(2^{g-1})[\frac{a}{\sqrt{g}},\frac{b}{\sqrt{g}}]$.
\end{proposition}

By the surgery of collapsing this map is one-to-one and denote the opening surgery by $F_1$.
\begin{figure}[h]
    \centering
    \includegraphics[width=0.6\linewidth]{opening}
    \caption{The resulting surface}
    \label{opening surface}
\end{figure}

\begin{remark}
We can also choose $\sigma^{\pm}$ to make the angle between $\gamma$ and them are $2\pi$,
it suffices to make sure the opening operation can obtain an abelian differential and a smooth loop homotopic to $\gamma$.
\end{remark}
\subsection{Move zero along closed curve and pinch}
We have constructed the mapping $F_1$,
which takes a closed saddle connection $\gamma$ at a double zero to two saddle connections $\gamma_1$ and $\gamma_2$
sharing same endpoints $p$ and $q$ which are both simple zeros.
Moreover, the angles between the two saddle connections at the two simple zeros (on both sides) are all $2\pi$,
and if we denote by $\ell_\omega(\gamma)=L$, we have $\ell_{\omega'}(\gamma_1)=L$, $\ell_{\omega'}(\gamma_2)=2L$, where $(X',\omega')=F_1(X,\omega)$.
If we want to collapse $\gamma_1$ and $\gamma_2$ simultaneously, we have to make a surgery to move $q$ along $\gamma_2$ to adjust the length of $\gamma_2$.

\subsubsection{Move the zero locally by period mapping.}

First we need to choose a special basis of $H^1(X,\Sigma;\mathbb{C})$, where $\Sigma=\{p,q,p_2,\cdots,p_{g-1}\}$
is the zero set of $(X',\omega')=F_1(X,\omega)$.
From Claim \ref{clm1}, the homology class $[\gamma]=[\gamma_1+\gamma_2]$ is non-separable.
So we can choose a basis of $H^1(X,\mathbb{C})$
$$(\alpha_1,\beta_1,\cdots,\alpha_g,\beta_g),$$
where $\alpha_1=[\gamma]$.

Next we choose the relative homology class
$$(pq,pp_2\cdots,pp_{g-1}),$$
where $[\gamma_1]=pq$, and $pp_i$ is freely homotopic to $0$ in $H^1(X,\mathbb{C})$.
Together they compose a basis of $H^1(X,\Sigma;\mathbb{C})$,
and their holonomy gives a period mapping around $(X',\omega')$ locally.

Let $\omega'(\alpha_1)=(x_1,y_1)$, $\omega'(\beta_1)=(x_2,y_2)$, $\omega'(\alpha_2)=(x_3,y_3)$
$\omega'(\beta_2)=(x_4,y_4)$.
Now choose a relative cohomology class $\upsilon \in H^1(X,\Sigma;\mathbb{C})$ ,
which can be considered as a tangent vector in $T_{\H_g(1,1,\cdots,2)}(X',\omega')$ such that
\begin{equation}\label{moving}
\upsilon(\alpha_1)=(-x_1,-y_1), \upsilon(\alpha_2)=(-\frac{y_2x_1}{y_4},\frac{x_2y_1}{x_4})
\end{equation}
and on the other basis we assign zero to $\upsilon$.

\begin{remark}\label{explain}
Here $\upsilon(\alpha_1)$ is chosen to guarantee that along the curve $(X',\omega')+t\upsilon$ in moduli space the zero moves along $\gamma$,
and $\upsilon(\alpha_2)$ is chosen to guarantee the resulting surface has area $1$, since the area of $(X,\omega)$ can be written by
$$\int_X|\omega|^2=\frac{i}{2}\sum_i(\int_{\alpha_i}\omega \int_{\beta_i}\overline{\omega}-\int_{\beta_i}\omega \int_{\alpha_i}\overline{\omega}).$$
\end{remark}

Consider the curve in $\H_g(1,1,2,,\cdots,2)$: $(X',\omega')+t\upsilon, t\in [0,1]$.
If $\gamma_1$ and $\gamma_2$ don't degenerate along the curve, then we obtain
$$F_2:\mathcal{C}^{1,0}_{g,Per}(2^{g-1})[\frac{a}{\sqrt{g}},\frac{b}{\sqrt{g}}]  \to \H_g(1^2,2^{g-1})$$
such that $F_2[(X,\omega)]=F_1[(X,\omega)]+\upsilon$.
From the construction above the resulting surface has two saddle connections
in the relative homology class $\gamma_1$ and $\gamma_2$ with equal length.

\subsubsection{$F_2$ is well-defined almost everywhere}
We have defined a mapping if the saddle connections $\gamma_1$ and $\gamma_2$ are preserved along the curve $(X',\omega')+t\upsilon, t\in [0,1]$
 on $\mathcal{C}^{1,0}_{g,Per}(2^{g-1})$.
Next we will see the mapping can be defined for almost every translation surface in $\mathcal{C}^{1,0}_{g,Per}(2^{g-1})$.

From above we know the moving surgery can be realized except for some of $\gamma_1$ or $\gamma_2$ degenerate along $(X',\omega')+t\upsilon, t\in [0,1]$.
Suppose $T$ is the first time when the geodesic in $[\gamma_1]$ can be represented by $pp_i-p_ip_j-\cdots-p_kq$,
since the saddle connection is smooth on $(X',\omega')+t\upsilon,t\in [0,T]$, the corner of $pp_i-p_ip_j-\cdots-p_kq$ must be $\pi$.
But the holonomy of $pp_i$ is not changed on $(X',\omega')+t\upsilon,t\in [0,T]$
This implies there exists some $pp_i$ such that its holonomy on $(X',\omega')$ has the same direction with $p_0p_1$.
And by the relation of holonomy under $F_1$, this implies $p_0p_i$ has the same direction with the closed curve on $(X,\omega)$.
Under period mapping, on each local chart this is a measure-zero set.
So all such $(X,\omega)$ is a subset of measure zero in $ \H_g(2^{g-1})$, also in $\mathcal{C}^{1,0}_{g,Per}(2^{g-1})[\frac{a}{\sqrt{g}},\frac{b}{\sqrt{g}}]$.

\subsubsection{Pinching}
On the subset of $\mathcal{C}^{1,0}_{g,Per}(2^{g-1})[\frac{a}{\sqrt{g}},\frac{b}{\sqrt{g}}]$ where $F_2$ is well-defined,
we can cut along $\gamma_1+\gamma_2$ and denote the two copies of $p$ and $q$ by $q^+$, $q^-$, $p^+$, $p^-$,
the two copies of $\gamma_1$ and $\gamma_2$ by $\gamma^+_1$, $\gamma^-_1$, $\gamma^+_2$, $\gamma^-_2$.
Notice that the total angles at $q^+$, $q^-$, $p^+$, $p^-$ are all $2\pi$,
which means $\gamma^\pm_1$ have same direction with $\gamma^\pm_2$,
then glue $\gamma^+_1$ with $\gamma^+_2$, $\gamma^-_1$ with $\gamma^-_2$, which make $q^+$, $q^-$, $p^+$, $p^-$ become regular points.
Since $\gamma_1+\gamma_2$ is non-separable, this surgery reduces the genus by one, and obtains a new abelian differential
on which we mark two points $p^+$ and $p^-$ so as to find its inverse.
Considering the choice of which zero is selected after applying the inverse map, we finally obtain:
$$F_3: \mathcal{C}^{1,0}_{g,Per}(2^{g-1})[\frac{a}{\sqrt{g}},\frac{b}{\sqrt{g}}]\to M\times \H_{g-1}(2^{g-2},0,0)\times A_{[\frac{a}{\sqrt{g}},\frac{b}{\sqrt{g}}]},$$
where $M$ is the combinatorial data of choosing one from $g-1$ zeros.

\begin{figure}[h]
    \centering
    \includegraphics[width=0.6\linewidth]{movingandcut}
    \caption{Moving and cut}
    \label{moving and cut}
\end{figure}

\begin{figure}[h]
    \centering
    \includegraphics[width=0.6\linewidth]{pinching}
    \caption{Pinching}
    \label{pinching}
\end{figure}

\subsubsection{$F_3$ is measure-preserving.}
This surgery can be inverse except for some directions which has saddle connections, which is a measure zero subset $A_0(X,\omega)$
in $A_{[\frac{a}{\sqrt{g}},\frac{b}{\sqrt{g}}]}$ for every surface in $\H_{g-1}(2^{g-2},0,0)$:
choose a vector $\kappa$ in $A_{[\frac{a}{\sqrt{g}},\frac{b}{\sqrt{g}}]}$,
consider the two rays from $p^+$ and $p^-$ whose holonomy is $\kappa$, if the rays exist, then cut the surface along the two rays
to obtain two closed loci and glue them.
Let
$$\tilde{\H}=\{(X,\omega, \tau): (X,\omega)\in \H_{g-1}(2^{g-2},0,0), \tau \in A_0(X,\omega)\},$$
which is a measure zero subset of $\H_{g-1}(2^{g-2},0,0)\times A_{[\frac{a}{\sqrt{g}},\frac{b}{\sqrt{g}}]}$.
So we have
$$F_3( \mathcal{C}^{1,0}_{g,Per}(2^{g-1}))\subset [\H_{g-1}(2^{g-2},0,0)\times (A_{[\frac{a}{\sqrt{g}},\frac{b}{\sqrt{g}}]})]\setminus \tilde{\H}.$$

Moreover, under local chart determined by the basis we choose,
$$F_3(z_1,\cdots,z_n)=(z_1,z_i+\upsilon(z_i)),$$
whose Jacobian has deteminant one and thus is measure-preserving.

\section{Collapse closed saddle connections simultaneously}
From the construction above we can collapse one closed saddle connection on some $(X,\omega)\in \mathcal{C}^{1,0}_{g}(2^{g-1}))$ and obtain a new surface
$(X',\omega')\in \H_{g-1}(2^{g-2},0,0)$. if we want to collapse any $k$ closed saddle connections,
we have to ensure the locus of surgery for these saddle connections are disjoint.
For a given interval $[\frac{a}{\sqrt{g}},\frac{b}{\sqrt{g}}]$,
define the \emph{exception set} $\mathcal{C}^{exc}_{g,Per}(2^{g-1})[\frac{a}{\sqrt{g}},\frac{b}{\sqrt{g}}]$ to be the subset of
$\mathcal{C}^{1,0}_{g,Per}(2^{g-1})[\frac{a}{\sqrt{g}},\frac{b}{\sqrt{g}}]$
such that on every $(X,\omega)\in \mathcal{C}^{exc}_{g,Per}(2^{g-1})[\frac{a}{\sqrt{g}},\frac{b}{\sqrt{g}}]$ there are two closed saddle connections with length in
$[\frac{a}{\sqrt{g}},\frac{b}{\sqrt{g}}]$ whose surgery loci intersect.
First we will show that the measure of exception set goes to zero as genus goes to infinity.

\subsection{The measure of exception set}

For $(X,\omega)\in \mathcal{C}^{exc}_{g,Per}(2^{g-1})[\frac{a}{\sqrt{g}},\frac{b}{\sqrt{g}}]$, there are three situations:
\begin{itemize}
  \item[1]  There exist two closed saddle connections on $(X,\omega)$ with lengths in $[\frac{a}{\sqrt{g}},\frac{b}{\sqrt{g}}]$
            that do not share a zero. And the loci of opening surgery intersect or the two closed saddle connections intersect.
  \item[2]  There exist two closed saddle connections on $(X,\omega)$ with lengths in $[\frac{a}{\sqrt{g}},\frac{b}{\sqrt{g}}]$ sharing one zero.
\end{itemize}

Note that in the first situation, we can find a curve connecting the two zeros with length no more than $\frac{2b}{\sqrt{g}}$.

Fix $B\in \mathbb{R}^+$, define $\mathcal{C}_g^{1}(\frac{B}{\sqrt{g}})$ be the set of $(X,\omega)\in \mathcal{C}^{1,0}_{g,Per}(2^{g-1})[0,\frac{B}{\sqrt{g}}]$
such that there exist two closed saddle connections on $(X,\omega)$
that do not share a zero, moreover the lengths of the two closed saddle connections and the distance between the two zeros are less than $\frac{B}{\sqrt{g}}$.
Define $\mathcal{C}_g^{2}(\frac{B}{\sqrt{g}})$ be the set
of $(X,\omega)\in \mathcal{C}^{1,0}_{g,Per}(2^{g-1})[0,\frac{B}{\sqrt{g}}]$ such that there exists two closed saddle connections with lengths less than $\frac{B}{\sqrt{g}}$ sharing one zero.
Obviously for $B\geq 2b$, we have
$$ \mathcal{C}^{exc}_{g,Per}(2^{g-1})[\frac{a}{\sqrt{g}},\frac{b}{\sqrt{g}}]\subset \mathcal{C}_g^{1}(\frac{B}{\sqrt{g}}) \cup \mathcal{C}_g^{2}(\frac{B}{\sqrt{g}}).$$

We need to compute the measure of $\mathcal{C}_g^{1}(\frac{B}{\sqrt{g}})$ and $\mathcal{C}_g^{2}(\frac{B}{\sqrt{g}})$ when $g\to \infty$.
First consider $\mathcal{C}_g^{2}(\frac{B}{\sqrt{g}})$, we have
\begin{proposition}\label{chain}
$$\lim_{g\to \infty}\frac{\V(\mathcal{C}_g^{2}(\frac{B}{\sqrt{g}}))}{\V(\H_g(2^{g-1}))} = 0.$$
\end{proposition}

\begin{proof}
For $(X,\omega) \in \mathcal{C}_g^{2}(\frac{B}{\sqrt{g}})$,
let $p$ be a zero on $(X,\omega)$ and $\gamma_1, \gamma_2$ be the closed saddle connections with lengths less than $\frac{B}{\sqrt{g}}$ at $p$.

