\label{cor:Grasssmannian}
Fix integers $1 \leq \ell < n$ and let $X = \mathrm{Gr}_{\ell,n}(\mathbb{R})$ be the Grassmann variety of $\ell$-dimensional subspaces in $\mathbb{R}^n$. Then there exists an explicit constant $\varkappa > 0$ and $\varepsilon > 0$ such that for $\sigma_X$-almost every $x \in X$, as $T \rightarrow + \infty$,
\begin{equation} \label{cor:eq_Grassmannian}
\mathcal{N}_{c, \beta}(x, T) = \varkappa \, c^d \, \ln (T) \left ( 1 + O_x(\ln(T)^{- \varepsilon}) \right ).
\end{equation}

\begin{thmx} [Effective equidistribution of maximal compact subgroup orbits] \label{thm:Effective}
Suppose the action of $G$ on $\Omega = G/\G$ has a spectral gap. Let $A = \{a(y) : y \in \R_+^{\times} \}$ be an $\Ad$-diagonalizable one-parameter subgroup of $G$ such that $a(e)$ projects non-trivially to each simple factor of $G$. Then there exist a constant $c > 0$ and an integer $r \geq 1$ such that for every compact subset $Q \subset \Omega$, for all $f \in C^{\infty}(K)$, $\phi \in C_c^\infty(\Omega)$, $x \in Q$, and $y \geq 1$, we have
\begin{equation} \label{eq:Single-Eq}
\int_K f(k) \phi \bigl(a(y) k x \bigr) \dd \mu_K(k) = \int_K f \dd \mu_K \int_{\Omega} \phi \dd \mu_{\Omega} + O \big ( y^{-c} \cS_r(f) \cS_r(\phi) \big ),
\end{equation}
and, for all $f \in C^{\infty}(K)$, $\phi_1, \phi_2 \in C_c^\infty(\Omega)$, $x_1, x_2 \in Q$,
and $y_2 \geq y_1 \geq 1$, we have
\begin{multline} \label{eq:Double-Eq}
\int_K f(k) \phi_{1}\bigl(a(y_1) k x_1 \bigr) \phi_{2} \bigl(a(y_2) k x_2 \bigr) d \mu_K(k) =  \int_K f \dd \mu_K \int_{\Omega} \phi_1 \dd \mu_{\Omega}  \int_{\Omega} \phi_2 \dd \mu_{\Omega} \\ + O \big ( \min \{y_1, y_2/y_1 \}^{-c} \cS_r(f) \cS_r(\phi_1) \cS_r(\phi_2) \big ),
\end{multline}
where the implicit constant in $O(\cdot)$ depends only on $Q$.
\end{thmx}

\begin{equation} \label{eq:SchmidtCountingFunction}
\cN_\psi(x,T) = \# \left \{ (p,q) \in \Z \times \N : 0 \leq q x - p < \psi(q), \, 1 \leq q < T \right \}
\end{equation}

\begin{equation} \label{eq:primitive_Siegel_Transform}
\forall \, g \in \mathrm{SL}_n(\mathbb{R}), \qquad S f(g\Z^n) = \sum_{\bm{v} \in \cP(\Z^n)} f (g \bm{v}).
\end{equation}

\begin{thmx} [$L^2$-integrability] \label{thm:L2}
Suppose that the Siegel transform $S_\chi$ maps $B_c^{\infty}(\widetilde{X})$ into $L^{2}(\Omega)$. Then
\[
\forall \, w \in W, \quad X^*(\bL_w^\circ)_{\Q} = \{1\}.
\]
In particular, the parabolic $\Q$-subgroup $\bP$ is maximal and the simple root $\alpha \in \Delta$, satisfying that $\bP = \bP_{\Delta \smallsetminus \{\alpha \}}$, has at most one neighbor in the associated Dynkin diagram.
\end{thmx}

\begin{equation} \label{eq:Norm}
\cS_r(\phi) = \sum_{\deg(\cD) \leq r} \left \|  \cD \phi \right \|_{\infty},
\end{equation}

\begin{equation} \label{eq:primitive_Siegel_Transform_variance}
\int_{\Omega} \left| S f - \frac{1}{\zeta(n)} \int_{\R^n} f \, \dd \lambda_{\R^n} \right|^2 \dd \mu_{\Omega} \, \ll \, \int_{\R^n} |f|^2 \, \dd \lambda_{\R^n}.
\end{equation}

The Siegel transform, introduced in 1945 by Siegel \cite{Siegel45}, maps a function of sufficient decay on the Euclidean space $\mathbb{R}^n$ to a function on the moduli space of unimodular lattices $\Omega = \mathrm{SL}_n(\mathbb{R}) / \mathrm{SL}_n(\mathbb{Z})$. Let ${\mathcal P}(\mathbb{Z}^n)$ denote the set of primitive elements of $\mathbb{Z}^n$ and let $B_c^{\infty}(\mathbb{R}^n)$ be the space of Borel measurable bounded compactly supported functions $f : \mathbb{R}^n \rightarrow \mathbb{C}$. Then, for every $f \in B_c^{\infty}(\mathbb{R}^n)$, the primitive Siegel transform $S f : \Omega \rightarrow \mathbb{C}$ of $f$ is defined by
\begin{equation} \label{eq:primitive_Siegel_Transform}
\forall \, g \in \mathrm{SL}_n(\mathbb{R}), \qquad S f(g\mathbb{Z}^n) = \sum_{\bm{v} \in {\mathcal P}(\mathbb{Z}^n)} f (g \bm{v}).
\end{equation}
Let $\mu_{\Omega}$ be the unique $\SL_{n}(\mathbb{R})$-invariant probability measure on $\Omega$, let $\zeta$ be the Riemann zeta function and let $\lambda_{\mathbb{R}^n}$ be the usual Lebesgue measure on $\mathbb{R}^n$. Siegel's mean value formula \cite{Siegel45} expresses the average of $Sf$ in terms of the average of $f$:
\begin{equation} \label{eq:primitive_Siegel_Transform_formula}
\int_{\Omega} S f \, \,\mathrm{d} \mu_{\Omega} = \frac{1}{\zeta(n)} \int_{\mathbb{R}^n} f \, \,\mathrm{d} \lambda_{\mathbb{R}^n}.
\end{equation}
Later, extending Siegel’s result, Rogers \cite{Rogers55} proved a $k$-th moment formula for the Siegel transform for $k$ up to $n-1$. A remarkable application of the second moment formula to the geometry of numbers was given by Schmidt \cite{Schmidt60b}, who derived an asymptotic formula for counting lattice points in an expanding family of sets in $\mathbb{R}^n$ from the variance bound
\begin{equation} \label{eq:primitive_Siegel_Transform_variance}
\int_{\Omega} \left| S f - \frac{1}{\zeta(n)} \int_{\mathbb{R}^n} f \, \,\mathrm{d} \lambda_{\mathbb{R}^n} \right|^2 \,\mathrm{d} \mu_{\Omega} \, \ll \, \int_{\mathbb{R}^n} |f|^2 \, \,\mathrm{d} \lambda_{\mathbb{R}^n}.
\end{equation}
In other words, the \emph{centered} Siegel transform $\overline{S} f = Sf - 1/\zeta(n) \int_{\mathbb{R}^n} f \, \,\mathrm{d} \lambda_{\mathbb{R}^n}$ extends to a bounded linear operator
\[
\overline{S} : L^2(\mathbb{R}^n) \rightarrow L^2(\Omega).
\]
The variance bound \eqref{eq:primitive_Siegel_Transform_variance} also yields an alternative proof of Schmidt’s strengthening \cite{Schmidt60a} of Khintchine’s theorem \cite{Khintchine26} in metric Diophantine approximation on $\mathbb{R}^n$ and its projective counterpart $\mathbb{P}(\mathbb{R}^n)$. There has been an active line of research extending classical results in Diophantine approximation from Euclidean space to other varieties, such as spheres \cite{AG22, KM15, KY23, Ouaggag23}, projective quadrics \cite{SK18, deSaxce22b, FKMS22}, Grassmannians \cite{deSaxce22a}, and more general flag varieties \cite{deSaxce20}.

The purpose of this paper is to study fundamental integrability properties of a natural extension of the Siegel transform \eqref{eq:primitive_Siegel_Transform} from the Euclidean space to the setting of generalized flag varieties. Our results have an application to metric Diophantine approximation on rank-one flag varieties. The proof of this application relies in addition on the effective single and double equidistribution property for expanding orbits of maximal compact subgroups, a result of independent interest.

\subsection{Main results}
Let ${\bf G}$ be a connected simply-connected almost $\mathbb{Q}$-simple $\mathbb{Q}$-group and let ${\bf P}$ be a proper parabolic $\mathbb{Q}$-subgroup of ${\bf G}$. We denote algebraic varieties defined over $\mathbb{Q}$ by bold letters and their sets of real points by ordinary letters. For instance, we write $G = {\bf G}(\mathbb{R})$ to denote the group of real points of ${\bf G}$. Let $\Gamma \subset {\bf G}(\mathbb{Q})$ be an arithmetic subgroup of $G$. Let $\pi_{\chi} : {\bf G} \rightarrow \GL(\bV_{\chi})$ be an irreducible representation defined over $\mathbb{Q}$ which is generated by a line $\bD_{\chi}$ defined over $\mathbb{Q}$ of highest weight $\chi$ such that ${\bf P} = \mathrm{Stab}_{{\bf G}} (\bD_{\chi})$ (Section~\ref{sec:Reps}). In particular, the space of real points $X = {\bf X}(\mathbb{R})$ of the generalized flag variety ${\bf X} = {\bf G} / {\bf P}$ embeds into the projective space $\mathbb{P}(V_{\chi})$. We fix a highest weight vector $\bm{e}_{\chi} \in \bD_{\chi}(\mathbb{Q})$ and define $\widetilde{X}$ to be the orbit $\widetilde{X} = G \, \bm{e}_{\chi} \subset V_{\chi}$. We refer to $\widetilde{X}$ as the \emph{cone over $X$ relative to $\chi$}. Fix a $\Gamma$-stable lattice $\bV_{\chi}(\mathbb{Z}) \subset \bV_{\chi}(\mathbb{Q})$ of $V_{\chi}$ and denote by $\cP_{\chi}$ the set of primitive elements of $\bV_{\chi}(\mathbb{Z}) \cap \widetilde{X}$. Let $B_c^{\infty}(\widetilde{X})$ be the space of Borel measurable bounded compactly supported complex-valued functions $f : \widetilde{X} \rightarrow \mathbb{C}$.
\begin{definition} [Siegel transform]
For every $f \in B_c^{\infty}(\widetilde{X})$, we define the \emph{Siegel transform} $S_{\chi} f : \Omega \rightarrow \mathbb{C}$ of $f$ by
\begin{equation*} \label{def:Siegel-Transform}
\forall \, g \in G, \quad S_{\chi} f(g \Gamma) = \sum_{\bm{v} \in \cP_{\chi}} f(g \bm{v}).
\end{equation*}
\end{definition}

Let $\mu_{\Omega}$ be the unique $G$-invariant Borel probability measure on the homogeneous space $\Omega = G/\Gamma$. We will answer the question: \emph{For any $p = 1,2, \infty$, what are necessary and sufficient conditions for $S_{\chi}$ to map $B_c^{\infty}(\widetilde{X})$ into $L^p(\Omega)$?}

In our first result, the equivalences $(1)$ - $(4)$ are likely known to experts; the formula \eqref{eq:WeilIntegrationFormula*} below is a consequence of a general integration formula due to Weil \cite[Theorem~2.51]{Folland15}. Let ${\bf L} = \mathrm{Stab}_{{\bf G}} (\bm{e}_{\chi}) \subset {\bf P}$.

\begin{thmx} [$L^1$-integrability] \label{thm:L1}
The following assertions are equivalent.
\begin{enumerate}[label=(\arabic*)]
\item The Siegel transform $S_\chi$ maps $B_c^{\infty}(\widetilde{X})$ into $L^1(\Omega)$.
\item There exists a unique (up to scaling) $G$-invariant Radon measure $\lambda_{\widetilde{X}}$ on $\widetilde{X}$, the Siegel transform $S_\chi$ extends to a bounded operator $S_{\chi} : L^1(\widetilde{X}) \rightarrow L^1(\Omega)$, and $\lambda_{\widetilde{X}}$ can be normalized so that we have a convergent mean value formula:
\begin{equation} \label{eq:WeilIntegrationFormula*}
\forall \, f \in L^1(\widetilde{X}), \quad \int_{\Omega} S_\chi f \, \,\mathrm{d} \mu_{\Omega} = \int_{\widetilde{X}} f \, \,\mathrm{d} \lambda_{\widetilde{X}}.
\end{equation}
\item The Lie group $L = {\bf L}(\mathbb{R})$ is unimodular and $\G_L = \Gamma \cap L$ is a lattice in $L$.
\item The parabolic $\mathbb{Q}$-subgroup ${\bf P}$ of ${\bf G}$ is maximal.
\item There exists $\varepsilon > 0$ such that $S_\chi$ maps $B_{c}^{\infty}(\widetilde{X})$ into $L^{1+\varepsilon}(\Omega)$.
\end{enumerate}
\end{thmx}

\begin{thmx} [$L^{\infty}$-integrability] \label{thm:Linfty}
The following assertions are equivalent.
\begin{enumerate} [label=(\arabic*)]
\item The Siegel transform $S_\chi$ maps $B_c^{\infty}(\widetilde{X})$ into $L^{\infty}(\Omega)$.
\item The $\mathbb{Q}$-rank of $\mathbf{G}$ is $1$.
\item The discrete group $\G_L$ is a cocompact lattice in $L$.
\end{enumerate}
\end{thmx}