As above we can collapse $\gamma_1$ and get a new translation surface in $\H_{g-1}(2^{g-2},0,0)$.
After the surgery $\gamma_2$ will become a segment connecting two marked regular points which are from the pinching surgery.
So the image of mapping $F_3$ on $\mathcal{C}_g^{2}(\frac{B}{\sqrt{g}})$ is in
$\H_{g-1,\frac{B}{\sqrt{g}}}(2^{g-2},0,0)\times D(\frac{B}{\sqrt{g}})$,
where $\H_{g-1,\frac{B}{\sqrt{g}}}(2^{g-2},0,0)$ is the subset that there exists a segment connnecting the two marked points of length less than
$\frac{B}{\sqrt{g}}$ and $D(\frac{B}{\sqrt{g}})$ is the disk of radius $\frac{B}{\sqrt{g}}$.
From \cite[Theorem 1.2]{aggarwal2019large}, the Siegel-Veech constant of saddle connections connecting two fix zeros of order $m_1$ and $m_2$ is
$$c=(m_1+1)(m_2+1)(1+O(\frac{1}{g})).$$
So we have
$$\V(\mathcal{C}_g^{2}(\frac{B}{\sqrt{g}}))\leq (g-1)\frac{B^2}{g}\V(\H_{g-1,\frac{B}{\sqrt{g}}}(2^{g-2},0,0)) \leq c(g-1)\frac{B^4}{g^2}\V(\H_{g-1}(2^{g-2},0,0)),$$
where the coefficient $g-1$ is the choice of $p$.
Then by \ref{Volume}, we have
$$\lim_{g\to \infty}\frac{\V(\mathcal{C}_g^{2}(\frac{B}{\sqrt{g}}))}{\V(\H_g(2^{g-1}))}\leq \lim_{g\to \infty}3c\frac{B^4}{g}=0$$
\end{proof}

For $\mathcal{C}_g^{1}(\frac{B}{\sqrt{g}})$ we also have
\begin{proposition}\label{intersect}
$$\lim_{g\to \infty}\frac{\V(\mathcal{C}_g^{1}(\frac{B}{\sqrt{g}}))}{\V(\H_g(2^{g-1}))} = 0.$$
\end{proposition}

\begin{proof}
For $(X,\omega) \in \mathcal{C}_g^{1}(\frac{B}{\sqrt{g}})$, let $p_1, p_2$ be the two zeros on $(X,\omega)$ with distance less than $\frac{B}{\sqrt{g}}$
and $\gamma_1, \gamma_2$ be the two closed saddle connections connecting $p_1, p_2$.
For every $k\in \mathbb{N}^+$, denote by $\mathcal{C}_g^{1,k}(\frac{B}{\sqrt{g}})$ the subset of $\mathcal{C}_g^{1}(\frac{B}{\sqrt{g}})$
that the geodesic between $p_1$ and $p_2$ is a concatenation of $k$ open saddle connections.
Their lengths are less than $\frac{B}{\sqrt{g}}$.
Choose the shortest saddle connection, using the surgery in \cite{masur2024lengths} we can collapse it and get a new translation surface in
$\H_g(3,1,2^{g-3})$. This is because the locus, that is the extension of the shortest saddle connection can not intersect the whole geodesic.
Repeat the surgery until $p_1$ and $p_2$ are collapsed to one zero and we get a map
$$F:\mathcal{C}_g^{1,k}(\frac{B}{\sqrt{g}})\to \mathcal{C}_g^{2}(2+k,1^k,2^{g-2-k})(\frac{B}{\sqrt{g}})\times D^k(\frac{B}{\sqrt{g}}),$$
where $\mathcal{C}_g^{2}(2+k,1^k,2^{g-2-k})(\frac{B}{\sqrt{g}})$ is the subset of $\H_g(2+k,1^k,2^{g-2-k})$ such that there are
two closed saddle connections $\gamma_1, \gamma_2$ connecting the zero of order $2+k$ with lengths less than $\frac{B}{\sqrt{g}}$ .

Similarly one can collapse $\gamma_1$ and the other will become a saddle connection connecting two zeros.
Note that the order is larger than $2$, although the collapsing surgery can also be realized, we need to require the angles
on the two sides of the closed saddle connections collapsed, which decide the order of zeros after collapsing.
We will explain this situation in detail in Section \ref{general case}.

Let $\mathcal{C}_g^{2,b',b''}(2+k,1^k,2^{g-2-k})(\frac{B}{\sqrt{g}})$ be the subset of
$\mathcal{C}_g^{2}(2+k,1^k,2^{g-2-k})(\frac{B}{\sqrt{g}})$ such that
the two angles on both sides of $\gamma_1$ are $(2b'+1)\pi$ and $(2b''+1)\pi$, where $b'+b''=2+k$.
Then after collapsing we get a map
$$F':\mathcal{C}_g^{2,b',b''}(2+k,1^k,2^{g-2-k})(\frac{B}{\sqrt{g}}) \to \H_{g-1,\frac{B}{\sqrt{g}}}(b'-1,b''-1,1^k,2^{g-2-k})\times D(\frac{B}{\sqrt{g}}).$$

Similar to Proposition \ref{chain}, we have
$$\V(\mathcal{C}_g^{2,b',b''}(2+k,1^k,2^{g-2-k})(\frac{B}{\sqrt{g}}))\leq b'b''\frac{B^4}{g^2}\V(\H_{g-1}(b'-1,b''-1,1^k,2^{g-2-k})).$$

Sum all $k$ and $(b',b'')$ we have
$$\V(\mathcal{C}_g^{1}(\frac{B}{\sqrt{g}}))\leq (g-1)(g-2)\sum_k \sum_{(b',b'')}b'b''\frac{B^4}{g^2}\V(\H_{g-1}(b'-1,b''-1,1^k,2^{g-2-k}))(\frac{B^2}{g})^k,$$
where the coefficient $(g-1)(g-2)$ is the choice of $p_1$ and $p_2$.
Then when $g\to \infty$ we have
$$\frac{\V(\mathcal{C}_g^{1}(\frac{B}{\sqrt{g}}))}{\V(\H_g(2^{g-1}))}\leq
\sum_k \sum_{(b',b'')}b'b''B^4\frac{\V(\H_{g-1}(b'-1,b''-1,1^k,2^{g-2-k}))}{\V(\H_g(2^{g-1}))}(\frac{B^2}{g})^k=O(\frac{1}{g}).$$
\end{proof}

Combine the two proposition we have
\begin{corollary}
$$\frac{\V(\mathcal{C}^{exc}_{g,Per}(2^{g-1})[\frac{a}{\sqrt{g}},\frac{b}{\sqrt{g}}])}{\V(\H_g(2^{g-1}))}=O(\frac{1}{g}).$$
\end{corollary}

\subsection{Collapsing simultaneously}
Now fix a positive integer $K$ and $n_1,\cdots,n_k$ a partition of $K$.
We want to compute the limit of the expectation
$$(L_{g,[a_1,b_1]})_{n_1}\cdots (L_{g,[a_k,b_k]})_{n_k},$$
for which we will collapse $K$ closed saddle connections in order.

Define $\hat{\mathcal{C}}^{1,0}_{g}(2^{g-1})=(X,\omega,\Gamma_1,\cdots,\Gamma_k)$,
where $(X,\omega)\in \mathcal{C}^{1,0}_{g}(2^{g-1})$,
and $\Gamma_i$ is an ordered list of $n_i$ closed saddle connections $(\gamma_{1,i},\cdots,\gamma_{n_i,i})$
with lengths in $[\frac{a_i}{\sqrt{g}},\frac{b_i}{\sqrt{g}}]$.
Suppose $a_1\leq b_1 \leq a_2\leq b_2 \cdots \leq a_k \leq b_k$, let $B=b_k$.
Then we have
$$\bigcup_i \mathcal{C}^{exc}_{g,Per}(2^{g-1})[\frac{a_i}{\sqrt{g}},\frac{b_i}{\sqrt{g}}]\subset \mathcal{C}_g^{1}(\frac{B}{\sqrt{g}})\cup \mathcal{C}_g^{2}(\frac{B}{\sqrt{g}}).$$

Let $\mathcal{C}'_{g,Per}=\mathcal{C}^{1,0}_{g,Per}(2^{g-1})\setminus (\mathcal{C}_g^{1}(\frac{B}{\sqrt{g}})\cup \mathcal{C}_g^{2}(\frac{B}{\sqrt{g}}))$,
we will adjust the surgery from last section on $\mathcal{C}'_{g,Per}$ to collapse the closed saddle connections in
$\Gamma_1,\cdots,\Gamma_k$ in order.
The opening and pinching operations are same to the situation of one curve. The only modification is the equation \ref{moving}:
by remark \ref{explain}, we need to choose the moving vector $\upsilon$ to ensure the area unchanged,
hence we can choose a basis containing $\gamma'_1,\gamma_1\cdots,\gamma'_K,\gamma_K$,
where $\gamma'_i$ is a saddle connection from opening $\gamma_i$.
Then the moving vector $\upsilon$ can be constructed like equation \ref{moving}: after defining the value on $\gamma_i$,
we can choose another relative homotopy class and give a value of $\upsilon$ on it to ensure the area unchanged.
This means we open and move the closed saddle connections simultaneously and then pinch them.

Finally  we obtain a mapping
$$\hat{F}: \hat{\mathcal{C}}'_{g,Per}(2^{g-1}) \to M_K\cdot \tilde{\H}_{g-K}(2^{g-1-K},0^{2K}) \times \prod^k_{i=1} (A_{[\frac{a_i}{\sqrt{g}},\frac{b_i}{\sqrt{g}}]})^{n_i},$$
where $\hat{\mathcal{C}}'_{g,Per}(2^{g-1})$ is defined as $\hat{\mathcal{C}}^{1,0}_{g}(2^{g-1})$
and $M_K$ is a combinatorial data consisting of choosing $K$ ordered zeros from $g-1$ zeros to label. As the discussion above,
we have
\begin{equation*}
\begin{aligned}
&\int_{\hat{\mathcal{C}}'_{g,Per}(2^{g-1})}(L_{g,[a_1,b_1]})_{n_1}\cdots (L_{g,[a_k,b_k]})_{n_k}d\mu_{MV}\\&=\V(\hat{\mathcal{C}}'_{g}(2^{g-1})))
\\&=|M_K|\prod^k_{i=1}[\frac{\pi(b^2-a^2)}{g}]^{n_i}\V(\H_{g-k}(2^{g-1-K}))\\
&\to \prod^k_{i=1}[\pi(b_i^2-a_i^2)]^{n_i}\V(\H_{g-K}(2^{g-1-K})), g\to \infty,
\end{aligned}
\end{equation*}
where the limit is because $|M_K|=(g-1)\cdots (g-K)$ and
$$\lim_{g\to \infty}\frac{(g-1)\cdots (g-K)}{g^{n_1+\cdots+n_k}}=\frac{(g-1)\cdots (g-K)}{g^K}=1.$$

Let $L'_{g,[a_i,b_i]}$ be the restriction of $L_{g,[a_i,b_i]}$ on $\mathcal{C}'_{g,Per}(2^{g-1})$.
Then the limit of factorial moment of $(L'_{g,[a_i,b_i]})_i$ is
$$\lim_{g\to \infty}\mathbb{E}[(L'_{g,[a_1,b_1]})_{n_1}\cdots (L'_{g,[a_k,b_k]})_{n_k}]=\lim_{g\to \infty}\frac{\V(\H_{g-K}(2^{g-1-K}))}{\V(\mathcal{C}'_{g,Per}(2^{g-1}))}\prod^k_{i=1}[\pi(b_i^2-a_i^2)]^{n_i}.$$
From Proposition \ref{chain} and \ref{intersect}, we have
$$\lim_{g\to \infty}\frac{\V(\mathcal{C}'_{g,Per}(2^{g-1}))}{\V(\H_g(2^{g-1}))}=1.$$

From Theorem \ref{Volume}, we have
$$\lim_{g\to \infty}\frac{\V(\H_{g-K}(2^{g-1-K}))}{\V(\H_{g}(2^{g-1}))}=3^K.$$
So we have
$$\lim_{g\to \infty}\mathbb{E}[(L'_{g,[a_1,b_1]})_{n_1}\cdots (L'_{g,[a_k,b_k]})_{n_k}]=\prod^k_{i=1}[3\pi(b_i^2-a_i^2)]^{n_i}.$$
By theorem \ref{moment}, $(L'_{g,[a_1,b_1]},\cdots,L'_{g,[a_k,b_k]})$ converges jointly in distribution to a vector of random variables with Poisson distributions of means $\lambda_{[a_i,b_i]}$, where
$$\lambda_{[a_i,b_i]}=3\pi(b_i^2-a_i^2).$$

From the property of Poisson distribution, we have
$$\mathbb{P}(L'_{g,[a_1,b_1]}=n_1,\cdots,L'_{g,[a_k,b_k]}=n_k)=\prod_{i=1}^k\frac{\lambda_{[a_i,b_i]}^{n_i}e^{-\lambda_{[a_i,b_i]}}}{n_i!}.$$
On the other hand
$$\mathbb{P}(L'_{g,[a_1,b_1]}=n_1,\cdots,L'_{g,[a_k,b_k]}=n_k)=\frac{\V(\{(X,\omega):L'_{g,[a_i,b_i]}(X,\omega)=n_i,i=1,\cdots,k\})}{\V(\mathcal{C}'_{g,Per}(2^{g-1}))}.$$
Note that
\begin{equation*}
\begin{aligned}
&\V(\{(X,\omega):L_{g,[a_i,b_i]}(X,\omega)=n_i,i=1,\cdots,k\})-
\V(\mathcal{C}_g^{1}(\frac{B}{\sqrt{g}})\cup \mathcal{C}_g^{2}(\frac{B}{\sqrt{g}}))\\ &\leq \V(\{(X,\omega):L'_{g,[a_i,b_i]}(X,\omega)=n_i,i=1,\cdots,k\})\\&\leq
\V(\{(X,\omega):L_{g,[a_i,b_i]}(X,\omega)=n_i,i=1,\cdots,k\}).
\end{aligned}
\end{equation*}
From Proposition \ref{chain} and \ref{intersect}, we have
$$\lim_{g\to \infty}\frac{\V(\mathcal{C}'_{g,Per}(2^{g-1}))}{\V(\H_g(2^{g-1}))}=1,\\ \lim_{g\to \infty}\frac{\V(\mathcal{C}_g^{1}(\frac{B}{\sqrt{g}})\cup \mathcal{C}_g^{2}(\frac{B}{\sqrt{g}}))}{\V(\H_g(2^{g-1}))}=0.$$
So
\begin{equation*}
\begin{aligned}
&\mathbb{P}(L_{g,[a_1,b_1]}=n_1,\cdots,L_{g,[a_k,b_k]}=n_k)-
\frac{\V(\mathcal{C}_g^{1}(\frac{B}{\sqrt{g}})\cup \mathcal{C}_g^{2}(\frac{B}{\sqrt{g}}))}{\V(\H_g(2^{g-1}))}\\ &\leq \mathbb{P}(L'_{g,[a_1,b_1]}=n_1,\cdots,L'_{g,[a_k,b_k]}=n_k)\frac{\V(\mathcal{C}'_{g,Per}(2^{g-1}))}{\V(\H_g(2^{g-1}))}\\&\leq
\mathbb{P}(L_{g,[a_1,b_1]}=n_1,\cdots,L_{g,[a_k,b_k]}=n_k).
\end{aligned}
\end{equation*}

Let $g \to \infty$ we have
$$\mathbb{P}(L_{g,[a_1,b_1]}=n_1,\cdots,L_{g,[a_k,b_k]}=n_k)=\mathbb{P}(L'_{g,[a_1,b_1]}=n_1,\cdots,L'_{g,[a_k,b_k]}=n_k).$$
So $(L_{g,[a_1,b_1]},\cdots,L_{g,[a_k,b_k]})$ also converges jointly in distribution to a vector of random variables
with Poisson distributions of means $\lambda_{[a_i,b_i]}$, where
$$\lambda_{[a_i,b_i]}=3\pi(b_i^2-a_i^2).$$

Since we don't need to deal with simple zeros in the collapsing surgery,
samilarly to the proof of theorem \ref{closed}, we can prove Corollary \ref{general}.

\section{General stratum}\label{general case}
This section we consider the stratum $\H_g(m^{O(g)},1^{2g-2-mO(g)})$, where $m\geq 3$.

If a stratum has zeros of order more than $3$, the angles at both sides are not certain.
But we can consider some fixed configuration of closed saddle connection.
For a configuration $(J,b'_k,b''_k,a'_i,a''_i)$ of closed saddle connection.
We have known it suffices to consider multiplicity $1$ without cylinders, then the configuration become $(b',b'')$, where $b'+b''=m$,
which means the angles of the closed saddle connection are $(2b'+1)\pi$ on one side and $(2b''+1)\pi$ on the other side.
Now we choose all such configuration at each $m$-order zeros.

For $k$ such closed saddle connections, we can use the surgery in Section \ref{surgery} to collapse them.
Note that in this situation the opening surgery will replace the zero of order $m$ to two zeros of order $1$ and $m-1$,
and the angles at both sides of the zero of $m-1$ order are $(2b')\pi$ and $(2b'')\pi$. Then after moving and pinching,
the simple zero become two regular points and the zero of $m-1$ order become two zeros of order $b'-1$ and $b''-1$.
Finally we obtain a mapping
$$\hat{F}: \hat{\mathcal{C}}'_{g,Per}(m^{O(g)},1^{O'(g)})\to \tilde{\H}_{g-K}((b'-1)^K,(b''-1)^K,m^{O(g)-K},1^{O'(g)})\times \prod^k_{i=1} (A_{[\frac{a_i}{\sqrt{g}},\frac{b_i}{\sqrt{g}}]})^{n_i},$$
where $O'(g)=2g-2-mO(g)$.
Again we have
$$\lim_{g\to \infty}\mathbb{E}[(L'_{g,[a_1,b_1]})_{n_1}\cdots (L'_{g,[a_k,b_k]})_{n_k}]=\prod^k_{i=1}(\frac{(m+1)\pi(b_i^2-a_i^2)}{b'b''})^{n_i}.$$

And similar to the proof of theorem \ref{closed} we have
\begin{theorem}
For the stratum $\H(m^{O(g)},1^{2g-2-mO(g)})$, a configuration $(b',b'')$ and $a,b\in \mathbb{R}$,
let the random variable
$$L_{\mathcal{C},g,[a,b]}:\H(m^{O(g)},1^{2g-2-mO(g)}) \to \mathbb{N}_0$$
be the number of closed saddle connections on $(X,\omega)$ satisfying the configuration with lengths in $[\frac{a}{\sqrt{g}},\frac{b}{\sqrt{g}}]$.
Then for disjoint intervals $[a_1,b_1],\cdots,[a_k,b_k]$, when $g\to \infty$, the random variable sequence
$$(L_{\mathcal{C},g,[a_1,b_1]},\cdots,L_{\mathcal{C},g,[a_k,b_k]}):\H(m^{O(g)},1^{2g-2-mO(g)}) \to \mathbb{N}^k_0$$
converges jointly in distribution to a vector of random variables with Poisson distributions of means $\lambda_{[a_i,b_i]}$, where
$$\lambda_{[a_i,b_i]}=\frac{(m+1)\pi(b_i^2-a_i^2)}{b'b''}.$$

\end{theorem}

In 2024, Masur, Rafi and Randecker study the distribution of open saddle connections on a random translation surface in \cite{masur2024lengths}.
In this paper we study the distribution of closed saddle connections.

A translation surface can be denoted by $(X,\omega)$ where $X$ is a closed Riemann surface of genus $g$ and $\omega$ is an Abelian differential on $X$.
$(X,\omega)$ has a flat structure defined by the conical metric $|\omega|$, with singularities at the zeros of $\omega$.
If an $|\omega|$-geodesic of $(X,\omega)$ connects two zeros (or one zero to itself) of $\omega$ and has no other zeros in its interior,
we call it an open (or closed) saddle connection.

Let $\H_g$ be the moduli space of unit-area translation surfaces of genus $g$.
$\H_g$ has a natural stratification according to the zero order of Abelian differentials.
Particularly, the stratum consisting of all abelian differentials in $\H_g$ which have only simple zeros is called principal stratum,
denoted by $\H_g(1^{2g-2})$. For convenience, if a stratum consists of Abelian differentials on which the number of zeros of order $d_i$ is $\kappa_i$
we denote it by
$$\H_g(d_1^{\kappa_1},\cdots,d_k^{\kappa_k})$$
where $\sum^{k}_{i=1}\kappa_id_i=2g-2$.

Together with a finite natural measure which is introduced by Masur and Veech denoted by $\mu_{MV}$,
every stratum $\H_g(\kappa)$ of $\H_g$ becomes a probability space
with probability measure $\frac{\mu_{MV}}{\operatorname{Vol}{\H_g(\kappa)}}$, where $\operatorname{Vol}{\H_g(\kappa)}$ is the measure of total stratum,
which we call the Masur-Veech volume of $\H_g(\kappa)$.

Given $(X,\omega)\in \H_g(1^{2g-2})$ and $a,b\in \mathbb{R}^+$,
let $N_{g,[a,b]}(X,\omega)$ be the number of open saddle connections with lengths in the interval $[\frac{a}{g},\frac{b}{g}]$ on $(X,\omega)$.
Then $N_{g,[a,b]}:\H_g(1^{2g-2})\to \mathbb{N}_0$ becomes a random variable.
In \cite{masur2024lengths}, Masur, Rafi and Randecker proved :

Together with a finite natural measure which is introduced by Masur and Veech denoted by $\mu_{MV}$,
every stratum $\H_g(\kappa)$ of $\H_g$ becomes a probability space
with probability measure $\frac{\mu_{MV}}{\V{\H_g(\kappa)}}$, where $\V{\H_g(\kappa)}$ is the measure of total stratum,
which we call the Masur-Veech volume of $\H_g(\kappa)$.

Their method is based on the relationship between Poisson distribution and its factorial moment \cite{bollobas2001random},
which was used first in the work of Mirzakhani–Petri \cite{mirzakhani2019lengths} to study the distribution of closed hyperbolic geodesics on random hyperbolic surfaces in Teichm\"uller space with respect to Weil-Peterson measure.

\begin{theorem}\label{closed}
For the stratum $\H(2^{g-1})$ and disjoint intervals $[a_1,b_1],\cdots,[a_k,b_k]$, when $g\to \infty$, the random variable sequence
$$(L_{g,[a_1,b_1]},\cdots,L_{g,[a_k,b_k]}):\H(2^{g-1}) \to \mathbb{N}^k_0$$
converges jointly in distribution to a vector of random variables with Poisson distributions of means $\lambda_{[a_i,b_i]}$, where
$$\lambda_{[a_i,b_i]}=3\pi(b_i^2-a_i^2).$$
\end{theorem}
\begin{remark}
We will expalin the question is trivial for the principal stratum in Section \ref{claim}:
the probability measure of $\mathcal{C}_g(1^{2g-2})[\frac{a}{\sqrt{g}},\frac{b}{\sqrt{g}}]$ is zero when $g\to \infty$.
For principal stratum, the length interval to be considered should be $[a,b]$.
\end{remark}

Recall that a random variable $X:\Omega \to \mathbb{N}_0$ is Poisson distributed with mean $\lambda$ if
$$\mathbb{P}(X=k)=\frac{\lambda^k e^{-\lambda}}{k!}.$$
The following theorem from \cite{bollobas2001random} gives the relationship between factorial moment and distribution for Poisson distribution:
\begin{theorem}[The method of moment from \cite{bollobas2001random}, Theorem 1.23]\label{moment}
Let $\left\{(\Omega_i,\mathbb{P}_i)\right\}_{i\in \mathbb{N}}$ be a sequence of probability spaces.
For $m\in \mathbb{N}$, let $N_{1,i},\cdots,N_{m,i}:\Omega_i\to \mathbb{N}_0$ be random variables for all $i\in \mathbb{N}$
and suppose there exist $\lambda_1,\cdots,\lambda_m\in(0,\infty)$ such that:
$$\lim_{i\to \infty}\mathbb{E}[(N_{1,i})_{k_1}\cdots(N_{m,i})_{k_m}]=\lambda_1^{k_1}\cdots\lambda_m^{k_m}$$
for all $k_1\cdots k_m\in \mathbb{N}$. Then
$$\lim_{i\to \infty}\mathbb{P}[(N_{1,i})=n_1,\cdots,(N_{m,i})=n_m]=\prod^k_{j=1}\frac{\lambda_j^{n_j} e^{-\lambda_j}}{n_j!}.$$
That is, $(N_{1,i},\cdots,N_{m,i}):\Omega_i \to \mathbb{N}^m_0$ converges jointly in distribution to a vector which is independently Poisson distributed with means
$\lambda_1,\cdots,\lambda_m$.