Let $\bP_0$ be a minimal parabolic $\mathbb{Q}$-subgroup of ${\bf G}$ contained in ${\bf P}$ and let ${\bf T}$ be a maximal $\mathbb{Q}$-split torus of ${\bf G}$ contained in $\bP_0$. Let $\Phi$ be the root system of ${\bf G}$ relative to ${\bf T}$ and let $\Delta \subset \Phi$ be the corresponding set of simple roots. For each subset $\theta$ of $\Delta$, write $\bP_{\theta}$ for the associated standard parabolic $\mathbb{Q}$-subgroup of ${\bf G}$. Let $W$ be the Weyl group of ${\bf G}$ relative to ${\bf T}$. For every $w \in W$, we define $\bL_w = {\bf L} \cap x_w {\bf L} x_w^{-1}$, where $x_w \in \cN_{{\bf G}}({\bf T})(\mathbb{Q})$ is a representative of $w$, and denote by $X^*(\bL_w^{\circ})_{\mathbb{Q}}$ the group of $\mathbb{Q}$-characters of its identity component.

\begin{thmx} [$L^2$-integrability] \label{thm:L2}
Suppose that the Siegel transform $S_\chi$ maps $B_c^{\infty}(\widetilde{X})$ into $L^{2}(\Omega)$. Then
\[
\forall \, w \in W, \quad X^*(\bL_w^\circ)_{\mathbb{Q}} = \{1\}.
\]
In particular, the parabolic $\mathbb{Q}$-subgroup ${\bf P}$ is maximal and the simple root $\alpha \in \Delta$, satisfying that ${\bf P} = \bP_{\Delta \smallsetminus \{\alpha \}}$, has at most one neighbor in the associated Dynkin diagram.
\end{thmx}

We were unable to determine whether the converse statement is true: \emph{Assuming that for every $w \in W$, we have $X^*(\bL_w^\circ)_{\mathbb{Q}} = \{1\}$, does the Siegel transform $S_\chi$ map $B_c^{\infty}(\widetilde{X})$ into $L^2(\Omega)$?}

For every $f \in B_c^{\infty}(\widetilde{X})$, define the \emph{centered Siegel transform} $\overline{S}_{\chi} f : \Omega \rightarrow \mathbb{C}$ of $f$ by
\[
\forall \, g \in G, \qquad \overline{S}_{\chi} f(g\Gamma) = S_{\chi}f(g\Gamma) - \int_{\widetilde{X}} f \, \,\mathrm{d} \lambda_{\widetilde{X}}.
\]
Beyond the case $p=q=1$ in Theorem~\ref{thm:L1}, it is natural to ask for which pairs $p,q\in[1,+\infty]$ the Siegel transform, or its centered counterpart, extends to a bounded linear operator $L^p(\widetilde{X}) \to L^q(\Omega)$. More specifically, we would like to include the following question, due to Saxc\'e, suggesting a fractional version of the variance bound \eqref{eq:primitive_Siegel_Transform_variance} that also takes into account point~(5) of Theorem~\ref{thm:L1} as well as Theorem~\ref{thm:L2}:
\emph{Assuming that the parabolic $\mathbb{Q}$-subgroup ${\bf P}$ is maximal, does there exist $\varepsilon>0$ such that the centered Siegel transform $\overline{S}_{\chi}$ extends to a bounded linear operator}
\[
\overline{S}_{\chi} : L^{1+\varepsilon}(\widetilde{X}) \to L^{1+\varepsilon}(\Omega)\,?
\]

\subsection{Effective equidistribution of maximal compact subgroup orbits}
The fact that the Siegel transform $S_{\chi}$ maps $B_c^{\infty}(\widetilde{X})$ into $L^{1+\varepsilon}(\Omega)$ for some small $\varepsilon > 0$ when the parabolic subgroup ${\bf P}$ is maximal (Theorem~\ref{thm:L1}), together with the effective single and double equidistribution property for translated orbits of maximal compact subgroups (Theorem~\ref{thm:Effective}), are the key analytic inputs for our Schmidt-type counting theorem at the Diophantine exponent on rank-one flag varieties. Before describing this application, let us state here our equidistribution result, which will be derived from an effective multiple equidistribution result for expanding translates of horospherical orbits due to Shi (see \cite[Theorem~1.5]{Shi21}). Let $K \subset G$ be a maximal compact subgroup, equipped with the Haar probability measure $\mu_K$. We write $\cS_r$ ($r \in \mathbb{N}^*)$ for the degree $r$ Sobolev norms on $C_c^{\infty}(\Omega)$ and $C^{\infty}(K)$ as defined in \eqref{eq:Norm}.

\begin{thmx} [Effective equidistribution of maximal compact subgroup orbits] \label{thm:Effective}
Suppose the action of $G$ on $\Omega = G/\Gamma$ has a spectral gap. Let $A = \{a(y) : y \in \R_+^{\times} \}$ be an $\Ad$-diagonalizable one-parameter subgroup of $G$ such that $a(e)$ projects non-trivially to each simple factor of $G$. Then there exist a constant $c > 0$ and an integer $r \geq 1$ such that for every compact subset $Q \subset \Omega$, for all $f \in C^{\infty}(K)$, $\phi \in C_c^\infty(\Omega)$, $x \in Q$, and $y \geq 1$, we have
\begin{equation} \label{eq:Single-Eq}
\int_K f(k) \phi \bigl(a(y) k x \bigr) \,\mathrm{d} \mu_K(k) = \int_K f \,\mathrm{d} \mu_K \int_{\Omega} \phi \,\mathrm{d} \mu_{\Omega} + O \big ( y^{-c} \cS_r(f) \cS_r(\phi) \big ),
\end{equation}
and, for all $f \in C^{\infty}(K)$, $\phi_1, \phi_2 \in C_c^\infty(\Omega)$, $x_1, x_2 \in Q$,
and $y_2 \geq y_1 \geq 1$, we have
\begin{multline} \label{eq:Double-Eq}
\int_K f(k) \phi_{1}\bigl(a(y_1) k x_1 \bigr) \phi_{2} \bigl(a(y_2) k x_2 \bigr) d \mu_K(k) =  \int_K f \,\mathrm{d} \mu_K \int_{\Omega} \phi_1 \,\mathrm{d} \mu_{\Omega}  \int_{\Omega} \phi_2 \,\mathrm{d} \mu_{\Omega} \\ + O \big ( \min \{y_1, y_2/y_1 \}^{-c} \cS_r(f) \cS_r(\phi_1) \cS_r(\phi_2) \big ),
\end{multline}
where the implicit constant in $O(\cdot)$ depends only on $Q$.
\end{thmx}

\subsection{Application to Diophantine approximation on flag varieties}
Let us now state our Schmidt-type counting theorem at the Diophantine exponent on rank-one flag varieties, which uses Theorems~\ref{thm:L1} and \ref{thm:Effective} as inputs. Many classical results in Diophantine approximation on the real line $\mathbb{R}$ or in Euclidean space $\mathbb{R}^n$ admit a dynamical reinterpretation in terms of properties of certain diagonal orbits in the space of lattices $\Omega = \SL_n(\mathbb{R}) / \SL_n(\mathbb{Z})$; this is known as Dani's correspondence \cite{Dani85}. Via this dynamical reinterpretation and building on influential work of Margulis, Kleinbock and others \cite{KM96, KM98, KM99, FKMS22}, Saxc\'e \cite{deSaxce20} extended analogues of classical results to generalized flag varieties ${\bf X} = {\bf G} / {\bf P}$ defined over $\mathbb{Q}$. First examples of such varieties include projective $n$-space $\mathbb{P}^n(\mathbb{R})$, the Grassmann variety $\mathrm{Gr}_{\ell,n}(\mathbb{R})$ of $\ell$-dimensional subspaces of $\mathbb{R}^n$, projective quadric hypersurfaces (that is, the solution set in $\mathbb{P}^n(\mathbb{R})$ of a non-degenerate rational quadratic form in $n+1$ variables), and more general flag varieties, parametrizing flags of subspaces of a Euclidean space.

Let $\psi : \mathbb{N} \rightarrow (0,+\infty)$ be a non-increasing function. By Khintchine's theorem~\cite{Khintchine26}, the inequality
\[
0 \leq q x - p < \psi(q)
\]
has infinitely (resp. at most finitely) many solutions $(p,q) \in \mathbb{Z} \times \mathbb{N}$ for almost every $x \in \mathbb{R}$, if the series $\sum_{q = 1}^{\infty} \psi(q)$ diverges (resp. converges). In the divergence case, Schmidt~\cite{Schmidt60a} strengthened Khintchine's theorem. More precisely, for every $x \in \mathbb{R}$ and $T \geq 1$, he considered the counting function
\begin{equation} \label{eq:SchmidtCountingFunction}
\cN_\psi(x,T) = \# \left \{ (p,q) \in \mathbb{Z} \times \mathbb{N} : 0 \leq q x - p < \psi(q), \, 1 \leq q < T \right \}
\end{equation}
and showed that for almost every $x \in \mathbb{R}$, $\cN_\psi(x,T)$ is asymptotically equal to $\sum_{1\leq q < T} \psi(q)$ as $T$ goes to infinity, with an explicit error term. In fact, Schmidt's result holds not only for the real line, but also for the Euclidean space $\mathbb{R}^n$ of any dimension $n \geq 1$.

Our goal is to prove a version of this theorem, where the Euclidean space $\mathbb{R}^n$ is replaced by the space of real points $X = {\bf X}(\mathbb{R})$ of the generalized flag variety ${\bf X} = {\bf G} / {\bf P}$ defined over $\mathbb{Q}$. We assume that ${\bf P}$ is a maximal parabolic $\mathbb{Q}$-subgroup of ${\bf G}$ with abelian unipotent radical. In particular, ${\bf X}$ has $\mathbb{Q}$-rank $1$ and there exists a unique simple root $\alpha \in \Delta$ such that ${\bf P} = \bP_{\Delta \smallsetminus \{\alpha\}}$. Let $Y$ be the unique element in the Lie algebra of ${\bf T}(\mathbb{R})$ such that
\[
\alpha(Y) = -1 \quad \text{and} \quad \beta(Y) = 0 \quad \text{for all } \beta \in \Delta \smallsetminus \{\alpha\}.
\]
We suppose that the element $\exp(Y)$ projects non-trivially to each simple factor of $G$. Let $K$ be a maximal compact subgroup of $G$. Let $\sigma_X$ be the unique $K$-invariant probability measure on $X$. We equip $X$ with a $K$-invariant Riemannian distance $d(\cdot, \cdot)$ and the set of rational points ${\bf X}(\mathbb{Q})$ with a height function $H_\chi$ associated to an irreducible rational representation $\pi_{\chi} : {\bf G} \rightarrow \GL(\bV_{\chi})$ which is generated by a unique rational line $\bD_{\chi}$ of highest weight $\chi$ such that $\mathrm{Stab}_{{\bf G}} (\bD_{\chi}) = {\bf P}$ (see Section \ref{sec:Reps}). By \cite[Th\'eor\`emes 2.4.5 et 3.2.1]{deSaxce20}, there exists a rational number $\beta_\chi \in \Q_{>0}$ such that, for every $c > 0$ and for $\sigma_X$-almost every $x \in X$, the inequality
\begin{equation} \label{eq:DiophantineExpo}
d(x,v) < c \, H_\chi (v)^{-\tau}
\end{equation}
admits infinitely (resp. at most finitely) many solutions $v \in {\bf X}(\mathbb{Q})$, if $\tau \leq \beta_\chi$ (resp. $\tau > \beta_\chi$). We refer to $\beta_\chi$ as the \emph{Diophantine exponent} of $X$ relative to $\chi$ and to $\tau \in [0,\beta_{\chi}]$ as an \emph{approximation exponent}.