From Theorem \ref{Volume}, we have
$$\lim_{g\to \infty}\frac{\V(\H_{g-K}(2^{g-1-K}))}{\V(\H_{g}(2^{g-1}))}=3^K.$$
So we have
$$\lim_{g\to \infty}\mathbb{E}[(L'_{g,[a_1,b_1]})_{n_1}\cdots (L'_{g,[a_k,b_k]})_{n_k}]=\prod^k_{i=1}[3\pi(b_i^2-a_i^2)]^{n_i}.$$
By theorem \ref{moment}, $(L'_{g,[a_1,b_1]},\cdots,L'_{g,[a_k,b_k]})$ converges jointly in distribution to a vector of random variables with Poisson distributions of means $\lambda_{[a_i,b_i]}$, where
$$\lambda_{[a_i,b_i]}=3\pi(b_i^2-a_i^2).$$

From the property of Poisson distribution, we have
$$\mathbb{P}(L'_{g,[a_1,b_1]}=n_1,\cdots,L'_{g,[a_k,b_k]}=n_k)=\prod_{i=1}^k\frac{\lambda_{[a_i,b_i]}^{n_i}e^{-\lambda_{[a_i,b_i]}}}{n_i!}.$$
On the other hand
$$\mathbb{P}(L'_{g,[a_1,b_1]}=n_1,\cdots,L'_{g,[a_k,b_k]}=n_k)=\frac{\V(\{(X,\omega):L'_{g,[a_i,b_i]}(X,\omega)=n_i,i=1,\cdots,k\})}{\V(\mathcal{C}'_{g,Per}(2^{g-1}))}.$$
Note that
\begin{equation*}
\begin{aligned}
&\V(\{(X,\omega):L_{g,[a_i,b_i]}(X,\omega)=n_i,i=1,\cdots,k\})-
\V(\mathcal{C}_g^{1}(\frac{B}{\sqrt{g}})\cup \mathcal{C}_g^{2}(\frac{B}{\sqrt{g}}))\\ &\leq \V(\{(X,\omega):L'_{g,[a_i,b_i]}(X,\omega)=n_i,i=1,\cdots,k\})\\&\leq
\V(\{(X,\omega):L_{g,[a_i,b_i]}(X,\omega)=n_i,i=1,\cdots,k\}).
\end{aligned}
\end{equation*}
From Proposition \ref{chain} and \ref{intersect}, we have
$$\lim_{g\to \infty}\frac{\V(\mathcal{C}'_{g,Per}(2^{g-1}))}{\V(\H_g(2^{g-1}))}=1,\\ \lim_{g\to \infty}\frac{\V(\mathcal{C}_g^{1}(\frac{B}{\sqrt{g}})\cup \mathcal{C}_g^{2}(\frac{B}{\sqrt{g}}))}{\V(\H_g(2^{g-1}))}=0.$$
So
\begin{equation*}
\begin{aligned}
&\mathbb{P}(L_{g,[a_1,b_1]}=n_1,\cdots,L_{g,[a_k,b_k]}=n_k)-
\frac{\V(\mathcal{C}_g^{1}(\frac{B}{\sqrt{g}})\cup \mathcal{C}_g^{2}(\frac{B}{\sqrt{g}}))}{\V(\H_g(2^{g-1}))}\\ &\leq \mathbb{P}(L'_{g,[a_1,b_1]}=n_1,\cdots,L'_{g,[a_k,b_k]}=n_k)\frac{\V(\mathcal{C}'_{g,Per}(2^{g-1}))}{\V(\H_g(2^{g-1}))}\\&\leq
\mathbb{P}(L_{g,[a_1,b_1]}=n_1,\cdots,L_{g,[a_k,b_k]}=n_k).
\end{aligned}
\end{equation*}

Let $g \to \infty$ we have
$$\mathbb{P}(L_{g,[a_1,b_1]}=n_1,\cdots,L_{g,[a_k,b_k]}=n_k)=\mathbb{P}(L'_{g,[a_1,b_1]}=n_1,\cdots,L'_{g,[a_k,b_k]}=n_k).$$
So $(L_{g,[a_1,b_1]},\cdots,L_{g,[a_k,b_k]})$ also converges jointly in distribution to a vector of random variables
with Poisson distributions of means $\lambda_{[a_i,b_i]}$, where
$$\lambda_{[a_i,b_i]}=3\pi(b_i^2-a_i^2).$$

And similar to the proof of theorem \ref{closed} we have
\begin{theorem}
For the stratum $\H(m^{O(g)},1^{2g-2-mO(g)})$, a configuration $(b',b'')$ and $a,b\in \mathbb{R}$,
let the random variable
$$L_{\mathcal{C},g,[a,b]}:\H(m^{O(g)},1^{2g-2-mO(g)}) \to \mathbb{N}_0$$
be the number of closed saddle connections on $(X,\omega)$ satisfying the configuration with lengths in $[\frac{a}{\sqrt{g}},\frac{b}{\sqrt{g}}]$.
Then for disjoint intervals $[a_1,b_1],\cdots,[a_k,b_k]$, when $g\to \infty$, the random variable sequence
$$(L_{\mathcal{C},g,[a_1,b_1]},\cdots,L_{\mathcal{C},g,[a_k,b_k]}):\H(m^{O(g)},1^{2g-2-mO(g)}) \to \mathbb{N}^k_0$$
converges jointly in distribution to a vector of random variables with Poisson distributions of means $\lambda_{[a_i,b_i]}$, where
$$\lambda_{[a_i,b_i]}=\frac{(m+1)\pi(b_i^2-a_i^2)}{b'b''}.$$

Their method is based on the relationship between Poisson distribution and its factorial moment \cite{bollobas2001random},
which was used first in the work of Mirzakhani–Petri \cite{mirzakhani2019lengths} to study the distribution of closed hyperbolic geodesics on random hyperbolic surfaces in Teichm\"uller space with respect to Weil-Peterson measure.

As for translation surface, to use the method,
\cite{masur2024lengths} introduces a surgery to collapse the open saddle connections to obtain a new general translation surface in a new stratum.
This surgery does not work for closed saddle connections and they proposed the distribution question in the situation of closed saddle connections,
which inspires this work.
This paper gives a new surgery to collapse a closed saddle connection, which ensures the same method can be used for closed saddle connections.

Let $L_{g,[a,b]}(X,\omega)$ be the number of closed saddle connections on $(X,\omega)$ with lengths in $[\frac{a}{\sqrt{g}},\frac{b}{\sqrt{g}}]$.
Denote by $\mathcal{C}_g(\kappa)[\frac{a}{\sqrt{g}},\frac{b}{\sqrt{g}}]$ the subset of $\H_g(\kappa)$ consisting of translation
surfaces on which there exist closed saddle connections
with lengths in $[\frac{a}{\sqrt{g}},\frac{b}{\sqrt{g}}]$.
The main result of this paper is the following theorem

\begin{theorem}\label{closed}
For the stratum $\H(2^{g-1})$ and disjoint intervals $[a_1,b_1],\cdots,[a_k,b_k]$, when $g\to \infty$, the random variable sequence
$$(L_{g,[a_1,b_1]},\cdots,L_{g,[a_k,b_k]}):\H(2^{g-1}) \to \mathbb{N}^k_0$$
converges jointly in distribution to a vector of random variables with Poisson distributions of means $\lambda_{[a_i,b_i]}$, where
$$\lambda_{[a_i,b_i]}=3\pi(b_i^2-a_i^2).$$
\end{theorem}
\begin{remark}
We will expalin the question is trivial for the principal stratum in Section \ref{claim}:
the probability measure of $\mathcal{C}_g(1^{2g-2})[\frac{a}{\sqrt{g}},\frac{b}{\sqrt{g}}]$ is zero when $g\to \infty$.
For principal stratum, the length interval to be considered should be $[a,b]$.
\end{remark}

\begin{corollary}\label{general}
Let $\H_g(2^{O(g)},1^{2g-2-2O(g)})$ be a stratum with $O(g)$ double zeros, then the result in Theorem \ref{closed} is also true.
\end{corollary}

Their method is based on the relationship between Poisson distribution and its factorial moment \cite{bollobas2001random}, which was used first in the work of Mirzakhani--Petri \cite{mirzakhani2019lengths}. For translation surfaces, \cite{masur2024lengths} introduces a surgery to collapse the open saddle connections to obtain a new general translation surface in a new stratum. "This surgery does not work for closed saddle connections"; the paper "gives a new surgery to collapse a closed saddle connection, which ensures the same method can be used for closed saddle connections."

\label{open}
Let $[a_1,b_1],\cdots,[a_k,b_k]$ be $k$ disjoint intervals. Then as $g\to \infty$, the vector of random variables
$$(N_{g,[a_1,b_1]},\cdots,N_{g,[a_k,b_k]}):\H_g(1^{2g-2})\to \mathbb{N}_0^k$$
converges jointly in distribution to a vector of random variables with Poisson distributions of means $\lambda_{[a_i,b_i]}$, where
$$\lambda_{[a_i,b_i]}=8\pi(b_i^2-a_i^2)$$
for $i=1,\cdots,k$. That is,
$$\mathbb{P}(N_{g,[a_1,b_1]}=n_1,\cdots,N_{g,[a_k,b_k]}=n_k)=\prod_{i=1}^k\frac{\lambda_{[a_i,b_i]}^{n_i}e^{-\lambda_{[a_i,b_i]}}}{n_i!}$$,

Question: Let H_g(1^{2g-2}) denote the principal stratum of unit-area translation surfaces of genus g, consisting of Abelian differentials with 2g-2 simple zeros, equipped with the normalized Masur–Veech probability measure. For (X,ω) in H_g(1^{2g-2}) and an interval [a,b] with a,b>0, let N_{g,[a,b]}(X,ω) be the number of open saddle connections on (X,ω), meaning |ω|-geodesic segments joining two distinct zeros of ω and containing no zero in their interior, whose lengths lie in [a/g,b/g]. If [a_1,b_1],...,[a_k,b_k] are k pairwise disjoint intervals, then as g→∞, which limiting statement holds for the random vector (N_{g,[a_1,b_1]},...,N_{g,[a_k,b_k]})?

As g→∞, (N_{g,[a_1,b_1]},...,N_{g,[a_k,b_k]}) converges jointly in distribution to a vector of independent Poisson random variables with means λ_[a_i,b_i] = 8π(b_i^2-a_i^2) for i=1,...,k. Equivalently, for every fixed n_1,...,n_k in N_0, lim_{g→∞} P(N_{g,[a_1,b_1]}=n_1,...,N_{g,[a_k,b_k]}=n_k) = ∏_{i=1}^k [λ_[a_i,b_i]^{n_i} e^{-λ_[a_i,b_i]} / n_i!].

Choices:
B. As g→∞, (N_{g,[a_1,b_1]},...,N_{g,[a_k,b_k]}) converges jointly in distribution to a vector of independent Poisson random variables with means λ_[a_i,b_i] = 8π(b_i-a_i) for i=1,...,k. Equivalently, for every fixed n_1,...,n_k in N_0, lim_{g→∞} P(N_{g,[a_1,b_1]}=n_1,...,N_{g,[a_k,b_k]}=n_k) = ∏_{i=1}^k [λ_[a_i,b_i]^{n_i} e^{-λ_[a_i,b_i]} / n_i!].
C. As g→∞, each individual random variable N_{g,[a_i,b_i]} converges in distribution to a Poisson random variable with mean λ_[a_i,b_i] = 8π(b_i^2-a_i^2) for i=1,...,k.
D. As g→∞, (N_{g,[a_1,b_1]},...,N_{g,[a_k,b_k]}) converges jointly in distribution to a vector of Poisson random variables with means λ_[a_i,b_i] = 8π(b_i^2-a_i^2) for i=1,...,k, but in general the limiting coordinates need not be independent. Equivalently, for every fixed n_1,...,n_k in N_0, lim_{g→∞} P(N_{g,[a_1,b_1]}=n_1,...,N_{g,[a_k,b_k]}=n_k) exists and has Poisson marginals with those means.
E. As g→∞, (N_{g,[a_1,b_1]},...,N_{g,[a_k,b_k]}) converges jointly in distribution to a vector of independent Poisson random variables with means λ_[a_i,b_i] = 3π(b_i^2-a_i^2) for i=1,...,k. Equivalently, for every fixed n_1,...,n_k in N_0, lim_{g→∞} P(N_{g,[a_1,b_1]}=n_1,...,N_{g,[a_k,b_k]}=n_k) = ∏_{i=1}^k [λ_[a_i,b_i]^{n_i} e^{-λ_[a_i,b_i]} / n_i!].

quadratic area-scaling in the Poisson mean

joint convergence/independence of counts across disjoint intervals

factorial-moment method yields independent joint Poisson law, not merely Poisson marginals

mean constant specific to open saddle connections in the principal stratum

The stem defines the objects and asks for the limiting law, but it does not explicitly state the correct conclusion or uniquely signal option A. No direct answer leakage is present.

The item is close to a theorem-recall question: it essentially asks for the exact asymptotic statement of a known result. The alternatives do introduce meaningful variations, but the prompt is still a fairly direct request for the theorem’s conclusion.

Some reasoning is required to distinguish the full joint independent Poisson limit from weaker marginal convergence, incorrect dependence claims, and wrong mean formulas/constants. However, solving it is driven more by precise recall/recognition than by substantial generative mathematical reasoning.

The distractors are strong and mathematically plausible: one uses the wrong scaling in the mean, one gives a weaker true statement, one removes independence, and one changes the constant. These align well with realistic confusion points.

A solid MCQ with no answer leakage and high-quality distractors, but it functions mainly as theorem recognition rather than a deeply generative reasoning task.


\begin{equation}\label{k cond S}
    |k_{i}-\mu_{K}|\leq C_{g} \textrm{ and } k_{1}+k_{2}+k_{3}\equiv M\md{g-1},
\end{equation}

\begin{theorem}\label{Main theorem 1}
   For any $g\geq 3$, the set of base-$g$ Niven numbers is an asymptotic basis of order 3.
\end{theorem}

A central question in additive number theory is to establish whether a given set of integers $\mathcal{S}$ is an \emph{asymptotic basis} for the integers, that is, to determine whether there exists a natural number $k$ such that any sufficiently large integer can be written as the sum of $k$ elements of $\mathcal{S}$. Here, $k$ denotes the \textit{order} of the basis.

Famously, Lagrange's theorem gives that the squares are a basis of order 4, and Waring's problem, solved by Hilbert, shows that $k$\textsuperscript{th } powers are also an additive basis. Some interesting variants of Waring's problem consider $k$\textsuperscript{th } powers of integers which have restrictions on their digits in some base. For example, Pfeiffer and Thuswaldner \cite{pfeiffer2007waring} show that the $k$\textsuperscript{th } powers of integers with certain congruence conditions on their sums of digits in different bases is an asymptotic basis. More recently, Green \cite{green2025waring} established that, given any two digits which are coprime, the integers whose base-$g$ expansions consists of only these digits satisfy Waring's problem. Further references for additive bases coming from sets of integers with digit restrictions are given in the introduction of \cite{sanna2021additive}.

A \emph{base-$g$ Niven number} is a natural number that is divisible by its base-$g$ sum of digits. Such integers are also referred to as \textit{Harshad numbers}. It is shown in \cite{de2003counting}, and independently in \cite{mauduit2005distribution}, that the number of base-$g$ Niven numbers less than $x$ is asymptotically $\eta_{g}x/\log x$ for some constant $\eta_g>0$.