In analogy to \eqref{eq:SchmidtCountingFunction}, for every constant $c > 0$, approximation exponent $\tau \in [0, \beta_{\chi}]$, element $x \in X$, and parameter $T \geq 1$, we define
\[
\cN_{c,\tau}(x,T) = \# \left \{ v \in {\bf X}(\mathbb{Q}) : d(x,v) < c \, H_{\chi}(v)^{-\tau}, \, 1 \leq H_{\chi}(v) < T \right \}.
\]
In \cite{Pfitscher24}, we provided an almost-sure asymptotic formula for $\cN_{c,\tau}(x,T)$ as $T \rightarrow +\infty$, with an explicit error term in the case where $\tau \in [0, \beta_{\chi})$. Our method did not yield an effective estimate when counting \emph{at the Diophantine exponent}, that is, when $\tau = \beta_{\chi}$. In our application, we upgrade our previous result to an effective asymptotic estimate. Our approach is inspired by a recent effective counting result due to Ouaggag \cite[Theorem~1.2]{Ouaggag23} for spheres, and our result may be viewed as a substantial generalization thereof.

\begin{thmx}[Effective counting at the Diophantine exponent] \label{thm:Critical}
Let $d = \dim X$ be the dimension of $X$ and let $c > 0$. Then there exists an explicit constant $\varkappa > 0$ and $\varepsilon > 0$ such that for $\sigma_X$-almost every $x \in X$, as $T \rightarrow + \infty$,
\begin{equation} \label{eq:Thm_Critical}
\mathcal{N}_{c, \beta_{\chi}}(x, T) = \varkappa \, c^d \, \ln (T) \left ( 1 + O_x(\ln(T)^{- \varepsilon}) \right ).
\end{equation}
\end{thmx}

\subsection{Proof sketch of Theorem \ref{thm:Critical}}
Let us illustrate Theorem~\ref{thm:Critical} and its proof in the special case of the Grassmann variety $X = \mathrm{Gr}_{\ell, n}(\mathbb{R})$ ($\ell, n \in \mathbb{N}, \, 1 \leq \ell < n$), parametrizing $\ell$-dimensional subspaces of the Euclidean space $\mathbb{R}^n$; this theorem also applies to projective quadric hypersurfaces and we refer the reader to \cite[Sections $1.2$ and $8$]{Pfitscher24} and the references therein. The argument involves introducing the Siegel transform in this specific setting, studying its analytic properties, and establishing equidistribution of expanding translates of orbits of maximal compact subgroups.

Let ${\bf G} = \SL_n$, let ${\bf T} \leq {\bf G}$ be the maximal $\mathbb{Q}$-split $\mathbb{Q}$-torus given by the subgroup of ${\bf G}$ consisting of all diagonal matrices, and let $\bP_0$ be the Borel subgroup of ${\bf G}$ consisting of all upper-triangular matrices. Let $\Phi = \Phi({\bf G}, {\bf T})$ be the associated root system with ordering induced by $\bP_0$, $\Delta = \{\alpha_1, \dots, \alpha_{n-1}\}$ the set of simple roots, and $\{\lambda_{1}, \dots, \lambda_{n-1} \}$ the set of fundamental $\mathbb{Q}$-weights. Fix $\alpha_{\ell} \in \Delta$ and let $\chi = \lambda_{\ell}$ be the associated fundamental $\mathbb{Q}$-weight. Recall that for all $a = \diag(a_1, \dots, a_n) \in {\bf T}$, we have $\chi(a) = a_1 \cdots a_{\ell}$. Let ${\bf P} = \bP_{\Delta \smallsetminus \{\alpha_{\ell}\}}$ be the corresponding standard parabolic $\mathbb{Q}$-subgroup. Then ${\bf P}$ is the stabilizer in ${\bf G}$ of the rational line spanned by the pure tensor $\bm{e}_{\chi} = \bm{e}_1 \wedge \dots \wedge \bm{e}_{\ell}$ in the $\ell$-th exterior power of the standard representation of ${\bf G}$. The Siegel transform in this case is defined as follows. Let $\widetilde{X} = G \, \bm{e}_{\chi} \subset \bigwedge^\ell \mathbb{R}^n$: this is the set of all non-zero pure tensors of $\bigwedge^\ell \mathbb{R}^n$. Let ${\bf L} = \mathrm{Stab}_{{\bf G}} ( \bm{e}_{\chi})$, $\Gamma = \SL_n(\mathbb{Z})$ and let $\cP_{\chi}$ be the set of all primitive elements of $\bigwedge^\ell \mathbb{Z}^n$ that are contained in $\widetilde{X}$. The group $\Gamma$ acts transitively on $\cP_{\chi}$: $\cP_{\chi} = \Gamma \, \bm{e}_{\chi} \cong \Gamma/\G_L$. Therefore, for every $f \in B_c^{\infty}(\widetilde{X})$, the Siegel transform $S_{\chi} f : G / \Gamma \rightarrow \mathbb{C}$ is given by
\[
\forall \, g \in G, \qquad S_{\chi} f(g \Gamma) = \sum_{\bm{v} \in \cP_{\chi}} f(g \bm{v}) = \sum_{\gamma \in \Gamma / \G_L} f(g \gamma \bm{e}_{\chi}).
\]
Then $X = G/P$, viewed as a subvariety of $\mathbb{P}(\bigwedge^\ell \mathbb{R}^n)$ via the embedding $g P \mapsto g [\bm{e}_{\chi}]$ (here $[\bm{e}_{\chi}]$ denotes the projectivization of $\bm{e}_{\chi}$), is the Grassmann variety $\mathrm{Gr}_{\ell,n}(\mathbb{R})$ of $\ell$-dimensional subspaces of $\mathbb{R}^n$. This is in accordance with Schmidt's paper \cite{Schmidt67}, where he used the Pl\"ucker embedding to define the height $H(v)$ of a rational subspace $v$ of $\mathbb{R}^n$: for $v \in \mathrm{Gr}_{\ell,n}(\mathbb{Q})$ pick $\bm{v} \in \cP_{\chi}$ with $v = [\bm{v}]$ and set $H(v) = \|\bm{v}\|$, where $\|\cdot \|$ denotes the $\SO_n(\mathbb{R})$-invariant norm on $\bigwedge^{\ell} \mathbb{R}^n$ induced from the standard Euclidean norm on $\mathbb{R}^n$. The distance used on $X$ is the usual Riemannian distance and we equip $X$ with the unique probability measure $\sigma_X$ invariant under the action of the maximal compact subgroup $K = \SO_n(\mathbb{R}) \leq G$. We study the approximation of a real subspace chosen randomly according to $\sigma_X$ by rational subspaces. Write $d$ for the dimension $\dim_\mathbb{R} X = \ell(n-\ell)$. The Diophantine exponent of $X = \mathrm{Gr}_{\ell,n}(\mathbb{R})$ with respect to $\chi = \lambda_{\ell}$ is given by $\beta_{\chi} = \frac{n}{\ell(n-\ell)}$ (see \cite[Th\'eor\`eme~1]{deSaxce22a}). We wish to determine the asymptotic behavior of the counting function
\begin{equation} \label{eq:CountingFunction}
\cN_{c,\beta_{\chi}}(x,T) = \# \left \{ v \in \mathrm{Gr}_{\ell,n}(\mathbb{Q}) : d(x,v) < c \, H(v)^{-\beta_{\chi}}, \, H(v) < T \right \}
\end{equation}
as $T \rightarrow + \infty$, for $\sigma_X$-almost every $x \in X$. In fact, Theorem \ref{thm:Critical} takes the following form in this special case.

Let $\psi : \mathbb{N} \rightarrow (0,+\infty)$ be a non-increasing function. By Khintchine's theorem~\cite{Khintchine26}, the inequality
\[
0 \leq q x - p < \psi(q)
\]
has infinitely (resp. at most finitely) many solutions $(p,q) \in \mathbb{Z} \times \mathbb{N}$ for almost every $x \in \mathbb{R}$, if the series $\sum_{q = 1}^{\infty} \psi(q)$ diverges (resp. converges). In the divergence case, Schmidt~\cite{Schmidt60a} strengthened Khintchine's theorem. More precisely, for every $x \in \mathbb{R}$ and $T \geq 1$, he considered the counting function
\begin{equation} \label{eq:SchmidtCountingFunction}
\cN_\psi(x,T) = \# \left \{ (p,q) \in \mathbb{Z} \times \mathbb{N} : 0 \leq q x - p < \psi(q), \, 1 \leq q < T \right \}
\end{equation}
and showed that for almost every $x \in \mathbb{R}$, $\cN_\psi(x,T)$ is asymptotically equal to $\sum_{1\leq q < T} \psi(q)$ as $T$ goes to infinity, with an explicit error term. In fact, Schmidt's result holds not only for the real line, but also for the Euclidean space $\mathbb{R}^n$ of any dimension $n \geq 1$.

\begin{thmx}[Effective counting at the Diophantine exponent] \label{thm:Critical}
Let $d = \dim X$ be the dimension of $X$ and let $c > 0$. Then there exists an explicit constant $\varkappa > 0$ and $\varepsilon > 0$ such that for $\sigma_X$-almost every $x \in X$, as $T \rightarrow + \infty$,
\begin{equation} \label{eq:Thm_Critical}
\mathcal{N}_{c, \beta_{\chi}}(x, T) = \varkappa \, c^d \, \ln (T) \left ( 1 + O_x(\ln(T)^{- \varepsilon}) \right ).
\end{equation}
\end{thmx}

\subsection{Proof sketch of Theorem \ref{thm:Critical}}
Let us illustrate Theorem~\ref{thm:Critical} and its proof in the special case of the Grassmann variety $X = \mathrm{Gr}_{\ell, n}(\mathbb{R})$ ($\ell, n \in \mathbb{N}, \, 1 \leq \ell < n$), parametrizing $\ell$-dimensional subspaces of the Euclidean space $\mathbb{R}^n$; this theorem also applies to projective quadric hypersurfaces and we refer the reader to \cite[Sections $1.2$ and $8$]{Pfitscher24} and the references therein. The argument involves introducing the Siegel transform in this specific setting, studying its analytic properties, and establishing equidistribution of expanding translates of orbits of maximal compact subgroups.

Let ${\bf G} = \SL_n$, let ${\bf T} \leq {\bf G}$ be the maximal $\mathbb{Q}$-split $\mathbb{Q}$-torus given by the subgroup of ${\bf G}$ consisting of all diagonal matrices, and let $\bP_0$ be the Borel subgroup of ${\bf G}$ consisting of all upper-triangular matrices. Let $\Phi = \Phi({\bf G}, {\bf T})$ be the associated root system with ordering induced by $\bP_0$, $\Delta = \{\alpha_1, \dots, \alpha_{n-1}\}$ the set of simple roots, and $\{\lambda_{1}, \dots, \lambda_{n-1} \}$ the set of fundamental $\mathbb{Q}$-weights. Fix $\alpha_{\ell} \in \Delta$ and let $\chi = \lambda_{\ell}$ be the associated fundamental $\mathbb{Q}$-weight. Recall that for all $a = \diag(a_1, \dots, a_n) \in {\bf T}$, we have $\chi(a) = a_1 \cdots a_{\ell}$. Let ${\bf P} = \bP_{\Delta \smallsetminus \{\alpha_{\ell}\}}$ be the corresponding standard parabolic $\mathbb{Q}$-subgroup. Then ${\bf P}$ is the stabilizer in ${\bf G}$ of the rational line spanned by the pure tensor $\bm{e}_{\chi} = \bm{e}_1 \wedge \dots \wedge \bm{e}_{\ell}$ in the $\ell$-th exterior power of the standard representation of ${\bf G}$. The Siegel transform in this case is defined as follows. Let $\widetilde{X} = G \, \bm{e}_{\chi} \subset \bigwedge^\ell \mathbb{R}^n$: this is the set of all non-zero pure tensors of $\bigwedge^\ell \mathbb{R}^n$. Let ${\bf L} = \mathrm{Stab}_{{\bf G}} ( \bm{e}_{\chi})$, $\Gamma = \SL_n(\mathbb{Z})$ and let $\cP_{\chi}$ be the set of all primitive elements of $\bigwedge^\ell \mathbb{Z}^n$ that are contained in $\widetilde{X}$. The group $\Gamma$ acts transitively on $\cP_{\chi}$: $\cP_{\chi} = \Gamma \, \bm{e}_{\chi} \cong \Gamma/\G_L$. Therefore, for every $f \in B_c^{\infty}(\widetilde{X})$, the Siegel transform $S_{\chi} f : G / \Gamma \rightarrow \mathbb{C}$ is given by
\[
\forall \, g \in G, \qquad S_{\chi} f(g \Gamma) = \sum_{\bm{v} \in \cP_{\chi}} f(g \bm{v}) = \sum_{\gamma \in \Gamma / \G_L} f(g \gamma \bm{e}_{\chi}).
\]
Then $X = G/P$, viewed as a subvariety of $\mathbb{P}(\bigwedge^\ell \mathbb{R}^n)$ via the embedding $g P \mapsto g [\bm{e}_{\chi}]$ (here $[\bm{e}_{\chi}]$ denotes the projectivization of $\bm{e}_{\chi}$), is the Grassmann variety $\mathrm{Gr}_{\ell,n}(\mathbb{R})$ of $\ell$-dimensional subspaces of $\mathbb{R}^n$. This is in accordance with Schmidt's paper \cite{Schmidt67}, where he used the Pl\"ucker embedding to define the height $H(v)$ of a rational subspace $v$ of $\mathbb{R}^n$: for $v \in \mathrm{Gr}_{\ell,n}(\mathbb{Q})$ pick $\bm{v} \in \cP_{\chi}$ with $v = [\bm{v}]$ and set $H(v) = \|\bm{v}\|$, where $\|\cdot \|$ denotes the $\SO_n(\mathbb{R})$-invariant norm on $\bigwedge^{\ell} \mathbb{R}^n$ induced from the standard Euclidean norm on $\mathbb{R}^n$. The distance used on $X$ is the usual Riemannian distance and we equip $X$ with the unique probability measure $\sigma_X$ invariant under the action of the maximal compact subgroup $K = \SO_n(\mathbb{R}) \leq G$. We study the approximation of a real subspace chosen randomly according to $\sigma_X$ by rational subspaces. Write $d$ for the dimension $\dim_\mathbb{R} X = \ell(n-\ell)$. The Diophantine exponent of $X = \mathrm{Gr}_{\ell,n}(\mathbb{R})$ with respect to $\chi = \lambda_{\ell}$ is given by $\beta_{\chi} = \frac{n}{\ell(n-\ell)}$ (see \cite[Th\'eor\`eme~1]{deSaxce22a}). We wish to determine the asymptotic behavior of the counting function
\begin{equation} \label{eq:CountingFunction}
\cN_{c,\beta_{\chi}}(x,T) = \# \left \{ v \in \mathrm{Gr}_{\ell,n}(\mathbb{Q}) : d(x,v) < c \, H(v)^{-\beta_{\chi}}, \, H(v) < T \right \}
\end{equation}
as $T \rightarrow + \infty$, for $\sigma_X$-almost every $x \in X$. In fact, Theorem \ref{thm:Critical} takes the following form in this special case.

\begin{thmx}[Effective counting at the Diophantine exponent] \label{thm:Critical}
Let $d = \dim X$ be the dimension of $X$ and let $c > 0$. Then there exists an explicit constant $\varkappa > 0$ and $\varepsilon > 0$ such that for $\sigma_X$-almost every $x \in X$, as $T \rightarrow + \infty$,
\begin{equation} \label{eq:Thm_Critical}
\mathcal{N}_{c, \beta_{\chi}}(x, T) = \varkappa \, c^d \, \ln (T) \left ( 1 + O_x(\ln(T)^{- \varepsilon}) \right ).
\end{equation}
\end{thmx}

\begin{proof}
We shall need the following consequence of the proof of \cite[Theorem~C]{Pfitscher24}. For every $T \geq 1$, consider the function
\[
\cN(T) = \# \left \{v \in \bX(\Q) : H_{\chi}(v) < T \right \}
\]
counting rational points in $X$ of height $< T$. Let $\beta_{\chi} \in \Q_{>0}$ be the Diophantine exponent of $X$ with respect to $\chi$ (see \cite[D\'efinition~2.4.1 et Th\'eor\`eme~2.4.5]{deSaxce20}) and let $d = \dim X$ be the dimension of $X$. Then, as $T \rightarrow + \infty$, we have $\cN(T) \sim T^{\beta_{\chi} d}$. Since there is a one-to-one correspondence between points in $\bX(\Q)$ and lines passing through $\cP_{\chi}$, by the definition of the height function $H_{\chi}$, we also have that, as $T \rightarrow + \infty$
\begin{equation} \label{eq:Number_Primitive_Bounded_Height}
\# \left \{\bm{v} \in \cP_{\chi} : \|\bm{v}\| < T \right \} \asymp T^{\beta_{\chi} d}.
\end{equation}
Fix $f \in B_{c}^{\infty}(\widetilde{X})$ and pick $r = r(\supp(f)) \geq 1$ such that $\supp(f)$ is contained in $B_{\widetilde{X}}(r) = \{\bm{v} \in \widetilde{X} : \|\bm{v}\| < r \}$. The proof now proceeds using reduction theory as presented, for instance, in \cite[Section~12, Theorem~13.1]{Borel69}. By a slight abuse of notation, we let $\mathfrak{a}$ be the Lie algebra of $T^{\circ}$ and, for every $\tau \geq 0$, let $\mathfrak{a}_{\tau} = \{Y \in \mathfrak{a} : \forall \, \beta \in \Delta, \, \beta(Y) \leq \tau \}$. We set $A_{\tau} = \exp \, \mathfrak{a}_{\tau}$ and an note that $\mathfrak{a}^- = \mathfrak{a}_0$ is the negative Weyl chamber of $\mathfrak{a}$ with respect to $\Delta$. Let $\bM_0$ be the largest $\Q$-anisotropic $\Q$-subgroup of the centralizer $\cZ_{\bG}(\bT)^{\circ}$ in $\bG$ of $\bT$ and let $\bU_0$ be the unipotent radical of the minimal parabolic $\Q$-subgroup $\bP_0$. There exist $\tau > 0$, a compact subset $\bm{\omega}$ of $M_0 U_0$, and a finite subset $C \subset \bG(\Q)$ such that the Siegel set $\mathfrak{S} = K \, A_{\tau} \, \bm{\omega}$ satisfies
\[
G = \mathfrak{S} \, C \, \G.
\]
In particular, we can express, though not uniquely, each $g \in G$ as $g = k a n c \gamma$ with $k \in K$, $a \in A_{\tau}$, $n \in \bm{\omega}$, $c \in C$, and $\gamma \in \G$. Fix any norm $\|\cdot \|_{\mathfrak{a}}$ on $\mathfrak{a}$ and, for $r_0 > 0$, let $B_{\mathfrak{a}}(r_0)$ denote the corresponding ball centered at the origin with radius $r_0$. Let $r_0 > 0$ be such that $\mathfrak{a}_{\tau}$ is contained in $\mathfrak{a}^- + B_{\mathfrak{a}}(r_0)$. Let $k \in K$, $n \in \bm{\omega}$, $a \in A_{\tau}$, $c \in C$, and $\gamma \in \G$. We express $a = a^- \, \exp(O(1))$ with $a^- \in \exp(\mathfrak{a}^-)$. Using that $\lambda_\chi$ is right $\G$-invariant, that $K$ is compact, that $\bigcup_{a \in A_{\tau}} a \bm{\omega}a^{-1}$ is relatively compact (see \cite[Lemma~12.2]{Borel69}), and that $C \subset \bG(\Q)$ consists of rational elements and is finite, we have
\begin{equation} \label{eq:Proof-lem:Upper-Bound-Siegel}
\lambda_\chi(k a n c \gamma \G) \asymp \lambda_\chi(a^- \G).
\end{equation}
By the description of the $\Q$-weights of the representation $\pi_{\chi}$ in \eqref{eq:Q-weights}, for every $\Q$-weight $\mu$ of $\pi_{\chi}$, we have
\[
\chi(a^-) \leq \mu(a^-).
\]
Hence, since we assumed $\bV_{\chi}(\Z)$ to be spanned over $\Z$ by an orthonormal basis consisting of weight vectors for the action of $T$, we have $\lambda_\chi(a^-) = \chi(a^-)$. Thus, for every $\bm{v} \in V_{\chi}$, we have $\lambda_\chi(a^-) \| \bm{v} \| \leq \|a^- \bm{v} \|$. Using that the norm $\| \cdot \|$ on $V_{\chi}$ is $K$-invariant, that $\bigcup_{a \in A_{\tau}} a \bm{\omega}a^{-1}$ is relatively compact, and that $C \subset \bG(\Q)$ is finite, there exists a constant $C_0 \geq 1$, independent of $f$, such that, for every $g \in G$ with Siegel decomposition $g = k a n c \gamma$ (and writing $a = a^- \exp(O(1))$ as above), we have
\begin{align*}
|S_{\chi} f(g \G)| &\leq \|f\|_{\infty} \, \# \big \{ \bm{v} \in \cP_{\chi} : \| g \bm{v} \| < r \big \} \\
&\leq \|f\|_{\infty} \, \# \big \{ \bm{v} \in \cP_{\chi} : \| \bm{v} \| < C_0 \, \lambda_\chi(a^-)^{-1} \, r \big \}.
\end{align*}
By the estimate in \eqref{eq:Number_Primitive_Bounded_Height}, we further have
\[
\# \big \{ \bm{v} \in \cP_{\chi} : \| \bm{v} \| < C_0 \, \lambda_\chi(a^-)^{-1} \, r \big \} \, \ll_{\supp(f)} \, \lambda_\chi(a^-)^{-\beta_{\chi} d}.
\]
This together with \eqref{eq:Proof-lem:Upper-Bound-Siegel} now implies that
\[
|S_{\chi} f(g\G)| \ll_{\supp(f)} \|f\|_{\infty} \, \lambda_\chi(g \G)^{-\beta_\chi d},
\]
finishing the proof of the lemma.
\end{proof}

Let us now define these sets that approximate $\cE_{\beta_{\chi}}(T)$. We recall from Section \ref{sec:Distance} that the map $\ku^- \rightarrow X$ sending $u \mapsto \exp(u) x_0$ restricts to a diffeomorphism from a neighborhood of $1 \in \ku^-$ to a neighborhood of $x_0 \in X$. In particular, any $\bm{v} \in \widetilde{X}$, such that $[\bm{v}]$ is close to $x_0$, defines an element $u_{\bm{v}}^-$ in the Lie algebra $\ku^-$ by $[\bm{v}] = \exp(u_{\bm{v}}^-) x_0$. The adjoint action of $a(y) \in A$ on $\ku^- = T_{x_0} X$ acts by scalar multiplication: for all $y \in \R_+^\times$, $\Ad(a(y)) u^- = y \, u^-$. Observe that
\[
[a(y) \bm{v}] = a(y) [\bm{v}] = a(y) \exp(u_{\bm{v}}^-) a(y) a(y)^{-1} x_0 = \exp(\Ad(a(y)) u_{\bm{v}}^-) x_0.
\]
But we also have $[a(y) \bm{v}] = \exp(u_{a(y) \bm{v}}^- ) x_0$. By uniqueness, this gives the relation
\begin{equation} \label{eq:LieAlgebraDiagonal}
u_{a(y) \bm{v}}^- = y \, u_{\bm{v}}^-.
\end{equation}
Moreover, by the distance estimate \eqref{eq:Distance_Estimate}, there exists a constant $C_0 > 0$ such that
\[
\left | d(x_0, [\bm{v}]) - \|u_{\bm{v}}^-\|_{\ku^-} \right | \leq C_0 \, \|u_{\bm{v}}^-\|_{\ku^-}^2.
\]
Let $\pi^+ : V_\chi \rightarrow V_\chi$ be the orthogonal projection onto $\R\bm{e}_{\chi}$ and we abbreviate $\pi^+(\bm{v})$ by $\bm{v}^+$. For every $T \geq 1$ and $c > 0$ close to $1$, we will work with the sets
\[
\cE_{T, c}^+ = \{\bm{v} \in \widetilde{X} : \|u_{\bm{v}}^-\|_{\ku^-} < c \, \|\bm{v}^+\|^{-\beta_{\chi}}, 1 \leq \|\bm{v}^+\| < c \, T \}.
\]
By enlarging $C_0$ if necessary, we can assume that $\|\bm{v}^+\| \geq C_0^{-1} \|\bm{v}\|$ as soon as $d(x_0, [\bm{v}]) < 1$. For every integer $\ell \geq 1$, we let
\[
Q_{\ell} = \{\bm{v} \in \widetilde{X} : \|\bm{v}\| \leq C_0 \, \ell \}
\]
and we define
\[
\widehat{c}_{\ell} = \left ( 1 + C_0 \, \ell^{-\beta_{\chi}} \right )^{-(1+\beta_{\chi})} \in (0,1).
\]
In particular, we have $\widehat{c}_{\ell} \nearrow 1$ as $\ell \rightarrow + \infty$.