It is conjectured that the set of base-$g$ Niven numbers is an asymptotic basis of order 2. In \cite{sanna2021additive} Sanna established, conditionally upon a certain generalisation of the Riemann Hypothesis, that the set of base-$g$ Niven numbers is an asymptotic basis with order growing linearly in $g$. Our result is the following unconditional statement.

A \emph{base-$g$ Niven number} is a natural number that is divisible by its base-$g$ sum of digits. Such integers are also referred to as \textit{Harshad numbers}. It is shown in \cite{de2003counting}, and independently in \cite{mauduit2005distribution}, that the number of base-$g$ Niven numbers less than $x$ is asymptotically $\eta_{g}x/\log x$ for some constant $\eta_g>0$.

Therefore $|\mathcal{E}_{0}|\geq |\mathcal{E}_{y}|\geq R$. Recall the definition of the tuple $\mathbf{a}$ from \cref{a tuple defn}; as $x=0$ we have that \[\mathbf{a}=(|\mathcal{E}_{0}|,0,\ldots,0,|\mathcal{E}_{y}|,0,\ldots,0 ).\]
Our aim is to bound $\Psi(\mathbf{a};0)$ using \cref{psi 0 bound} with $m=R$ and $t=y$. To satisfy the assumptions of \cref{psi 0 bound} we need $R\ll_{C} |\mathcal{E}_{0}|^{1/4}$, which follows from the fact that $|\mathcal{E}_{0}|\geq (K-|\mathcal{D}|)/2\geq (K-LR)/2$. As $LR\leq CK^{1/4}$ by assumption, we have that $|\mathcal{E}_{0}|\gg_{C} K$, and certainly $R\leq CK^{1/4}$, giving $R\ll_{C} |\mathcal{E}_{0}|^{1/4} $. Applying \cref{psi 0 bound} gives the bound
\begin{equation*}\label{psi cancellation}
 \Psi(\mathbf{a};0)\ll_{C,g} g^{-R}+R^{2}K^{-3/2}.
\end{equation*}
This gives \cref{main term cancellation psi}, using that $R\geq \tfrac{3}{2}\log_{g}K$.
\end{proof}
\subsection{Decoupling the averages over $\xbd$ and $\xbee$}\label{section decoupling}
In this section we prove the decoupling result, \cref{decoupling lemma}. This lemma allows us to replace the condition on the digits of $X$, $s(\xbd)+s(\xbee)=\xi$, with the condition $s(\xbee)=0$, even as the digits in $\xbd$ vary.
\begin{proof}[Proof of \cref{decoupling lemma}] Let $(g-1)\theta$ have centred base-$g$ expansion given by \cref{theta digit exp}. Dividing through by $(g-1)$ in this expansion gives
\[\theta=\frac{1}{g-1}\sum_{j=1}^{w}\varepsilon_{n_{j}}g^{-n_{j}}+\frac{\eta}{g-1},\]
where $|\eta|<g^{-K}$ and $w\coloneq w_{K}((g-1)\theta)$. Thus we can separate the contribution to $e(\xb\theta)$ from each non-zero digit as follows,
\begin{equation}\label{decoupling proof eq 1}
 e(\xb\theta)=e\Big(\xb\frac{\eta}{g-1}\Big)\prod_{j=1}^{w} e\Big(\xb \frac{\varepsilon_{n_{j}}}{(g-1)g^{n_{j}}}\Big).
\end{equation}
We have that
\[\Big|\xb_{[0,n_{j}-R-1]}\frac{\varepsilon_{n_{j}}}{(g-1)g^{n_{j}}}\Big|\leq g^{-R} \textrm{ and } \xb_{[n_{j},K-1]}\frac{\varepsilon_{n_{j}}}{g^{n_{j}}}\equiv X_{[n_{j},K-1]}\varepsilon_{n_{j}}\md{g-1}, \]
where the latter statement uses \cref{trans inv}. Thus the contribution to $e(X\theta)$ from the $n_{j}$\sth digit of $(g-1)\theta$ is
 \begin{align}
 e\Big(\xb\frac{\varepsilon_{j}}{(g-1)g^{n_{j}}}\Big)=e\Big(\xb_{[n_{j}-R,n_{j}-1]}\frac{\varepsilon_{n_{j}}}{(g-1)g^{n_{j}}}\Big )\ru{\varepsilon_{n_{j}}\xb_{[n_{j},K-1]}}+O(g^{-R})\label{ni contribution}.
 \end{align}
The $\eta$ term gives a similar contribution: let $u\coloneq n_{w_{K}((g-1)\theta)+1}-R$, then
 \begin{equation}\label{eta contribution}
 e(\xb\eta/(g-1))=e(\xb_{[u,K-1]}\eta/(g-1))+O(g^{-R}).
 \end{equation}
 If $u\geq K$, then the interval $[u,K-1]\cap \{0,\ldots, K-1\}$ is empty and $e(X\eta/(g-1))=1+O(g^{-R})$. From \cref{decoupling proof eq 1}, \cref{ni contribution} and \cref{eta contribution}, we have
\begin{align}
 \E_{\xb}e(\xb\theta)\ind_{\sx{\xb}=\xi}&=\E_{\xb}e\Big(\xb\frac{\eta}{g-1}\Big)\prod_{i=1}^{w}e\Big (\xb\frac{\varepsilon_{n_{i}}}{(g-1)g^{n_{i}}}\Big)\ind_{\sx{\xb}=\xi}\nonumber\\
 &=\E_{\xb}e\Big(\xb_{[u,K-1]}\frac{\eta}{g-1}\Big)\prod_{j=1}^{w}e\Big (\xb_{[n_{j}-R,n_{j}-1]}\frac{\varepsilon_{n_{j}}}{(g-1)g^{n_{j}}}\Big)\nonumber\\&\hspace{12em}\times \ru{\xb_{[n_{j},K-1]}\varepsilon_{n_{j}}}\ind_{\sx{\xb}=\xi}+O_{C,g}(L g^{-R}).\label{decoupling eg 2}\end{align}
 Here, we have used that $L\leq CK^{1/4}$ and $R\geq \tfrac{3}{2}\log_{g}K$ to obtain this error term.
Let $\mathcal{E}\coloneq \{0,\ldots, K-1\}\setminus \mathcal{D}$, and note that $\mathcal{E} $ is partitioned into $\mathcal{E}= \mathcal{E}_{0}\cup \ldots \cup \mathcal{E}_{g-2}$ for $\mathcal{E}_{r}$ defined by \cref{E set defn}. With this notation, we can rewrite \cref{decoupling eg 2} as
\begin{equation}\label{decoupling eq 2.5}\E_{X}e(X\theta)\ind_{s(X)=\xi}=\E_{\xbd}e(\xbd\theta)\E_{\xbee}\prod_{j=1}^{w}\ru{\xb_{[n_{j},K-1]\cap \mathcal{E}}\varepsilon_{j}}\ind_{\sx{\xb}=\xi}+O_{C,g}(L g^{-R}).\end{equation}
As the terms $n_{j}$ index precisely the non-zero digits of $(g-1)\theta$, each random variable $X_{u}$ for $u\in \mathcal{E}$ occurs in the above product once for each non-zero digit with index $n_{j}\leq u$, weighted by the value of that non-zero digit, $\varepsilon_{n_{j}}$. Rearranging the above product gives
 \[\prod_{j=1}^{w}\ru{\xb_{[n_{j},K-1]\cap \mathcal{E}}\varepsilon_{n_{j}}}=\prod_{u\in \mathcal{E}}\ru{X_{u}g^{u}\sum_{r\leq u}\varepsilon_{r}}=\prod_{v=1}^{g-2}\ru{v\xb_{\mathcal{E}_{v}}},\]
where the final equality follows from the definition of the sets $\mathcal{E}_{v}$ given by \cref{E set defn}. Hence from \cref{decoupling eq 2.5} and the above equation, $\E_{X}e(X\theta)\ind_{s(X)=\xi}$ equals
 \begin{equation}\label{decoupling eq 3}
 \E_{\xbd}e(\xbd\theta) \E_{\xbee}\ru{\xb_{\mathcal{E}_{1}}+2\xb_{\mathcal{E}_{2}}+\ldots+(g-2)\xb_{\mathcal{E}_{g-2}}}\ind_{\sx{\xbee}=\xi-\sx{\xbd}}+O_{C,g}(L g^{-R}).
 \end{equation}
Let $\mathbf{a}_{0}\coloneq(|\mathcal{E}_{0}|,\ldots,|\mathcal{E}_{g-2}|)$. From \cref{Psi expectation form} the average over $\xbee$ equals
\begin{align*}\label{decoupling psi for a0}
 \E_{\xbee}\ru{\xb_{\mathcal{E}_{1}}+2\xb_{\mathcal{E}_{2}}+\ldots+(g-2)\xb_{\mathcal{E}_{g-2}}}\ind_{\sx{\xbee}=\xi-\sx{\xbd}}=\Psi(\mathbf{a}_{0};\xi-\sx{\xbd}),
\end{align*}
and thus \cref{decoupling eq 3} equals
\[\E_{\xbd}e(\xbd\theta) \Psi(\mathbf{a}_{0};\xi-\sx{\xbd})+O_{C,g}(L g^{-R}).\]
Our aim is to remove the dependence on $\xbd$ from the term $ \Psi(\mathbf{a}_{0};\xi-\sx{\xbd})$ by applying \cref{psi lemma}. However, in order to apply \cref{psi lemma} with the tuple $\mathbf{a}_{0}$, we require that the first coordinate of $\mathbf{a}_{0}$, $|\mathcal{E}_{0}|$, satisfies $|\mathcal{E}_{0}|\geq |\mathcal{E}_{j}|$ for all $1\leq j \leq g-2$, which may not be the case. To circumvent this issue, we exploit the fact that multiplying \cref{decoupling eq 3} through by factors of $\ru{X}$ increases the coefficients of each $X_{\mathcal{E}_{r}}$, essentially allowing us to cycle the coordinates of $\mathbf{a}_{0}$. To do this, we use the following relation. As $\xb\equiv \sx{\xb} \md{g-1}$, we have $\E_{X}\ru{\xb}\ind_{\sx{\xb}=\xi}=\ru{\xi}$, and thus for any integer $m$,
 \begin{equation}\label{decoupling eq 5}
 \E_{\xb}e(\xb\theta)\ind_{\sx{\xb}}=\ru{m\xi}\E_{\xb}e(\xb\theta)\ru{-m\xb }\ind_{\sx{\xb}=\xi}.
 \end{equation}
Let $x$ be defined as in the statement of \cref{decoupling lemma}, that is, $x\in\{0,\ldots, g-2\}$ is such that $\mathcal{E}_{x}$ is maximal among the $|\mathcal{E}_{r}|$, and let $\mathbf{a}$ be as defined by \cref{a tuple defn},
\[ \mathbf{a}\coloneq(|\mathcal{E}_{x}|,|\mathcal{E}_{x+1}|,\ldots, |\mathcal{E}_{g-2}|,|\mathcal{E}_{0}|,|\mathcal{E}_{1}|,\ldots, |\mathcal{E}_{x-1}|).\]
From \cref{decoupling eq 5}, multiplying \cref{decoupling eq 3} through by $\ru{-x\xbee}$ gives
\begin{align}
 \E_{\xb}e(X\theta)\ind_{s(X)=\xi}&=\ru{x\xi}\E_{X}e(X\theta)\ru{-xX}\ind_{s(X)=\xi}\nonumber\\
 &=\ru{x\xi}\E_{\xbd}e\Big (\xbd\Big (\theta-\frac{x}{g-1}\Big )\Big )\E_{\xbee}\prod_{j=0}^{g-2}\ru{(j-x)\xb_{\mathcal{E}_{j}}}\ind_{\sx{\xbee}=\xi-\sx{\xbd}}\nonumber\\
 &=\ru{x\xi}\E_{\xbd}e\Big (\xbd\Big (\theta-\frac{x}{g-1}\Big )\Big )\Psi(\mathbf{a};\xi-\sx{\xbd}),\label{decoupling eq 4}
\end{align}
using \cref{Psi expectation form} and the definition of $\mathbf{a}$ for the final equality. As $|\mathcal{E}_{x}|$ is maximal among the sets $\mathcal{E}_{r}$, the tuple $\mathbf{a}$ fulfils the requirement that its first coordinate is the largest.

We first count, via the circle method, the number of representations of a sufficiently large integer $M$ as the sum of three integers with a near-average digit sum. By showing that a proportion of such representations have each summand being a Niven number, we provide a lower bound for the number of representations of $M$ as the sum of three Niven numbers. To state these results more precisely, we need the following notation.