Let us now go through the main steps of the argument. For simplicity, we assume that $c = 1$ and we write $\cN_{\beta_{\chi}}(x,T) = \cN_{1,\beta_{\chi}}(x,T)$. The first observation is that the quantity $\cN_{\beta_{\chi}}(x,T)$ can be understood as the Siegel transform of the indicator function of a certain subset $\cE_{\beta_{\chi}}(T) \subset \widetilde{X}$ evaluated at a certain point in $\Omega = G/\Gamma$: we can associate to each $x \in X$ an element $k_{x} \in K$ such that
\[
\cN_{\beta_{\chi}}(x,T) = \# (\cP_{\chi} \cap k_{x} \, \cE_{\beta_{\chi}}(T)) = S_{\chi} \mathbbm{1}_{\cE_{\beta_{\chi}}(T)} (k_{x}^{-1} \Gamma).
\]
By Theorem \ref{thm:L1}, since ${\bf P} = \bP_{\Delta \smallsetminus \{\alpha_{\ell}\}}$ is maximal, the group $L = {\bf L}(\mathbb{R})$ is unimodular, $\widetilde{X} = G / L$ admits a unique up to scaling Radon measure $\lambda_{\widetilde{X}}$ and the expected value of $S_{\chi} \mathbbm{1}_{\cE_{\beta}(T)}$, viewed as a random variable on $\Omega$, is given by
\[
\int_{\Omega} S_{\chi} \mathbbm{1}_{\cE_{\beta_{\chi}}(T)} \, \,\mathrm{d} \mu_{\Omega} = \int_{\widetilde{X}} \mathbbm{1}_{\cE_{\beta_{\chi}}(T)} \, \,\mathrm{d} \lambda_{\widetilde{X}} = \lambda_{\widetilde{X}} (\cE_{\beta_{\chi}}(T)).
\]
The hope is that, for $\sigma_X$-almost every $x \in X$, the quantity $\cN_{\beta_{\chi}}(x,T)$ is asymptotically equal to the volume $\lambda_{\widetilde{X}} (\cE_{\beta_{\chi}}(T))$, as $T \rightarrow + \infty$, and this is what we will show. In fact, the main term on the right-hand side in \eqref{cor:eq_Grassmannian} is just the explicit value of the (main term of the) volume $\lambda_{\widetilde{X}} (\cE_{\beta_{\chi}}(T))$. In order to prove the desired asymptotic estimate, we will exploit the special geometry of the set $\cE_{\beta_{\chi}}(T)$. In fact, this set can be approximated by a set $\cE_{\beta_{\chi}}(T)^{+}$ that admits a simple decomposition under the action of the diagonal subgroup
\[
\forall \, y \in \R_+^\times, \quad a(y) = \diag \big (\underbrace{y^{-(n-\ell)/n}, \dots, y^{-(n-\ell)/n}}_{\text{$\ell$ times}}, \underbrace{y^{\ell/n}, \dots, y^{\ell/n}}_{\text{$n-\ell$ times}} \big ).
\]
Indeed, there exists a subset ${\mathcal F} \subset \widetilde{X}$ such that for all integers $N \geq 1$
\begin{equation} \label{eq:Proof_Sketch_Decomposition}
\cE_{\beta_{\chi}}(e^N)^+ = \bigsqcup_{i=0}^{N-1} a(e^{\beta_{\chi}})^{-i} {\mathcal F}.
\end{equation}

\begin{figure}[htbp]
\includegraphics[scale=0.6]{figure_counting.png}
\caption{The set $\cE_{\beta_{\chi}}(T)^+$ for the group $G = \SL_2(\mathbb{R})$, the flag variety the real projective line $X = \mathbb{P}^1(\mathbb{R}) = \mathrm{Gr}_{1,2}(\mathbb{R})$, the punctured affine cone $\widetilde{X} = \mathbb{R}^2 \setminus \{0\}$ above $X$, and the set $\cP_{\chi} = \mathcal{P}(\mathbb{Z}^2)$ of primitive elements of $\mathbb{Z}^2$. Rational approximations to a point $x \in X$ of height bounded by $T$ correspond to primitive lattice points of $\mathbb{Z}^2$ in the red region $k_x \cE_{\beta_{\chi}}(T)$, where $k_x \in \SO_2(\mathbb{R})$ is a rotation such that $x = k_x [\bm{e}_1]$. The action of $a(y) = \diag ( y^{-1/2}, y^{1/2})$ with $y > 1$ on $\widetilde{X} = \mathbb{R}^2 \setminus \{0\}$ contracts the line through $\bm{e}_\chi = \bm{e}_1$ and expands the line through $\bm{e}_2$. The domain $\cE_{\beta_{\chi}}(T)^+$ can be decomposed into translates of the elementary domain ${\mathcal F}$ under the action of $a(y)$. The hope is that for $x$ chosen randomly according to the Lebesgue measure on $X$ the number of primitive lattice points in the red region $k_x \cE_{\beta_{\chi}}(T)$, that is, the quantity $\# (\cP_{\chi} \cap k_x \, \cE_{\beta_{\chi}}(T))$, which is the classical primitive Siegel transform of the indicator function $\mathbbm{1}_{\cE_{\beta_{\chi}}(T)}$ evaluated at the rotated lattice $k_x^{-1} \mathbb{Z}^2$, is approximately given (up to a scalar) by the volume of $\cE_{\beta_{\chi}}(T)$.}
\label{fig:4}
\end{figure}

On the level of the Siegel transform this yields the sum decomposition
\[
S_{\chi} \mathbbm{1}_{\cE_{\beta_{\chi}}(T)^+}(k_x^{-1} \Gamma) = \sum_{i=0}^{N-1} \mathbbm{1}_{{\mathcal F}}(a(e^{\beta_{\chi}})^{i} k_x^{-1} \Gamma).
\]
From now on, we simply view $S_{\chi} \mathbbm{1}_{\cE_{\beta_{\chi}}(T)^+}(k \Gamma)$ as a random variable on the probability space $(K, \mu_K)$, where $\mu_K$ is the Haar probability measure of $K$.
Up to dividing the right-hand side by $N$, it is a Birkhoff sum, but we will not take this viewpoint. Instead, we shall try to bound a quantity related to the variance of $S_{\chi} \mathbbm{1}_{\cE_{\beta_{\chi}}(T)^+}$ and then conclude by a Borel-Cantelli argument. More specifically, we shall bound a $(1+\varepsilon)$-moment, for some $\varepsilon > 0$, of the centered Siegel transform $S_{\chi} \mathbbm{1}_{\cE_{\beta_{\chi}}(T)^+} - \lambda_{\widetilde{X}}(\cE_{\beta_{\chi}}(T)^+)$, viewed as a random variable on $(K,\mu_K)$: if we can show that for some $\varepsilon > 0$ and all $N \geq 1$,
\begin{equation} \label{eq:Proof_Sketch}
\int_K \left | S_{\chi} \mathbbm{1}_{\cE_{\beta_{\chi}}(e^N)^+}(k\Gamma) - \lambda_{\widetilde{X}}(\cE_{\beta_{\chi}}(e^N)^+) \right |^{1+\varepsilon} \, \,\mathrm{d} \mu_{K}(k) \, \ll \, N,
\end{equation}
then there exists $c > 0$ and $\nu(\varepsilon) \in (0,1)$ such that for $\mu_K$-almost every $k \in K$,
\[
S_{\chi} \mathbbm{1}_{\cE_{\beta_{\chi}}(e^N)^+}(k\Gamma) = c \, N \, \left ( 1 + O_x(N^{-\nu(\varepsilon)})\right ),
\]
as required. Due to integrability issues of the Siegel transform at this level of generality (see Theorems \ref{thm:L1} and \ref{thm:L2}), we are forced to work with $1 + \varepsilon$ for some small $\varepsilon > 0$ instead of $2$, which would represent the usual variance.
Using the decomposition \eqref{eq:Proof_Sketch_Decomposition}, we express the argument in the integral of \eqref{eq:Proof_Sketch} as
\begin{equation*}
S_{\chi} \mathbbm{1}_{\cE_{\beta_{\chi}}(e^N)^+}(k\Gamma) - \lambda_{\widetilde{X}}(\cE_{\beta_{\chi}}(e^N)^+)
= \sum_{i=0}^{N-1} \bigg ( S_{\chi} \mathbbm{1}_{{\mathcal F}} (a(e^{\beta_{\chi}}) k_x^{-1} \Gamma) - \lambda_{\widetilde{X}}({\mathcal F}) \bigg ).
\end{equation*}
and obtain the bound in \eqref{eq:Proof_Sketch} using the effective single and double equidistribution property of expanding translates of $K$-orbits. In particular, we will need to work with smooth compactly supported functions that, on translated $K$-orbits, approximate the Siegel transform $S_{\chi} \mathbbm{1}_{{\mathcal F}}$, which typically is neither smooth nor compactly supported.

\subsection{Notation and conventions}
We use the Landau notation $O(\cdot)$ and the Vinogradov symbol $\ll$. Given $A, B > 0$, we use the notation $A \gg B$ for $B \ll A$, and $A \asymp B$ for $A \ll B \ll A$. We use subscripts to indicate the dependence of the constant on parameters.
For simplicity of exposition, we will work with the set of complex points of an algebraic variety defined over $\mathbb{Q}$, and refer to it simply as the variety itself when no confusion arises. For instance, we write $G = {\bf G}(\mathbb{R})$ and ${\bf G} = {\bf G}(\mathbb{C})$ to denote the groups of real and complex points of ${\bf G}$, respectively. Given a discrete subgroup $\Gamma \leq G$ and a closed subgroup $H \leq G$, we write $\G_H$ for $\Gamma \cap H$. Discrete groups are always equipped with the counting measure.

\vspace{5mm}
\textbf{Acknowledgments}.
I am very grateful to Nicolas de Saxc\'e for introducing me to this topic, during my doctoral thesis under his supervision, and for sharing with me crucial insights that contributed to the proofs of Theorems \ref{thm:L2} and \ref{thm:Linfty}. I also thank Shucheng Yu for useful discussions, and Fr\'ed\'eric Paulin for numerous corrections and suggestions that led, in particular, to the removal of a restrictive hypothesis in Theorem \ref{thm:Critical} (namely, that $\beta_{\chi} \leq 1$).

Let us now go through the main steps of the argument (for simplicity, $c=1$). One first rewrites $\cN_{\beta_{\chi}}(x,T)$ as a Siegel transform: for $x\in X$ one associates $k_x\in K$ such that
\[
\cN_{\beta_{\chi}}(x,T)=\#(\cP_{\chi}\cap k_x\cE_{\beta_{\chi}}(T))=S_{\chi}\mathbbm{1}_{\cE_{\beta_{\chi}}(T)}(k_x^{-1}\Gamma).
\]
By Theorem \ref{thm:L1}, the expected value (on $\Omega$) of this random variable is
\[
\int_{\Omega} S_{\chi} \mathbbm{1}_{\cE_{\beta_{\chi}}(T)}\,d\mu_{\Omega}=\int_{\widetilde{X}}\mathbbm{1}_{\cE_{\beta_{\chi}}(T)}\,d\lambda_{\widetilde{X}}=\lambda_{\widetilde{X}}(\cE_{\beta_{\chi}}(T)).
\]
The goal is to show that for $\sigma_X$-almost every $x$, $\cN_{\beta_{\chi}}(x,T)$ is asymptotically equal to this volume as $T\to+\infty$; moreover, “the main term on the right-hand side in \eqref{cor:eq_Grassmannian} is just the explicit value of the (main term of the) volume $\lambda_{\widetilde{X}}(\cE_{\beta_{\chi}}(T))$.”

To get an effective almost-everywhere estimate, one exploits the geometry of $\cE_{\beta_{\chi}}(T)$ by approximating it with a set $\cE_{\beta_{\chi}}(T)^+$ that decomposes under the diagonal subgroup $a(y)$: there exists ${\mathcal F}\subset\widetilde{X}$ such that for integers $N\ge1$,
\[
\cE_{\beta_{\chi}}(e^N)^+=\bigsqcup_{i=0}^{N-1} a(e^{\beta_{\chi}})^{-i}{\mathcal F}. \tag{\ref{eq:Proof_Sketch_Decomposition}}
\]
On the Siegel-transform level this gives
\[
S_{\chi}\mathbbm{1}_{\cE_{\beta_{\chi}}(T)^+}(k_x^{-1}\Gamma)=\sum_{i=0}^{N-1}\mathbbm{1}_{{\mathcal F}}(a(e^{\beta_{\chi}})^i k_x^{-1}\Gamma),
\]
and one views $S_{\chi}\mathbbm{1}_{\cE_{\beta_{\chi}}(T)^+}(k\Gamma)$ as a random variable on $(K,\mu_K)$.

Rather than using a Birkhoff-sum viewpoint, the argument “bound[s] a quantity related to the variance … and then conclude[s] by a Borel–Cantelli argument,” namely a $(1+\varepsilon)$-moment of the centered Siegel transform. Concretely, if for some $\varepsilon>0$ and all $N\ge1$,
\[
\int_K\Big|S_{\chi}\mathbbm{1}_{\cE_{\beta_{\chi}}(e^N)^+}(k\Gamma)-\lambda_{\widetilde{X}}(\cE_{\beta_{\chi}}(e^N)^+)\Big|^{1+\varepsilon}\,d\mu_K(k)\ll N, \tag{\ref{eq:Proof_Sketch}}
\]
then one deduces that for $\mu_K$-almost every $k\in K$,
\[
S_{\chi}\mathbbm{1}_{\cE_{\beta_{\chi}}(e^N)^+}(k\Gamma)=c\,N\,(1+O_x(N^{-\nu(\varepsilon)})).
\]
Because of “integrability issues of the Siegel transform … we are forced to work with $1+\varepsilon$ … instead of $2$.” Using the decomposition \eqref{eq:Proof_Sketch_Decomposition}, the centered sum is written as a sum of $N$ terms involving $S_{\chi}\mathbbm{1}_{\mathcal F}$, and the bound \eqref{eq:Proof_Sketch} is obtained “using the effective single and double equidistribution property of expanding translates of $K$-orbits,” after approximating $S_{\chi}\mathbbm{1}_{\mathcal F}$ by “smooth compactly supported functions” on translated $K$-orbits.