Let $s_{g}(n)$ denote the base-$g$ digit sum of $n$. Let $K\geqslant 1$ be a sufficiently large integer, and let $M\in (g^{K-1},g^{K}]$. For $k\in \N$, let
\[S_{g}(k)\coloneq \{n<g^{K}\ensuremath{:} s_{g}(n)=k\} \textrm{ and }\mathcal{N}_{g}(k)\coloneq \{n\in S(k) \ensuremath{:} k\mid n\}.\]
Thus $\mathcal{N}_{g}(k)$ is the set of Niven numbers in $S_{g}(k)$, that is, Niven numbers of a certain size with fixed digit sum. Let \begin{equation*}
    \mu_{K}\coloneq \frac{(g-1)K}{2},
\end{equation*}
which is the average digit sum of $n<g^{K}$. For a fixed choice of $k_{1},k_{2},k_{3}\in \N$, let $S_{i}\coloneq S_{g}(k_{i})$ for $i=1,2,3$, and let $\mathcal{N}_{i}\coloneq \mathcal{N}_{g}(k_{i})$ for $i=1,2,3$. Suppose that $k_{1},k_{2},k_{3}\in \N$ are such that
\begin{equation}\label{k cond S}
    |k_{i}-\mu_{K}|\leqslant C_{g} \textrm{ and } k_{1}+k_{2}+k_{3}\equiv M\md{g-1},
\end{equation}
where $C_g := g(g-1)\prod_{p\leqslant 10g^{2}}p$, and the product is over primes.  The specific choice of constant here is unimportant, and any sufficiently large value would do, but this value is large enough to ensure that $k_{i}$ with further desired properties exist. Let $r_{S_{1}+S_{2}+S_{3}}(M)$ be the number of representations of $M=s_{1}+s_{2}+s_{3}$, for $s_{i}\in S_{i}$. Our main result is the following theorem, from which we can later deduce the corresponding result for $\mathcal{N}_{i}\subset S_{i}$.
\begin{theorem}\label{Main theorem S count} Let $g,K$ and $M$ be integers such that $g\geqslant 3$, $K$ is sufficiently large in terms of $g$, and $M\in (g^{K-1},g^{K}]$. Suppose that $k_{1},k_{2},k_{3}$ satisfy \cref{k cond S}. Then
    \[r_{S_{1}+S_{2}+S_{3}}(M)=\frac{(g-1)M^{2}}{2(2\pi\sigma^{2}K)^{3/2}}(1+O_{g}((\log K)^{4}K^{-1/4})),\]
where $\sigma^{2}=(g^{2}-1)/12$.
\end{theorem}
Let $r_{\mathcal{N}_{1}+\mathcal{N}_{2}+\mathcal{N}_{3}}(M)$ be the number of representations of $M=n_{1}+n_{2}+n_{3}$, where $n_{i}\in \mathcal{N}_{i}$ for $i=1,2,3$. In order to relate the quantity $r_{S_{1}+S_{2}+S_{3}}(M)$ to $r_{\mathcal{N}_{1}+\mathcal{N}_{2}+\mathcal{N}_{3}}(M)$, we require some further conditions on the choice of $k_{1},k_{2},k_{3}$. Throughout, $(a,b)$ denotes the greatest common divisor of $a$ and $b$. Suppose that $k_{1},k_{2},k_{3}\in\N$ also satisfy, in addition to \cref{k cond S},
\begin{align}\label{k cond N}
  (k_{i},k_{j})=1 \textrm{ and }(k_{i},g)=1 \textrm{ for }i,j=1,2,3 \textrm{ and } i\neq j.
 \end{align}
Let
\[c_{g}(k_{1},k_{2},k_{3})\coloneq \frac{4}{(g-1)^{2}}\prod_{i=1}^{3}(g-1,k_{i}).\]
Then we have the following theorem, from which \cref{Main theorem 1} is a corollary after showing that such a choice of $k_{1},k_{2}$ and $k_{3}$ exist.
\begin{theorem}\label{Main theorem N count}
Let $g,K$ and $M$ be integers such that $g\geqslant 3$, $K$ is sufficiently large in terms of $g$, and $M\in (g^{K-1},g^{K}]$. Suppose that $k_{1},k_{2},k_{3}$ fulfil \cref{k cond S} and \cref{k cond N}, then
\[r_{\mathcal{N}_{1}+\mathcal{N}_{2}+\mathcal{N}_{3}}(M)=\frac{(g-1)^{2}c_{g}(k_{1},k_{2},k_{3})}{4k_{1}k_{2}k_{3}}r_{S_{1}+S_{2}+S_{3}}(M)+O_{g}(M^{2}K^{-29/6}).\]
In particular,
 \[r_{\mathcal{N}_{1}+\mathcal{N}_{2}+\mathcal{N}_{3}}(M)=c_{g}(k_{1},k_{2},k_{3})\frac{M^{2}}{(2\pi\sigma^{2})^{3/2}K^{9/2}}+O_{g}(M^{2}(\log K)^{4}K^{-19/4}),\]
 where $\sigma^{2}=(g^{2}-1)/12$.
\end{theorem}
Our methods can be adapted to show the analogous versions of \cref{Main theorem 1}, \cref{Main theorem S count} and \cref{Main theorem N count} for base 2, however there is one main technical difference. For readability, we do not give the details here, but comment on the necessary changes in \cref{Section: minor arc}.
\subsection{Notation}\label{notation}
Throughout, we consider $g$ to be fixed. Let $\Vert x \Vert_{\R/\Z}$ denote the distance of a real number $x$ to the nearest integer. We write $e(x)$ for $\exp(2\pi ix)$, and $e_{g-1}(x)$ for $e(x/(g-1))$. We require the quantity $\ell$, defined to be
\begin{equation}\label{l choice}
    \ell\coloneqq \lceil 384g^{3}\log K\rceil.
\end{equation}
Throughout, $\sigma=\sqrt{(g^{2}-1)/12}.$
\stoptoc

To prove Theorem~\ref{Main theorem 1}, the introduction explains the following strategy.

1. "We first count, via the circle method, the number of representations of a sufficiently large integer $M$ as the sum of three integers with a near-average digit sum." Concretely, for $M\in (g^{K-1},g^{K}]$ and digit sums $k_i$ satisfying $|k_i-\mu_K|\le C_g$ and $k_1+k_2+k_3\equiv M\pmod{g-1}$, one studies
\[r_{S_1+S_2+S_3}(M)=\#\{(s_1,s_2,s_3)\in S_g(k_1)\times S_g(k_2)\times S_g(k_3): s_1+s_2+s_3=M\},\]
with the goal of obtaining an asymptotic formula (Theorem \ref{Main theorem S count}).

2. "By showing that a proportion of such representations have each summand being a Niven number, we provide a lower bound for the number of representations of $M$ as the sum of three Niven numbers." This is implemented by relating the above count to
\[r_{\mathcal N_1+\mathcal N_2+\mathcal N_3}(M)=\#\{(n_1,n_2,n_3)\in \mathcal N_g(k_1)\times\mathcal N_g(k_2)\times\mathcal N_g(k_3): n_1+n_2+n_3=M\}.\]
Under additional coprimality conditions on the digit sums, namely $(k_i,k_j)=1$ for $i\ne j$ and $(k_i,g)=1$, one obtains a comparison
\[r_{\mathcal{N}_{1}+\mathcal{N}_{2}+\mathcal{N}_{3}}(M)=\frac{(g-1)^{2}c_{g}(k_{1},k_{2},k_{3})}{4k_{1}k_{2}k_{3}}r_{S_{1}+S_{2}+S_{3}}(M)+O_g(M^{2}K^{-29/6})\]
(Theorem \ref{Main theorem N count}), hence a positive proportion of the $S_1+S_2+S_3$ representations contribute to Niven-summand representations.

3. Finally, the introduction states that Theorem~\ref{Main theorem 1} follows because Theorem \ref{Main theorem N count} implies it "after showing that such a choice of $k_1,k_2$ and $k_3$ exist" (i.e., satisfying the near-average digit-sum congruence condition and the extra coprimality conditions).

To prove the main theorem, the introduction explains the following strategy.

1. "We first count, via the circle method, the number of representations of a sufficiently large integer $M$ as the sum of three integers with a near-average digit sum." Concretely, for $M\in (g^{K-1},g^{K}]$ and digit sums $k_i$ satisfying $|k_i-\mu_K|\le C_g$ and $k_1+k_2+k_3\equiv M\pmod{g-1}$, one studies
\[r_{S_1+S_2+S_3}(M)=\#\{(s_1,s_2,s_3)\in S_g(k_1)\times S_g(k_2)\times S_g(k_3): s_1+s_2+s_3=M\},\]
with the goal of obtaining an asymptotic formula. We first prove the following theorem.

\begin{theorem}\label{Main theorem S count} Let $g,K$ and $M$ be integers such that $g\geq 3$, $K$ is sufficiently large in terms of $g$, and $M\in (g^{K-1},g^{K}]$. Suppose that $k_{1},k_{2},k_{3}$ satisfy \cref{k cond S}. Then
    \[r_{S_{1}+S_{2}+S_{3}}(M)=\frac{(g-1)M^{2}}{2(2\pi\sigma^{2}K)^{3/2}}(1+O_{g}((\log K)^{4}K^{-1/4})),\]
where $\sigma^{2}=(g^{2}-1)/12$.
\end{theorem}

2. "By showing that a proportion of such representations have each summand being a Niven number, we provide a lower bound for the number of representations of $M$ as the sum of three Niven numbers." This is implemented by relating the above count to
\[r_{\mathcal N_1+\mathcal N_2+\mathcal N_3}(M)=\#\{(n_1,n_2,n_3)\in \mathcal N_g(k_1)\times\mathcal N_g(k_2)\times\mathcal N_g(k_3): n_1+n_2+n_3=M\}.\]
Under additional coprimality conditions on the digit sums, namely $(k_i,k_j)=1$ for $i\ne j$ and $(k_i,g)=1$, one obtains the following comparison theorem.

\begin{theorem}\label{Main theorem N count}
Let $g,K$ and $M$ be integers such that $g\geq 3$, $K$ is sufficiently large in terms of $g$, and $M\in (g^{K-1},g^{K}]$. Suppose that $k_{1},k_{2},k_{3}$ fulfil \cref{k cond S} and \cref{k cond N}, then
\[r_{\mathcal{N}_{1}+\mathcal{N}_{2}+\mathcal{N}_{3}}(M)=\frac{(g-1)^{2}c_{g}(k_{1},k_{2},k_{3})}{4k_{1}k_{2}k_{3}}r_{S_{1}+S_{2}+S_{3}}(M)+O_{g}(M^{2}K^{-29/6}).\]
In particular,
 \[r_{\mathcal{N}_{1}+\mathcal{N}_{2}+\mathcal{N}_{3}}(M)=c_{g}(k_{1},k_{2},k_{3})\frac{M^{2}}{(2\pi\sigma^{2})^{3/2}K^{9/2}}+O_{g}(M^{2}(\log K)^{4}K^{-19/4}),\]
 where $\sigma^{2}=(g^{2}-1)/12$.
\end{theorem}

Hence a positive proportion of the $S_1+S_2+S_3$ representations contribute to Niven-summand representations.

3. Finally, the introduction states that the main theorem follows because the preceding theorem implies it after showing that such a choice of $k_1,k_2$ and $k_3$ exist (i.e., satisfying the near-average digit-sum congruence condition and the extra coprimality conditions).

\label{Main theorem 1}
For any $g\geqslant 3$, the set of base-$g$ Niven numbers is an asymptotic basis of order 3. This completes the proof of the main theorem.

Question: Fix an integer base $g\ge 3$, and let $\mathcal S_g$ be the set of base-$g$ Niven numbers, i.e. natural numbers $n$ such that $n$ is divisible by the sum of its digits in base $g$. Recall that a set $\mathcal S\subseteq \mathbb N$ is an asymptotic basis of order $3$ if there exists $N$ such that every integer $m\ge N$ can be written as $m=s_1+s_2+s_3$ with $s_1,s_2,s_3\in \mathcal S$. Under these assumptions, which asymptotic additive statement holds for $\mathcal S_g$?

For every integer $g\ge 3$, the set $\mathcal S_g$ of base-$g$ Niven numbers is an asymptotic basis of order $3$; equivalently, for each such $g$ there exists $N_g$ such that every integer $m\ge N_g$ can be expressed as the sum of three base-$g$ Niven numbers.

Choices:
B. For every integer $g\ge 3$, the set $\mathcal S_g$ of base-$g$ Niven numbers is an asymptotic basis of order $2$; equivalently, for each such $g$ there exists $N_g$ such that every integer $m\ge N_g$ can be expressed as the sum of two base-$g$ Niven numbers.
C. For every integer $g\ge 3$, the set $\mathcal S_g$ of base-$g$ Niven numbers is an asymptotic basis; equivalently, for each such $g$ there exists an integer $h_g\ge 1$ and $N_g$ such that every integer $m\ge N_g$ can be expressed as the sum of $h_g$ base-$g$ Niven numbers.
D. There exists an absolute integer $N$ such that for every base $g\ge 3$ and every integer $m\ge N$, one can write $m=s_1+s_2+s_3$ with $s_1,s_2,s_3\in \mathcal S_g$; that is, the base-$g$ Niven numbers form an asymptotic basis of order $3$ uniformly in $g$.
E. For every integer $g\ge 3$, there exists $N_g$ such that every integer $m\ge N_g$ can be expressed as the sum of three base-$g$ Niven numbers, and moreover each summand may be chosen to have the same base-$g$ digit sum.

three-summand representation count upgraded to two summands

fixed order $3$ dropped to existence of some order depending on $g$

threshold dependence on $g$ replaced by uniform absolute threshold

near-average digit-sum auxiliary choice misread as equal digit sums for the three Niven summands

The stem does not state the theorem's conclusion; it only defines base-g Niven numbers and what 'asymptotic basis of order 3' means. The correct choice is not explicitly leaked, though the terminology in the stem naturally frames the target result.

The item is close to a theorem-recall question: the correct answer essentially restates the claimed result in the same language as the stem. However, the presence of stronger, weaker, and quantifier-altered alternatives prevents it from being a pure tautology.

Some reasoning is needed to distinguish the exact statement from nearby variants such as order 2, existence of some order, or uniformity in g. Still, the item mainly tests recognition of the theorem's precise formulation rather than substantial mathematical generation or derivation.

The distractors are strong and mathematically meaningful: one is a stronger false claim (order 2), one is a weaker true statement, one alters quantifier dependence, and one adds an implausible extra constraint. These reflect common failure modes in reading theorem statements.

A solid theorem-statement discrimination MCQ with good distractors and little answer leakage, but it leans more toward precise recall of a result than genuine generative reasoning.