Let us now go through the main steps of the argument (for simplicity, $c=1$). One first rewrites $\cN_{\beta_{\chi}}(x,T)$ as a Siegel transform: for $x\in X$ one associates $k_x\in K$ such that
\[
\cN_{\beta_{\chi}}(x,T)=\#(\cP_{\chi}\cap k_x\cE_{\beta_{\chi}}(T))=S_{\chi}\mathbbm{1}_{\cE_{\beta_{\chi}}(T)}(k_x^{-1}\Gamma).
\]
We first recall the following theorem.
\begin{thmx} [$L^1$-integrability] \label{thm:L1}
The following assertions are equivalent.
\begin{enumerate}[label=(\arabic*)]
\item The Siegel transform $S_\chi$ maps $B_c^{\infty}(\widetilde{X})$ into $L^1(\Omega)$.
\item There exists a unique (up to scaling) $G$-invariant Radon measure $\lambda_{\widetilde{X}}$ on $\widetilde{X}$, the Siegel transform $S_\chi$ extends to a bounded operator $S_{\chi} : L^1(\widetilde{X}) \rightarrow L^1(\Omega)$, and $\lambda_{\widetilde{X}}$ can be normalized so that we have a convergent mean value formula:
\begin{equation} \label{eq:WeilIntegrationFormula*}
\forall \, f \in L^1(\widetilde{X}), \quad \int_{\Omega} S_\chi f \, \dd \mu_{\Omega} = \int_{\widetilde{X}} f \, \dd \lambda_{\widetilde{X}}.
\end{equation}
\item The Lie group $L = \bL(\R)$ is unimodular and $\G_L = \G \cap L$ is a lattice in $L$.
\item The parabolic $\Q$-subgroup $\bP$ of $\bG$ is maximal.
\item There exists $\varepsilon > 0$ such that $S_\chi$ maps $B_{c}^{\infty}(\widetilde{X})$ into $L^{1+\varepsilon}(\Omega)$.
\end{enumerate}
\end{thmx}
By this theorem, the expected value (on $\Omega$) of this random variable is
\[
\int_{\Omega} S_{\chi} \mathbbm{1}_{\cE_{\beta_{\chi}}(T)}\,d\mu_{\Omega}=\int_{\widetilde{X}}\mathbbm{1}_{\cE_{\beta_{\chi}}(T)}\,d\lambda_{\widetilde{X}}=\lambda_{\widetilde{X}}(\cE_{\beta_{\chi}}(T)).
\]
The goal is to show that for $\sigma_X$-almost every $x$, $\cN_{\beta_{\chi}}(x,T)$ is asymptotically equal to this volume as $T\to+\infty$; moreover, the main term in the volume asymptotic is given by the explicit formula
\begin{equation} \label{cor:eq_Grassmannian}
\mathcal{N}_{c, \beta}(x, T) = \varkappa \, c^d \, \ln (T) \left ( 1 + O_x(\ln(T)^{- \varepsilon}) \right ).
\end{equation}

To get an effective almost-everywhere estimate, one exploits the geometry of $\cE_{\beta_{\chi}}(T)$ by approximating it with a set $\cE_{\beta_{\chi}}(T)^+$ that decomposes under the diagonal subgroup $a(y)$: there exists ${\mathcal F}\subset\widetilde{X}$ such that for integers $N\ge1$,
\begin{equation} \label{eq:Proof_Sketch_Decomposition}
\cE_{\beta_{\chi}}(e^N)^+ = \bigsqcup_{i=0}^{N-1} a(e^{\beta_{\chi}})^{-i} \cF.
\end{equation}
On the Siegel-transform level this gives
\[
S_{\chi}\mathbbm{1}_{\cE_{\beta_{\chi}}(T)^+}(k_x^{-1}\Gamma)=\sum_{i=0}^{N-1}\mathbbm{1}_{{\mathcal F}}(a(e^{\beta_{\chi}})^i k_x^{-1}\Gamma),
\]
and one views $S_{\chi}\mathbbm{1}_{\cE_{\beta_{\chi}}(T)^+}(k\Gamma)$ as a random variable on $(K,\mu_K)$.

Rather than using a Birkhoff-sum viewpoint, the argument “bound[s] a quantity related to the variance … and then conclude[s] by a Borel–Cantelli argument,” namely a $(1+\varepsilon)$-moment of the centered Siegel transform. Concretely, if for some $\varepsilon>0$ and all $N\ge1$,
\begin{equation} \label{eq:Proof_Sketch}
\int_K \left | S_{\chi} \mathbbm{1}_{\cE_{\beta_{\chi}}(e^N)^+}(k\G) - \lambda_{\widetilde{X}}(\cE_{\beta_{\chi}}(e^N)^+) \right |^{1+\varepsilon} \, \dd \mu_{K}(k) \, \ll \, N,
\end{equation}
then one deduces that for $\mu_K$-almost every $k\in K$,
\[
S_{\chi}\mathbbm{1}_{\cE_{\beta_{\chi}}(e^N)^+}(k\Gamma)=c\,N\,(1+O_x(N^{-\nu(\varepsilon)})).
\]
Because of “integrability issues of the Siegel transform … we are forced to work with $1+\varepsilon$ … instead of $2$.” Using the decomposition above, the centered sum is written as a sum of $N$ terms involving $S_{\chi}\mathbbm{1}_{\mathcal F}$, and the bound above is obtained “using the effective single and double equidistribution property of expanding translates of $K$-orbits,” after approximating $S_{\chi}\mathbbm{1}_{\mathcal F}$ by “smooth compactly supported functions” on translated $K$-orbits.

Asymptotic or Limit
Existential–Universal

Question: Fix integers \(1\le \ell<n\), let \(X=\mathrm{Gr}_{\ell,n}(\mathbb{R})\) be the Grassmannian of \(\ell\)-dimensional subspaces of \(\mathbb{R}^n\), and let \(d=\dim X=\ell(n-\ell)\). Let \(\sigma_X\) be the natural measure on \(X\). For a fixed \(c>0\), let \(\mathcal N_{c,\beta}(x,T)\) denote the counting function for rational \(\ell\)-planes in \(X\) of height less than \(T\) that approximate \(x\) at the critical exponent \(\beta\). As \(T\to+\infty\), which asymptotic statement holds for \(\mathcal N_{c,\beta}(x,T)\) for \(\sigma_X\)-almost every \(x\in X\)?

There exist explicit constants \(\varkappa>0\) and \(\varepsilon>0\) such that for \(\sigma_X\)-almost every \(x\in X\),
\[
\mathcal N_{c,\beta}(x,T)=\varkappa\,c^d\,\ln(T)\left(1+O_x\bigl((\ln T)^{-\varepsilon}\bigr)\right)
\]
as \(T\to+\infty\).

Choices:
B.
There exist explicit constants \(\varkappa>0\) and \(\varepsilon>0\) such that for every \(x\in X\),
\[
\mathcal N_{c,\beta}(x,T)=\varkappa\,c^d\,\ln(T)\left(1+O_x\bigl((\ln T)^{-\varepsilon}\bigr)\right)
\]
as \(T\to+\infty\).
C.
There exists an explicit constant \(\varkappa>0\) such that for \(\sigma_X\)-almost every \(x\in X\),
\[
\mathcal N_{c,\beta}(x,T)\sim \varkappa\,c^d\,\ln(T)
\]
as \(T\to+\infty\).
D.
There exist explicit constants \(\varkappa>0\) and \(\varepsilon>0\) such that for \(\sigma_X\)-almost every \(x\in X\),
\[
\mathcal N_{c,\beta}(x,T)=\varkappa\,c^d\,T\left(1+O_x\bigl(T^{-\varepsilon}\bigr)\right)
\]
as \(T\to+\infty\).
E.
There exist explicit constants \(\varkappa>0\) and \(\varepsilon>0\) such that for \(\sigma_X\)-almost every \(x\in X\),
\[
\mathcal N_{c,\beta}(x,T)=\varkappa\,c^d\,\ln(T)\left(1+O_x\bigl(T^{-\varepsilon}\bigr)\right)
\]
as \(T\to+\infty\).

The stem does not state the asymptotic form, growth scale, quantifier, or error term. It only asks for the correct asymptotic at the critical exponent, so the answer is not leaked.

The item is close to theorem recall: it asks which asymptotic statement holds for the counting function. However, it is not a pure restatement because the options force distinctions between almost-everywhere vs everywhere, effective vs noneffective asymptotics, and logarithmic vs polynomial growth.

Some reasoning is required to choose the strongest valid conclusion among nearby alternatives, especially between the correct effective statement and the weaker true asymptotic. Still, the question mainly tests recognition/recall of a known theorem rather than deeper generative derivation.

The distractors are mathematically close and target realistic failure modes: overstrengthening the quantifier (B), choosing a weaker true conclusion instead of the strongest one (C), using the wrong main growth scale (D), and mismatching the decay variable in the error term (E). They are distinct and plausibly confusable.

A solid theorem-discrimination MCQ with no answer leakage and strong distractors, but it leans more toward precise theorem recall than genuinely generative mathematical reasoning.


\begin{theorem}\label{main theorem}
 	Let $q \ge 3$ be fixed. Uniformly for
 $
 	\tau \le \log n + \log_2 n - \theta(n),
 	$
 	where $2\leq \theta(n)\ll \log_3 n$ is any function tending arbitrarily slowly to infinity as $n \to \infty$, we have
 	\[
 	\phi_n(\tau)
 	:= \frac{1}{|\mathcal{H}_n|}
 	\sum_{\substack{D \in \mathcal{H}_n \\ L(1, \chi_D) \ge e^{\gamma}\tau}} 1
 	= \exp\!\left(
 	-\,C_1\!\left(q^{\{\log \kappa(\tau)\}}\right)
 	\,\frac{q^{\,\tau - C_0\!\left(q^{\{\log \kappa(\tau)\}}\right)}}{\tau}
 	\bigl(1 + o_\theta(1)\bigr)
 	\right),
 	\]
 	where the constants $C_0\!\left(q^{\{\log \kappa(\tau)\}}\right)$ and $C_1\!\left(q^{\{\log \kappa(\tau)\}}\right)$ depend on $\tau$, $q$, and $\kappa(\tau)$, yet remain bounded as the argument varies between $1$ and $q$.
 	Further details on these quantities can be found in~\cite[Theorem~1.3]{Lumley}.
 \end{theorem}

\begin{lemma}\label{character sum over nonsquare}
	Let $\ell=\ell_1 \ell_2^2\neq \square\in \mathbb{F}_q[t]$ with $\ell_1$ square-free. For any $\varepsilon>0$, we have
	\[
	\sum_{D\in \mathcal{H}_n}\chi_D(\ell)\ll  q^{(1/2+\varepsilon)n}\exp\left((d(\ell_1))^{1-\varepsilon}\right)\prod_{P\mid \ell_2}\left(1+|P|^{-1/2-\varepsilon}\right).
	\]

\end{lemma}

One of the central problems in analytic number theory is to determine the value distribution and obtain sharp bounds for the maximal size of Dirichlet $L$-functions on the line $\operatorname{Re}(s)=1$. These quantities are deeply intertwined with fundamental arithmetic invariants such as class numbers of number fields, and are closely linked to estimates for character sums via Fourier analysis. In particular, the tail behavior of the value distribution plays a decisive role in determining the true order of magnitude of extreme values.

Over the past few decades, significant progress has been made toward understanding the tails of these distributions and the maximal size of $L$-functions on $\operatorname{Re}(s)=1$. This line of research encompasses the Riemann zeta function and Dirichlet $L$-functions in the modulus aspect (see~\cite{AMM, AMMP, BS3, DOKIC, GSzeta}), as well as the family of quadratic Dirichlet $L$-functions $L(1,\chi_d)$ (see~\cite{DM, GS, Lumley}), and more generally, various families of automorphic $L$-functions (see~\cite{Lamzouri}).

An important $\mathrm{GL}(1)$ family of symplectic type is formed by the quadratic Dirichlet $L$-functions $L(s,\chi_d)$, where $d$ ranges over fundamental discriminants. The analytic study of this family is notably challenging due to the lack of a natural orthogonality relation among the characters. The tail behavior of $L(1,\chi_d)$ is closely tied to the distribution of class numbers of quadratic number fields. In this direction, Granville and Soundararajan~\cite{GS} carried out a seminal analysis of these quantities, and following their approach, Dahl and Lamzouri~\cite{DL} investigated an interesting subfamily, known as the \emph{Chowla family}, corresponding to real quadratic fields.

\subsection{Distribution of $L$-functions in the hyperelliptic ensemble}\label{large values section}
We now move our discussion to the setup of $L$-functions over function fields, a topic of interest in modern research in number theory; see, e.g., \cite{AT, AK, BF, BF2, DFL, DL, FR, Floreanegative, Florea, Jung, Lumley2}.
This includes investigations of various statistical properties of these $L$-functions, such as moments, non-vanishing, and value distribution.

Before we enunciate the main theorem of this paper, we need to present
 some basic notations on function fields.
 Let $\mathbb{F}_{q}$ be a finite field of odd cardinality and $\mathbb{F}_{q}[t]$ be the polynomial ring over $\mathbb{F}_{q}$ in variable $t$. Let $D\in \mathbb{F}_q[t]$ be a monic square-free polynomial. The quadratic character $\chi_D$ attached to $D$ is defined using quadratic residue symbol for $\mathbb{F}_{q}[t]$ by $\chi_{D}(f)=\left(\frac{f}{D}\right)$ and the corresponding Dirichlet $L$-function is denoted by $L(s, \chi_D)$.
 Let $\mathcal{H}_n$
 be the family of all curves given in affine form by $C_P: y^2=D(t)$, where $D$ be a monic square-free polynomial of degree $n$. These curve are non-singular of genus $g$ given by \eqref{lambda} and \eqref{genus}.