[Structural invariance of GGD thresholds {[Theorem.~\ref{thm:main-threshold}]{}}]\label{thm:invariance}
		Let \(\pi\colon \mathcal{X}\to Y\) be the coarse moduli map of a compact smooth analytic Deligne--Mumford stack (orbifold) \(\mathcal{X}\), and let \(A\) be an ample line bundle on \(Y\) endowed with a smooth positively curved metric.
		Set \(L=\pi^*A\).
		Then there exist integers \(m_0,q_0>0\) such that for all \(m\ge m_0\) and \(q\ge q_0\)$\colon$

		\begin{enumerate}[label=(\roman*)]
			\item all higher cohomology groups vanish,
			\[
			H^i(\mathcal{X},E^{\mathrm{inv}}_{k,m}\otimes L^{-q})=0,\quad i>0;
			\]
			\item consequently,
			\[
			h^0\!\big(\mathcal{X},E^{\mathrm{inv}}_{k,m}\otimes L^{-q}\big)
			=\chi\!\big(\mathcal{X},E^{\mathrm{inv}}_{k,m}\otimes L^{-q}\big)
			=\frac{1}{s}\!\int_Y\! \mathrm{ch}(E_{k,m})\,e^{m c_1(A)}\,\mathrm{Td}(TY)
			+O(m^{n-1}),
			\]
			where \(s=|\mathrm{Stab}_{\mathrm{gen}}|\);
			\item the minimal jet order and asymptotic slope at which invariant jet differentials exist depend only on the coarse Kähler class \([\omega_A]\).
			Equivalently,
			\[
			\text{$Y$ is GGD-positive}
			\quad\Longleftrightarrow\quad
			\text{$\mathcal{X}$ is GGD-positive}.
			\]
		\end{enumerate}
		Thus, orbifold compactification and rigidification neither alter nor shift the GGD threshold.

\label{sec:intro}

The Green--Griffiths--Demailly (GGD) program, initiated by Green and Griffiths~\cite{GreenGriffiths1979} and developed extensively by Demailly~\cite{Demailly1997,Demailly2007,Demailly2011}, provides a powerful analytic and cohomological framework for understanding algebraic degeneracy of entire curves in complex projective varieties.
 Its central idea is to construct invariant jet differentials along the Demailly--Semple (DS) tower and to evaluate their asymptotic Euler characteristics using curvature positivity and Riemann--Roch theory.
 The minimal jet order \(k_0\) and weight slope \(\lambda_0\) for which
 \[
 H^0\!\big(X,E^{\mathrm{inv}}_{k,m}\otimes A^{-q}\big)\neq0,
 \qquad q\simeq \lambda_0 m,
 \]
 quantify the onset of hyperbolicity-type behavior predicted by the Green--Griffiths--Lang conjecture.
 These numerical thresholds depend on the balance between vertical negativity and horizontal positivity in the curvature of the underlying Kähler class.

Orbifold and stack generalizations of this program have become increasingly relevant in modern geometry; see Campana–P\u{a}un~\cite{CampanaPaun2016}, Borghesi–Tomassini~\cite{BorghesiTomassini2017}, and Toën–Vezzosi~\cite{ToenVezzosi2008}.
 Deligne--Mumford stacks equipped with orbifold Kähler forms or log pairs, as in Abramovich–Olsson–Vistoli~\cite{AbramovichOlssonVistoli2008}, naturally arise in moduli theory and arithmetic geometry, where curvature and cohomological tools must be extended to finite quotient groupoids~\cite{MoerdijkPronk1997,Lerman2008,KeelMori1997,StacksProj04V2}.
 However, the analytic underpinnings of the GGD framework---Bochner identities, Hörmander-type \(L^2\) estimates, and Riemann--Roch asymptotics---were originally formulated for manifolds, not orbifolds.
 This raises the structural question:
\begin{question}
Does the passage from a smooth variety to a smooth orbifold or stack, with the same coarse Kähler class, alter the positivity thresholds that govern the existence of invariant jet differentials?
\end{question}
 At first glance, the answer need not be obvious.
 The orbifold Riemann--Roch theorem of Satake~\cite{Satake1956} and Kawasaki~\cite{Kawasaki1979,Kawasaki1981} involves contributions from twisted sectors with denominators determined by isotropy representations.
 Thus, the asymptotic expansion of the Euler characteristic
 \[
 \chi(\mathcal{X},E_{k,m}\otimes \pi^*A^{-q})
 \]
 where $\pi\colon \mathcal{X}\to Y$ is the coarse moduli map and $A$ is a line bundle on $Y$,
might, in principle, differ from its coarse counterpart on~$Y$.
 Furthermore, while the leading term of the Euler characteristic scales by \(1/s\), where \(s=|\mathrm{Stab}_{\mathrm{gen}}|\) is the generic stabilizer order, the actual number of global sections \(h^0\) could still depend on higher cohomology groups \(H^i(\mathcal{X},E_{k,m}\otimes L^{-q})\).
 To establish structural invariance, one must show that these higher cohomology terms are suppressed by curvature positivity in the orbifold setting, ensuring that the growth of \(h^0\) and \(\chi\) coincide asymptotically.

\begin{theorem}
		\label{thm:main-threshold}
		Let $\pi\colon \X\to Y$ be the coarse moduli map of a compact complex orbifold of complex dimension~$n$, and let $A$ be an ample line bundle on~$Y$ with Kähler form~$\omega_A$.
		Assume the curvature hypotheses \textup{(H1)}–\textup{(H4)} of \Cref{assump:curv}.
		Then$\colon $
		\begin{enumerate}[label=(\roman*), leftmargin=1.6em]
			\item The Green--Griffiths--Demailly hyperbolicity threshold depends only on the coarse Kähler class $[\omega_A]$ and not on the presence of orbifold or stack structure.
			\item Equivalently,
			\[
			\text{$Y$ is GGD--positive at some jet order $k_0$}
			\quad\Longleftrightarrow\quad
			\text{$\X$ is GGD--positive at the same order.}
			\]
			\item The minimal jet order and asymptotic growth rate of $h^0(E^{\mathrm{GG}}_{k,m}\otimes L^{-q})$ remain unchanged under coarse projection, finite quotient, or rigidification.
		\end{enumerate}
	\end{theorem}

Orbifold and stack generalizations of this program have become increasingly relevant in modern geometry; see Campana–P\u{a}un~\cite{CampanaPaun2016}, Borghesi–Tomassini~\cite{BorghesiTomassini2017}, and Toën–Vezzosi~\cite{ToenVezzosi2008}.
 Deligne--Mumford stacks equipped with orbifold Kähler forms or log pairs, as in Abramovich–Olsson–Vistoli~\cite{AbramovichOlssonVistoli2008}, naturally arise in moduli theory and arithmetic geometry, where curvature and cohomological tools must be extended to finite quotient groupoids~\cite{MoerdijkPronk1997,Lerman2008,KeelMori1997,StacksProj04V2}.
 However, the analytic underpinnings of the GGD framework---Bochner identities, Hörmander-type \(L^2\) estimates, and Riemann--Roch asymptotics---were originally formulated for manifolds, not orbifolds.
 This raises the structural question:
\begin{question}
Does the passage from a smooth variety to a smooth orbifold or stack, with the same coarse Kähler class, alter the positivity thresholds that govern the existence of invariant jet differentials?
\end{question}
 At first glance, the answer need not be obvious.
 The orbifold Riemann--Roch theorem of Satake~\cite{Satake1956} and Kawasaki~\cite{Kawasaki1979,Kawasaki1981} involves contributions from twisted sectors with denominators determined by isotropy representations.
 Thus, the asymptotic expansion of the Euler characteristic
 \[
 \chi(\X,E_{k,m}\otimes \pi^*A^{-q})
 \]
 where $\pi\colon \X\to Y$ is the coarse moduli map and $A$ is a line bundle on $Y$,
might, in principle, differ from its coarse counterpart on~$Y$.
 Furthermore, while the leading term of the Euler characteristic scales by \(1/s\), where \(s=|\mathrm{Stab}_{\mathrm{gen}}|\) is the generic stabilizer order, the actual number of global sections \(h^0\) could still depend on higher cohomology groups \(H^i(\X,E_{k,m}\otimes L^{-q})\).
 To establish structural invariance, one must show that these higher cohomology terms are suppressed by curvature positivity in the orbifold setting, ensuring that the growth of \(h^0\) and \(\chi\) coincide asymptotically.

The proof combines two analytic–cohomological mechanisms.
 First, the curvature–negativity–positivity package shows that vertical negativity of \(\ev_0^*A^{-1}\) on the \(1\)-jet bundle produces fiberwise positivity of the tautological bundle \(\OO_{\X_1}(1)\) via the Chern curvature formula on projectivized bundles; this positivity then propagates along the DS tower, ensuring semipositivity horizontally and strict positivity vertically.
 Second, a chartwise version of the Satake–Kawasaki–Toën Riemann--Roch theorem expresses the Euler characteristic as
 \[
 \chi(\X,E_{k,m}\otimes L^{-q})
 =\frac{1}{s}\int_Y \mathrm{ch}(E_{k,m})\,e^{m c_1(A)}\mathrm{Td}(TY)
 +O(m^{n-1}),
 \]
 with all twisted-sector terms of order \(O(m^{n-1})\).
 Together with an orbifold Kodaira-type vanishing theorem for DS bundles, which ensures that higher cohomology groups vanish for \(m,q\gg0\), the asymptotic growth of \(h^0\) and \(\chi\) coincide, yielding the stated invariance.

\begin{proposition}[Generic stabilizer and normalization of the untwisted integral]
 \label{prop:generic-stab-normalization}
 Let $\pi\colon \mathcal X\to Y$ be the coarse moduli map and let $n=\dim_{\C}\mathcal X$. Then$\colon $
 \begin{enumerate}[label=(\roman*)]
 \item (Existence and constancy) There exists a Zariski open dense sub-orbifold $\mathcal X^\circ\subset \mathcal X$ on which $|\Aut(x)|$ is constant. Its common value
 \[
 s \; \coloneq \; |\Stab_{\mathrm{gen}}|
 \]
 is the \emph{generic stabilizer order}. The function $x\mapsto |\Aut(x)|$ is upper semicontinuous, hence $|\Aut(x)|\ge s$ on $\mathcal X$.
 \item (Normalization) For any top-degree form $\alpha$ supported on the untwisted sector,
 \[
 \int_{\mathcal X}\alpha \;=\; \frac{1}{s}\int_Y \pi_*\alpha.
 \]
 Equivalently, on $\mathcal X^\circ$ the map $\pi$ is \'etale of degree $s$, and the identity-sector integral picks up the factor $1/s$.
 \end{enumerate}
 \end{proposition}

\begin{proposition}[Sectorwise asymptotics and slope control]
 \label{prop:slope-control}
 Let $\mathcal{X}$ be a connected compact complex orbifold of complex dimension $n$,
 and let $L=\pi^*A$ be the pullback of an ample line bundle on the coarse space $Y$.
 Then
 \[
 \chi(\mathcal{X},E\otimes L^{\otimes m})
 = \frac{1}{s}\int_Y \ch(E)\,e^{m\,c_1(A)}\,\Td(TY)
 + O(m^{n-1}),
 \qquad s=|\Stab_{\mathrm{gen}}|.
 \]
 Consequently, the sign of the asymptotic slope
 \[
 \mu(L) \coloneq \lim_{m\to\infty}\frac{\chi(\mathcal{X},E\otimes L^{\otimes m})}{m^n/n!} =\frac{1}{s}\int_Y \rk(E)\,c_1(A)^n
 \]
 is identical to that of the coarse manifold term.
 Twisted--sector corrections are of strictly lower order
 and therefore cannot alter the positivity or negativity of $\mu(L)$.
 \end{proposition}

\begin{lemma}[Untwisted control of cohomology dimensions]
 \label{lem:cohomology-control}
 Assume $(L,h)$ is a Hermitian line bundle with Nakano--positive curvature
 and that $L=\pi^*A$ descends from the coarse space.
 Then for all sufficiently large $m$,
 \[
 H^q(\mathcal{X},E\otimes L^{\otimes m})=0\quad\text{for all }q>0,
 \]
 and consequently
 \[
 h^0(\mathcal{X},E\otimes L^{\otimes m})
 =\chi(\mathcal{X},E\otimes L^{\otimes m})
 =\frac{1}{s}\int_Y \ch(E)\,e^{m\,c_1(A)}\,\Td(TY)
 +O(m^{n-1}).
 \]
 Thus the asymptotic growth of global sections is governed entirely by the untwisted (identity) sector.
 \end{lemma}

\begin{lemma}[Orbifold HRR asymptotics]
 \label{lem:eq-HRR}
 Let $\X$ be a compact complex orbifold with generic stabilizer of order $s$, and let $L=\pi^*A$ for an ample line bundle $A$ on the coarse space $Y$.
 Let $E^{\mathrm{inv}}_{k,m}$ denote the invariant jet (or DS) bundle.
 Then, as $m\to\infty$,
 \[
 \chi\!\bigl(\X,E^{\mathrm{inv}}_{k,m}\otimes L^{-q}\bigr)
 =\frac{1}{s}\!\int_Y\!
 \ch(\underline{E}^{\mathrm{inv}}_{k,m})\,e^{-q\,c_1(A)}\,\Td(TY)
 +O(m^{n-1}),
 \]
 and all twisted-sector terms contribute only $O(m^{n-1})$ due to the fixed-locus dimension drop and the $m$–independence of the Kawasaki denominator.
 \end{lemma}

\begin{theorem}[Stack-theoretic GGD degeneracy]
 \label{thm:stack-ggd-final}
 Under Assumption~\ref{ass:equiv-setting3}, there exist integers $k\gg1$, $m\gg1$, and $q=q(k,m)>0$ such that$\colon $
 \begin{enumerate}[label=(\roman*), leftmargin=1.6em]
 \item $H^0(\X,E^{\mathrm{inv}}_{k,m}\otimes L^{-q})\neq 0$;
 \item the common zero locus
 \[
 \mathcal{G}_k
 =\bigcap_{\substack{m\gg1\\0\le q\le q_0(k)}}
 Z\!\big(H^0(\X,E^{\mathrm{inv}}_{k,m}\otimes L^{-q})\big)
 \]
 is a proper closed analytic substack $\mathcal{G}_k\subsetneq\X$;
 \item every nonconstant entire map $f\colon \C\to\X$ satisfies $f(\C)\subset\mathcal{G}_k$.
 \end{enumerate}
 \end{theorem}

\begin{theorem}
 \label{thm:main-threshold}
 Let $\pi\colon \X\to Y$ be the coarse moduli map of a compact complex orbifold of complex dimension~$n$, and let $A$ be an ample line bundle on~$Y$ with Kähler form~$\omega_A$.
 Assume the curvature hypotheses \textup{(H1)}–\textup{(H4)} of \Cref{assump:curv}.
 Then$\colon $
 \begin{enumerate}[label=(\roman*), leftmargin=1.6em]
 \item The Green--Griffiths--Demailly hyperbolicity threshold depends only on the coarse Kähler class $[\omega_A]$ and not on the presence of orbifold or stack structure.
 \item Equivalently,
 \[
 \text{$Y$ is GGD--positive at some jet order $k_0$}
 \quad\Longleftrightarrow\quad
 \text{$\X$ is GGD--positive at the same order.}
 \]
 \item The minimal jet order and asymptotic growth rate of $h^0(E^{\mathrm{GG}}_{k,m}\otimes L^{-q})$ remain unchanged under coarse projection, finite quotient, or rigidification.
 \end{enumerate}
 \end{theorem}

The proof combines two analytic–cohomological mechanisms.
	First, the curvature–negativity–positivity package shows that vertical negativity of \(\ev_0^*A^{-1}\) on the \(1\)-jet bundle produces fiberwise positivity of the tautological bundle \(\OO_{\X_1}(1)\) via the Chern curvature formula on projectivized bundles; this positivity then propagates along the DS tower, ensuring semipositivity horizontally and strict positivity vertically.
	Second, a chartwise version of the Satake–Kawasaki–Toën Riemann--Roch theorem expresses the Euler characteristic as
	\[
	\chi(\mathcal{X},E_{k,m}\otimes L^{-q})
	=\frac{1}{s}\int_Y \mathrm{ch}(E_{k,m})\,e^{m c_1(A)}\mathrm{Td}(TY)
	+O(m^{n-1}),
	\]
	with all twisted-sector terms of order \(O(m^{n-1})\).
	Together with an orbifold Kodaira-type vanishing theorem for DS bundles, which ensures that higher cohomology groups vanish for \(m,q\gg0\), the asymptotic growth of \(h^0\) and \(\chi\) coincide, yielding the stated invariance.

	This conclusion situates the structural invariance of GGD thresholds at the intersection of analytic and stack-theoretic geometry.
	It refines Demailly’s curvature approach~\cite{Demailly1997,Demailly2011} and P\u{a}un’s vector-field method~\cite{Paun2008}, while connecting Satake–Kawasaki index theory~\cite{Kawasaki1979,Kawasaki1981} and modern stack Riemann--Roch theorems~\cite{Toen1999,Vistoli1989}.
	By aligning orbifold HRR asymptotics with curvature positivity, it confirms that orbifold structures preserve, rather than disturb, the hyperbolicity thresholds predicted by the Green--Griffiths--Lang conjecture.
	In particular, higher orbifold Betti numbers or stabilizers may influence lower-order corrections but have no effect on the leading asymptotic behavior that determines the GGD threshold.

	The remainder of the paper proceeds as follows.
	Section~\ref{sec:orbifold-hrr-ds} establishes the chartwise Riemann--Roch formula compatible with orbifold descent and applies it to jet bundles on $\X_k$.
	Section~\ref{sec:stack-ggd} develops the curvature–positivity package on the DS tower, derives \(L^2\)-vanishing and Bochner inequalities on orbifold charts, and proves the stack-theoretic GGD degeneracy theorem.
	Finally, Section~\ref{sec:ggd-structural-invariance} combines HRR asymptotics, Kodaira-type vanishing, and slope control to show that the GGD thresholds depend only on the coarse Kähler class and remain invariant under orbifold or stack structures, concluding with examples and applications.

	Throughout, compact analytic Deligne--Mumford stacks are identified with compact complex orbifolds after rigidification.
	Integration on $\mathcal{X}$ is normalized by the generic stabilizer order \(s=|\mathrm{Stab}_{\mathrm{gen}}|\), so that identity-sector integrals correspond to those on \(Y\) up to the factor \(1/s\).
	All Demailly--Semple and jet constructions are performed chartwise and descend by finite-group equivariance.

	\subsection*{Acknowledgements}
A substantial part of this work was carried out while the second author was visiting Texas State University to give a talk in the topology seminar.
The authors thank the host of the seminar Prof. Christine Lee, as well as the department chair and faculty members, for their warm hospitality and stimulating discussions.

To prove Theorem~\ref{thm:invariance}, the argument “combines two analytic–cohomological mechanisms.”

1) **Curvature/positivity along the Demailly–Semple (DS) tower.** The “curvature–negativity–positivity package shows that vertical negativity of \(\ev_0^*A^{-1}\) on the \(1\)-jet bundle produces fiberwise positivity of the tautological bundle \(\OO_{\X_1}(1)\) via the Chern curvature formula on projectivized bundles; this positivity then propagates along the DS tower, ensuring semipositivity horizontally and strict positivity vertically.”

2) **Orbifold HRR plus control of twisted sectors.** A “chartwise version of the Satake–Kawasaki–Toën Riemann--Roch theorem” gives
\[
\chi(\mathcal{X},E_{k,m}\otimes L^{-q})
=\frac{1}{s}\int_Y \mathrm{ch}(E_{k,m})\,e^{m c_1(A)}\mathrm{Td}(TY)
+O(m^{n-1}),
\]
“with all twisted-sector terms of order \(O(m^{n-1})\).”

3) **Vanishing to identify \(h^0\) with \(\chi\).** “Together with an orbifold Kodaira-type vanishing theorem for DS bundles, which ensures that higher cohomology groups vanish for \(m,q\gg0\), the asymptotic growth of \(h^0\) and \(\chi\) coincide,” and this “yield[s] the stated invariance” of the GGD threshold (i.e., dependence only on the coarse Kähler class and invariance under orbifold/stack structures).

To prove the main theorem, the argument “combines two analytic–cohomological mechanisms.”

1) **Curvature/positivity along the Demailly–Semple (DS) tower.** The “curvature–negativity–positivity package shows that vertical negativity of \(\ev_0^*A^{-1}\) on the \(1\)-jet bundle produces fiberwise positivity of the tautological bundle \(\OO_{\X_1}(1)\) via the Chern curvature formula on projectivized bundles; this positivity then propagates along the DS tower, ensuring semipositivity horizontally and strict positivity vertically.”

2) **Orbifold HRR plus control of twisted sectors.** A “chartwise version of the Satake–Kawasaki–Toën Riemann--Roch theorem” gives
\[
\chi(\mathcal{X},E_{k,m}\otimes L^{-q})
=\frac{1}{s}\int_Y \mathrm{ch}(E_{k,m})\,e^{m c_1(A)}\mathrm{Td}(TY)
+O(m^{n-1}),
\]
“with all twisted-sector terms of order \(O(m^{n-1})\).”

3) **Vanishing to identify \(h^0\) with \(\chi\).** “Together with an orbifold Kodaira-type vanishing theorem for DS bundles, which ensures that higher cohomology groups vanish for \(m,q\gg0\), the asymptotic growth of \(h^0\) and \(\chi\) coincide,” and this “yield[s] the stated invariance” of the GGD threshold (i.e., dependence only on the coarse Kähler class and invariance under orbifold/stack structures).

[Structural invariance of GGD thresholds {In establishing the main theorem,}{}]\label{thm:invariance}
		Let \(\pi\colon \mathcal{X}\to Y\) be the coarse moduli map of a compact smooth analytic Deligne--Mumford stack (orbifold) \(\mathcal{X}\), and let \(A\) be an ample line bundle on \(Y\) endowed with a smooth positively curved metric.
		Set \(L=\pi^*A\).
		Then there exist integers \(m_0,q_0>0\) such that for all \(m\ge m_0\) and \(q\ge q_0\)$\colon$

		\begin{enumerate}[label=(\roman*)]
			\item all higher cohomology groups vanish,
			\[
			H^i(\mathcal{X},E^{\mathrm{inv}}_{k,m}\otimes L^{-q})=0,\quad i>0;
			\]
			\item consequently,
			\[
			h^0\!\big(\mathcal{X},E^{\mathrm{inv}}_{k,m}\otimes L^{-q}\big)
			=\chi\!\big(\mathcal{X},E^{\mathrm{inv}}_{k,m}\otimes L^{-q}\big)
			=\frac{1}{s}\!\int_Y\! \mathrm{ch}(E_{k,m})\,e^{m c_1(A)}\,\mathrm{Td}(TY)
			+O(m^{n-1}),
			\]
			where \(s=|\mathrm{Stab}_{\mathrm{gen}}|\);
			\item the minimal jet order and asymptotic slope at which invariant jet differentials exist depend only on the coarse Kehler class \([\omega_A]\).
			Equivalently,
			\[
			\text{$Y$ is GGD-positive}
			\quad\Longleftrightarrow\quad
			\text{$\mathcal{X}$ is GGD-positive}.
			\]
		\end{enumerate}
		Thus, orbifold compactification and rigidification neither alter nor shift the GGD threshold.

Existential–Universal
Biconditional or Equivalence

Question: Let \(\pi\colon \mathcal{X}\to Y\) be the coarse moduli map of a compact smooth analytic Deligne--Mumford stack (orbifold), let \(A\) be an ample line bundle on \(Y\) with a smooth positively curved metric, and set \(L=\pi^*A\). Say that a space is GGD-positive if some invariant jet differential bundle \(E^{\mathrm{inv}}_{k,m}\otimes L^{-q}\) has a nonzero global section for suitable jet order \(k\) and integers \(m,q\). Which statement is equivalent to saying that the coarse space \(Y\) is GGD-positive?

The orbifold/stack \(\mathcal{X}\) is GGD-positive. Equivalently, the minimal jet order and the asymptotic slope at which invariant jet differentials exist depend only on the coarse K\"ahler class \([\omega_A]\), so orbifold compactification and rigidification do not alter or shift the GGD threshold.

Choices:
B. The orbifold/stack \(\mathcal{X}\) is GGD-positive after possibly replacing the jet order by some larger \(k'\ge k\) and rescaling the asymptotic slope by the generic stabilizer factor \(1/s\). Equivalently, orbifold compactification preserves existence of invariant jet differentials only up to this stabilizer-dependent shift of the GGD threshold.
C. The orbifold/stack \(\mathcal{X}\) is GGD-positive whenever the coarse space \(Y\) is GGD-positive.
D.
The orbifold/stack \(\mathcal{X}\) is GGD-positive if and only if the orbifold Euler characteristic satisfies
\[
\chi\!\big(\mathcal{X},E^{\mathrm{inv}}_{k,m}\otimes L^{-q}\big)>0
\]
for all sufficiently large \(m\) and \(q\), since the twisted sectors contribute only the factor \(1/s\) and therefore automatically force nonzero global sections without any higher-cohomology vanishing hypothesis.
E.
The orbifold/stack \(\mathcal{X}\) is GGD-positive if and only if there exist integers \(m_0,q_0>0\) such that for all \(m\ge m_0\) and \(q\ge q_0\),
\[
H^i(\mathcal{X},E^{\mathrm{inv}}_{k,m}\otimes L^{-q})=0\quad (i>0),
\]
and
\[
h^0\!\big(\mathcal{X},E^{\mathrm{inv}}_{k,m}\otimes L^{-q}\big)
=\chi\!\big(\mathcal{X},E^{\mathrm{inv}}_{k,m}\otimes L^{-q}\big)
=\frac{1}{s}\!\int_Y \mathrm{ch}(E_{k,m})\,e^{m c_1(A)}\,\mathrm{Td}(TY)+O(m^{n-1}),
\]
so the GGD threshold is characterized exactly by this asymptotic formula.

dropped the converse equivalence and threshold invariance

need for higher-cohomology vanishing to identify h^0 with chi

equivalence between positivity and eventual vanishing/asymptotic formula for all large m,q

The stem does not explicitly state that GGD-positivity is preserved between the coarse space and the stack. It sets up the objects and definition, but the correct equivalence is not directly revealed.

The item is close to theorem recall: it asks for a statement equivalent to Y being GGD-positive, and the correct choice is essentially a reformulation of the expected invariance theorem. The extra threshold-language makes it slightly more than a verbatim restatement, but only mildly.

Some reasoning is needed to reject the weaker true implication in C and the technically embellished but false statements B, D, and E. However, the question mainly tests recognition of the precise theorem rather than substantial derivation.

The distractors are mathematically plausible and target realistic failure modes: confusing equivalence with one-way implication, inserting a stabilizer-dependent shift, conflating positivity of Euler characteristic with existence of sections, and overstating asymptotic vanishing claims.

A reasonably strong MCQ with good distractors and limited answer leakage, but it leans heavily on theorem recognition and is only mildly non-tautological.