In~\cite{Lumley}, Lumley studied the distribution of $L(1,\chi_D)$ as $D$ ranges over $\mathcal{H}_n$.
 The following theorem strengthens the range of uniformity for the tail of the distribution of $L(1,\chi_D)$ established in~\cite[Theorem~1.2]{Lumley}.
Throughout, $\log$ denotes the logarithm to base $q$, while $\ln$ is the natural logarithm, and $\log_j$ (respectively $\ln_j$) for the $j$-fold iterated logarithm.

\begin{align}\label{genus}
2g=d(D)-1-\lambda
\end{align}

\begin{align}\label{lambda}
\lambda=
\left\{
\begin{array}
[c]{ll}
1, & \;\text{if\, $d(D)$ even}, \\
0, &\; \text{if\, $d(D)$ odd},
\end{array}
\right.
\end{align}

Before we enunciate the main theorem of this paper, we need to present
 some basic notations on function fields.
 Let $\mathbb{F}_{q}$ be a finite field of odd cardinality and $\mathbb{F}_{q}[t]$ be the polynomial ring over $\mathbb{F}_{q}$ in variable $t$. Let $D\in \mathbb{F}_q[t]$ be a monic square-free polynomial. The quadratic character $\chi_D$ attached to $D$ is defined using quadratic residue symbol for $\mathbb{F}_{q}[t]$ by $\chi_{D}(f)=\left(\frac{f}{D}\right)$ and the corresponding Dirichlet $L$-function is denoted by $L(s, \chi_D)$.
 Let $\mathcal{H}_n$
 be the family of all curves given in affine form by $C_P: y^2=D(t)$, where $D$ be a monic square-free polynomial of degree $n$. These curve are non-singular of genus $g$ given by \eqref{lambda} and \eqref{genus}.

\subsection{Truncating the Euler product of $L(1, \chi_D)$}
We show that $L(1, \chi_D)$ can be well approximated by its short Euler product for almost all $D \in \mathcal{H}_n$. Let
\begin{align}\label{short euler product}
L(1,\chi_D; M):= \prod_{d(P)\le M}\left(1-\frac{\chi_D(P)}{|P|}\right)^{-1}=\sum_{\substack{f\in \mathcal{M}\\ P\mid f \implies d(P)\leq M}}\frac{\chi_D(f)}{|f|}.
\end{align}
 Let $M:=3\log n$. From \cite[Lemma 2.2]{Lumley}, for $D\in \mathcal{H}_n$, we have
 \begin{align}\label{asym for L(1)}
 L(1,\chi_{D})=L(1,\chi_D; M)\left(1+O\left(\frac{1}{n^{1/2}\log n}\right)\right).
 \end{align}
The following lemma provides a better approximation on the truncation length and the associated error chosen optimally for the proof of Theorem~\ref{main theorem}.
\begin{lemma}\label{full l to its truncation}
For sufficiently large $n$,
\[
L(1, \chi_D)=L(1, \chi_D; N)\left(1+O\left(\frac{1}{f(n)\log n}\right)\right)
\]
for all but at most $q^{\frac{7n}{10}}$ elements $D\in \mathcal{H}_n$, where $N=\log n+\log_2 n+3\log f(n)$, and $f(n) \to \infty$ arbitrarily slowly as $n \to \infty$, with $f(n) \ll \log n$.
 \end{lemma}
\begin{proof}
 By using \eqref{asym for L(1)}, we have
 \[
 L(1, \chi_D) =
 L(1, \chi_D; N)
 \exp \!\bigg(
 \sum_{\substack{N < d(P) \le M}}
 \bigg(\frac{\chi_D(P)}{|P|}
 + O\!\left( \frac{1}{|P|^2} \right)
 \bigg)\bigg)
 \bigg( 1 + O\!\bigg( \frac{1}{n^{\tfrac{1}{2}} \log n} \bigg) \bigg).
 \]
 Next, we identify a density zero subfamily of $\mathcal{H}_n$ for which
 \begin{align}\label{DP bound}
 \Bigg|\sum_{N < d(P) \le M} \frac{\chi_D(P)}{|P|}\Bigg| \ge \frac{1}{h(n)}
 \end{align}
 holds, where $
 \log n \ll h(n) \to \infty$ as $n \to \infty.$
 It is worth noting that $h(n)$ must grow faster than $\log n$ chosen appropriately later in order for the saddle point analysis to yield an asymptotic formula exhibiting double exponential decay (see Section~\ref{spa}).

The additional factor $1/\sqrt{\log n}$ in the choice of $c$ is essential for proving Theorem \ref{theorem on resonance method}.
Hence, using \eqref{asym for L(1)}, we can that there exists a $D\in \mathcal{H}_n$ such that
\[
L(1,\chi_D)\ge e^{\gamma}\left(\log n +\log_2 n+C_2(q)-\beta+o(1)\right).
\]
We use \eqref{asym for L(1)}, \eqref{ratio final}, \eqref{constant c} to obtain
\begin{align*}
&\bigg(\tau_{\beta, n}+\frac{1}{2\sqrt{\log n}}\bigg)\sum_{D\in \mathcal{H}_n}R_D^2\leq \sum_{D\in \mathcal{H}_n}L(1, \chi_D)R_D^2\\
&= \sum_{\substack{D\in \mathcal{H}_n\\ L(1, \chi_D)< \tau_{\beta, n}}}L(1, \chi_D)R_D^2+\sum_{\substack{D\in \mathcal{H}_n\\ L(1, \chi_D)\geq \tau_{\beta, n}}}L(1, \chi_D)R_D^2< \tau_{\beta, n} \sum_{D\in \mathcal{H}_n}R_D^2+ \sum_{\substack{D\in \mathcal{H}_n\\ L(1, \chi_D)\geq \tau_{\beta, n}}}L(1, \chi_D)R_D^2.
\end{align*}
From the above computation of $S_2$ together with \eqref{scx} gives us
\[
\sum_{\substack{D\in \mathcal{H}_n\\ L(1, \chi_D)\geq \tau_{\beta, n}}}L(1, \chi_D)R_D^2> \frac{e^{\gamma}}{2 \sqrt{\log n}}\sum_{D\in \mathcal{H}_n}R_D^2> \frac{e^{\gamma}}{2}\frac{q^{(1+c')n\left(1+O(\log_2 n/\log n)\right)}}{\sqrt{\log n}}.
\]
On the other hand, using \cite[Proposition 1.4]{Lumley} that
\[
\max_{D\in \mathcal{H}_n}L(1, \chi_D)\leq 2e^{\gamma}\log n+O_q(1),
\]
we have from \eqref{RD},
\begin{align*}
\sum_{\substack{D\in \mathcal{H}_n\\ L(1, \chi_D)\geq \tau_{\beta, n}}}L(1, \chi_D)R_D^2&\leq |\mathcal{H}_n| \left(\max_{D\in \mathcal{H}_n} R_D^2 \cdot \max_{D\in \mathcal{H}_n}L(1, \chi_D)\right)\phi_n(\beta)\\
&\leq \left(2e^{\gamma}\log n\right) q^{\left(1+\frac{2c\zeta_{\mathbb{A}}(2)}{\ln q}\right)\left(1+O\left(\frac{\log_2 n}{\log n}\right)\right)n} \phi_n(\beta).
\end{align*}
Comparing the above lower and upper bound, and using $c$ as expressed in \eqref{constant c}, we finally conclude that
\[
\phi_n(\beta)> \frac{1}{4(\log n)^{3/2}}q^{-\frac{c(3\ln 2-\pi/2)\zeta_{\mathbb{A}}(2)}{\ln q}\left(1+O\left(\log_2 n/\log n\right)\right)}\geq e^{-\frac{q^{-\beta}\left(1+O\left(1/\sqrt{\log n}\right)\right)\ln q}{2}}.
\]
\end{proof}

From the estimation of $\mathcal{S}_{c, N}$ and $\mathcal{R}_{c, N}$, we finally obtain
\begin{align}\label{ratio}
\frac{S_1}{S_2}\ge \frac{|\mathcal{H}_n| \mathcal{E}(N) \mathcal{E}_1(N) \prod_{d(P)< N} (1-|P|^{-1}) + O\left(q^{\left(\frac12+\frac{2c\zeta_{\mathbb{A}}(2)}{\ln q}+\frac{11\varepsilon}{6}\right)n}\right)}{|\mathcal{H}_n| \mathcal{E}(N) \mathcal{E}_2(N)\prod_{d(P)<N}h(P)^{-1} + O\left( (q^{\left(\frac12+\frac{2c\zeta_{\mathbb{A}}(2)}{\ln q}+2\varepsilon\right)n}\right)}.
\end{align}
 Noting $N=\log n+\log_2 n +\log c$, we first optimize $c$ from the ratio to the right-hand side of \eqref{ratio} by comparing the error terms. To do this, we first calculate the size of $\mathcal{E}(N)=\prod_{d(P)<N}\left(1-r_P^2 \right)^{-2} \left(1+r_P^2\right)$. Applying Abel's summation formula,
\begin{align*}
&\ln \mathcal{E}(N)=-2\sum_{n\leq N}\Pi_{q}(n)\ln\left(\tfrac{2q^n}{q^N}-\tfrac{q^{2n}}{q^{2N}}\right)
+\sum_{n\leq N}\Pi_{q}(n)\ln\left(2-\tfrac{2q^n}{q^N}+\tfrac{q^{2n}}{q^{2N}}\right)\\
&= \frac{\zeta_{\mathbb{A}}(2)}{\ln q}\left(2(2-\ln 4)+(\ln 2+\pi/2-2)\right)\frac{q^N}{N}\left(1+O\left(\frac{\log N}{N}\right)\right),
\end{align*}
where the first and second integrals are referenced from \cite[page $841$]{AMMP} and \cite[page $20$]{DM} respectively by using the prime polynomial theorem and change of variable $q^t=u$. So, from the choice of $N$, we have
\begin{align*}
\mathcal{E}(N)=q^{\widetilde{c}\left(1+O\left(\frac{\log_2 n}{\log n}\right)\right)n},
\end{align*}
where $\widetilde{c}=\left(c(2+\pi/2-3\ln 2)\zeta_{\mathbb{A}}(2)\right)/\ln q$.
Noticing that $\mathcal{E}_2(N)\asymp 1/n$, from \eqref{ratio}, we compare the first term of the denominator to the both error terms to restrict ourself
\begin{align}\label{optimzed constant}
\frac12+\frac{2c\zeta_{\mathbb{A}}(2)}{\ln q}< 1+\frac{c\left(2+\pi/2-3\ln 2\right)\zeta_{\mathbb{A}}(2)}{\ln q}\implies c<\frac{\ln q}{2(3\ln 2-\pi/2)\zeta_{\mathbb{A}}(2)}.
\end{align}
Also, from the above computations, we obtain
\begin{align}\label{scx}
\mathcal{S}_{c, N}=q^{n\widetilde{c}\left(1+O(\log_2 n/\log n)\right)}.
\end{align}
To extract the main term from the ratio on the right-hand side of \eqref{ratio}, it remains to estimate $\mathcal{E}_1(N)$, $\prod_{d(P)\le N} \left(1-|P|^{-1}\right)$, and $\mathcal{E}_2(N)\prod_{d(P)\le N} h(P)^{-1} $.
Using the classical estimate using the Euler--Maclaurin summation formula,
\[
\sum_{n\le N} \frac{1}{n} = \ln N + \gamma + \frac{1}{2N} + O\Bigl(\frac{1}{N^2}\Bigr),
\]
one readily obtains
\[
\prod_{d(P)\le N} \left(1-\frac{1}{|P|}\right)^{-1} = e^{\gamma} N \left(1 + \frac{1}{2N} + O\Bigl(\frac{1}{N^2}\Bigr)\right).
\]
Similarly, following the approach used above for evaluating Euler products, we find
\[
\mathcal{E}_1(N) = q^{\left( \left(c_2 - c_3 + O\bigl(\frac{\log N}{N}\bigr)\right) \frac{\zeta_{\mathbb{A}}(2)}{N \ln q} \right)},
\]
where the constants $c_2$ and $c_3$ are as in \cite[page 21]{DM}, and
\[
\mathcal{E}_2(N)\prod_{d(P)\le N} h(P)^{-1} = \prod_{d(P)\le N} \left( 1 + \frac{(1-r_P^2)^2}{|P|(1+r_P^2)} \right)
= q^{\left( \left(c_2 + O\bigl(\frac{\log N}{N}\bigr)\right) \frac{\zeta_{\mathbb{A}}(2)}{N \ln q} \right)}.
\]

We emphasize that Theorem~\ref{main theorem} subsequently improves the uniformity range of~\cite[Theorem~1.2]{Lumley} from
 $\log n - 2\log_2 n - \log_3 n$ to $\log n + \log_2 n - \theta(n)$, where $\theta(n)$ grows arbitrarily slowly to infinity as $n \to \infty$. A similar range of uniformity as in Lumley’s work is also found in the work of Dahl and Lamzouri~\cite[Theorem~1.2]{DL}.
 For a discussion on the significance of these uniformity ranges, see~\cite{MV}.
Motivated by the work of Granville and Soundararajan~\cite{GS}, Lumley conjectured that as $n\to \infty$,
 \[
 \max_{D\in \mathcal{H}_{n}} L(1,\chi_D) = \log n + \log_2 n + O(1).
 \]
 It is plausible that if the asymptotic formula in Theorem~\ref{main theorem} were to hold up to the boundary of the admissible range, then the above conjecture would follow.
 Our result, however, falls slightly short of this limit, requiring that $\theta(n)$ remain bounded as stated in Theorem~\ref{main theorem}.
 Nevertheless, this constitutes the first unconditional result approaching the conjectural range, comparing upon the works of~\cite{DDLL, DL, GS, GSzeta, Lamzouri, Lumley}.
 Conditionally on the Generalized Riemann Hypothesis (GRH), the corresponding result was established by Granville and Soundararajan~\cite[Theorem~4]{GS} in the number field case.

Finally, by employing a long resonator method, we provide further evidence that the double-exponential decay may persists up to the conjectured level, where $\theta(n)$ transitions from an arbitrarily slowly unbounded function to a bounded one.

\begin{theorem}\label{theorem on resonance method}
	Let $q \ge 3$ and for every $\beta > 0$, define $\tau_{\beta, n}
	= e^{\gamma}\!\left(\log n + \log_2 n + C_2(q) - \beta\right),$ where
	\[
	C_2(q)
	= \frac{1}{2}
	- \left(\frac{\pi}{4} - \frac{\ln 2}{2}\right)\frac{q}{q-1}
	+ \log\!\left(
	\frac{(q-1)\ln q}{2q(3\ln 2 - \pi/2)}
	\right).
	\]
	Then we have
	\[
\phi_n(\beta)
:= \frac{1}{|\mathcal{H}_n|}
\sum_{\substack{D \in \mathcal{H}_n \\ L(1, \chi_D) \ge \tau_{\beta, n}}} 1
	\ge \exp\!\left(
	-\frac{q^{-\beta}\ln q}{2}
	\left(1 + o(1)\right)
	\right).
	\]
\end{theorem}
The best previously known unconditional result in this direction over number fields was due to Granville and Soundararajan~\cite[Theorem~5b]{GS}, while the conditional analogue under the GRH was obtained by the author and Maiti~\cite[Theorem~2]{DM}.

During the proof of Theorem~\ref{theorem on resonance method}, we also observed that
\[
\max_{D\in \mathcal{H}_n} L(1,\chi_D)
\ge e^{\gamma}\Bigl(\log n + \log_2 n + C_2(q) + o(1)\Bigr),
\]
where \(C_2(q)\) is as defined in the statement of the theorem. This result refines~\cite[Theorem~1.6]{Lumley} by providing an explicit value for the constant \(C_2(q)\). More strikingly, one has \(C_2(q) > 0\) for \(q > 10\); for instance, \(C_2(17) \approx 0.04\). This is the first time a positive constant is observed, showing that the maximum of \(L(1,\chi_D)\) can exceed the range \(\log n + \log_2 n\) (see~\cite[Conjecture~2]{MV}).

\subsubsection{Applications} For a monic square-free polynomial $D \in \mathbb{F}_q[t]$, define
\[
h_D = \lvert \mathrm{Pic}(\mathcal{O}_D) \rvert,
\]
where $\mathrm{Pic}(\mathcal{O}_D)$ denotes the Picard group of the ring of integers $\mathcal{O}_D \subset \mathbb{F}_q[t](\sqrt{D(t)})$.
Artin~\cite{Artin} established a class number formula over the hyperelliptic ensemble that connects $h_D$ and $L(1, \chi_D)$:
\begin{equation*}
L(1, \chi_D) = \frac{\sqrt{q}}{\sqrt{|D|}}\, h_D = q^{-g} h_D
\quad \text{for } D \in \mathcal{H}_{2g+1}.
\end{equation*}
From Theorem~\ref{main theorem}, we directly extend the range of uniformity in \cite[Corollary~1.8]{Lumley}, showing that the tail of the distribution of large values of $h_D$ over $\mathcal{H}_{2g+1}$ decays doubly exponentially. An analysis towards maximal value for $h_D$ follows from Theorem~\ref{theorem on resonance method}.

For $n = 2g + 2$ and $D \in \mathcal{H}_{2g+2}$, Artin also proved that
\begin{equation*}
L(1, \chi_D) = \frac{q - 1}{\sqrt{|D|}}\, h_D R_D,
\end{equation*}
where $R_D$ denotes the regulator of $\mathcal{O}_D$ (see \cite[Chapter 14]{ROS}).
Analogous results hold for the tail of the distribution of $h_D R_D$ as $D$ varies over $\mathcal{H}_{2g+2}$, improving~\cite[Corollary~1.9]{Lumley} in a similar manner.
 \subsection{Essence of the paper} We employ two distinct methods to prove Theorems~\ref{main theorem} and~\ref{theorem on resonance method}.
 The study of the distribution of $L(1, \chi_d)$ over number fields through an underlying probabilistic model was initiated by Granville and Soundararajan~\cite{GS}.
 Lumley~\cite{Lumley} studied the tail of the distribution for $L(1, \chi_D)$ over function fields from the corresponding probabilistic model (see \cite[eq.~(1.7)]{Lumley}) and computing large moments og $L(1, \chi_D)$ over the family.
 To extend the range of uniformity stated in Theorem~\ref{main theorem}, we able to compute much higher moments for the short Euler product $L(1, \chi_D; N)$ and then utilize the saddle-point method.
 The Lemma~\ref{full l to its truncation} provides an optimal way to approximate $L(1, \chi_D)$ by $L(1, \chi_D; N)$, allowing us to taken a shorter truncation length $N$.
 This step relies crucially on a large sieve type estimate together with Chebyshev’s inequality.

 To prove Theorem~\ref{theorem on resonance method}, we employ the long resonance method for quadratic Dirichlet $L$-functions over number fields, as developed by the author and Maiti in~\cite{DM}.
 The character sum estimate in Lemma~\ref{character sum over nonsquare} plays a central role in both results: it extends the range of uniformity as in Theorem \ref{main theorem} via computing larger moments of $L(1, \chi_D; N)$ over the family, and it enables the control of long resonators in Theorem~\ref{theorem on resonance method}. These ideas appear difficult to implement for higher degree $L$-functions but may still be applicable to families of $\mathrm{GL}(1)$ $L$-functions associated with higher order characters.

To extend the range of uniformity stated in Theorem~\ref{main theorem}, we are able to compute much higher moments for the short Euler product $L(1, \chi_D; N)$ and then utilize the saddle-point method. The Lemma~\ref{full l to its truncation} provides an optimal way to approximate $L(1, \chi_D)$ by $L(1, \chi_D; N)$, allowing us to taken a shorter truncation length $N$. This step relies crucially on a large sieve type estimate together with Chebyshev’s inequality. The character sum estimate in Lemma~\ref{character sum over nonsquare} plays a central role: it extends the range of uniformity as in Theorem~\ref{main theorem} via computing larger moments of $L(1, \chi_D; N)$ over the family.

To extend the range of uniformity stated in the main theorem, we are able to compute much higher moments for the short Euler product $L(1, \chi_D; N)$ and then utilize the saddle-point method.

For sufficiently large $n$,
\[
L(1, \chi_D)=L(1, \chi_D; N)\left(1+O\left(\frac{1}{f(n)\log n}\right)\right)
\]
for all but at most $q^{\frac{7n}{10}}$ elements $D\in \mathcal{H}_n$, where $N=\log n+\log_2 n+3\log f(n)$, and $f(n) \to \infty$ arbitrarily slowly as $n \to \infty$, with $f(n) \ll \log n$. This provides an optimal way to approximate $L(1, \chi_D)$ by $L(1, \chi_D; N)$, allowing us to taken a shorter truncation length $N$. This step relies crucially on a large sieve type estimate together with Chebyshev’s inequality.

Let $\ell=\ell_1 \ell_2^2\neq \square\in \mathbb{F}_q[t]$ with $\ell_1$ square-free. For any $\varepsilon>0$, we have
\[
\sum_{D\in \mathcal{H}_n}\chi_D(\ell)\ll  q^{(1/2+\varepsilon)n}\exp\left((d(\ell_1))^{1-\varepsilon}\right)\prod_{P\mid \ell_2}\left(1+|P|^{-1/2-\varepsilon}\right).
\]
This character sum estimate plays a central role: it extends the range of uniformity in establishing the main theorem via computing larger moments of $L(1, \chi_D; N)$ over the family.

Question:
Let $q\ge 3$ be fixed, let $\mathbb F_q[t]$ be the polynomial ring over the finite field $\mathbb F_q$, and let $\mathcal H_n$ denote the set of monic square-free polynomials $D\in \mathbb F_q[t]$ of degree $n$. For each $D\in \mathcal H_n$, let $\chi_D$ be the quadratic character attached to $D$, and let $L(1,\chi_D)$ be the corresponding Dirichlet $L$-value. Define the tail proportion
\[
\phi_n(\tau):=\frac1{|\mathcal H_n|}\sum_{\substack{D\in\mathcal H_n\\ L(1,\chi_D)\ge e^{\gamma}\tau}}1,
\]
where $\gamma$ is Euler's constant. Assume $\log$ denotes the logarithm to base $q$, $\log_j$ denotes the $j$-fold iterated logarithm, and $\{x\}$ denotes the fractional part of $x$. Let $\theta(n)$ satisfy $2\le \theta(n)\ll \log_3 n$ and $\theta(n)\to\infty$ arbitrarily slowly as $n\to\infty$. Uniformly for
\[
\tau\le \log n+\log_2 n-\theta(n),
\]
which asymptotic statement holds for $\phi_n(\tau)$ as $n\to\infty$?

\[
\phi_n(\tau)=\exp\!\left(-\,C_1\!\left(q^{\{\log \kappa(\tau)\}}\right)\,\frac{q^{\,\tau-C_0\!\left(q^{\{\log \kappa(\tau)\}}\right)}}{\tau}\bigl(1+o_\theta(1)\bigr)\right),
\]
uniformly for $\tau\le \log n+\log_2 n-\theta(n)$, where $C_0\!\left(q^{\{\log \kappa(\tau)\}}\right)$ and $C_1\!\left(q^{\{\log \kappa(\tau)\}}\right)$ depend on $\tau$, $q$, and $\kappa(\tau)$, but remain bounded as their argument varies between $1$ and $q$.

Choices:
B.
\[
\phi_n(\tau)=\exp\!\left(-\,C_1\!\left(q^{\{\log \kappa(\tau)\}}\right)\,\frac{q^{\,\tau-C_0\!\left(q^{\{\log \kappa(\tau)\}}\right)}}{\tau}\bigl(1+o_\theta(1)\bigr)\right),
\]
uniformly for $\tau\le \log n+\log_2 n+\theta(n)$, where $C_0\!\left(q^{\{\log \kappa(\tau)\}}\right)$ and $C_1\!\left(q^{\{\log \kappa(\tau)\}}\right)$ depend on $\tau$, $q$, and $\kappa(\tau)$, but remain bounded as their argument varies between $1$ and $q$.
C.
\[
\phi_n(\tau)=\exp\!\left(-\,C_1\!\left(q^{\{\log \kappa(\tau)\}}\right)\,\frac{q^{\,\tau-C_0\!\left(q^{\{\log \kappa(\tau)\}}\right)}}{\tau}\bigl(1+o_\theta(1)\bigr)\right)
\]
for all $\tau$ in the stated range $\tau\le \log n+\log_2 n-\theta(n)$.
D.
\[
\phi_n(\tau)=\exp\!\left(-\,C_1\!\left(q^{\{\log \kappa(\tau)\}}\right)\,\frac{q^{\,\tau-C_0\!\left(q^{\{\log \kappa(\tau)\}}\right)}}{\tau}\bigl(1+o(1)\bigr)\right),
\]
uniformly for $\tau\le \log n+\log_2 n-\theta(n)$, where the $o(1)$ term is absolute and uniform in every admissible choice of $\theta(n)$.
E.
\[
\phi_n(\tau)=\exp\!\left(-\,C_1\!\left(q^{\{\log \kappa(\tau)\}}\right)\,\frac{q^{\,\tau-C_0\!\left(q^{\{\log \kappa(\tau)\}}\right)}}{\log n}\bigl(1+o_\theta(1)\bigr)\right),
\]
uniformly for $\tau\le \log n+\log_2 n-\theta(n)$, where $C_0\!\left(q^{\{\log \kappa(\tau)\}}\right)$ and $C_1\!\left(q^{\{\log \kappa(\tau)\}}\right)$ depend on $\tau$, $q$, and $\kappa(\tau)$, but remain bounded as their argument varies between $1$ and $q$.

The stem does not explicitly reveal the correct option. It specifies the setup and admissible range, but the exact asymptotic form, denominator, and error-term dependence still have to be selected from the choices.

This is essentially a theorem-recall item: the stem asks which asymptotic statement holds, and the correct answer is the full theorem statement itself rather than a conclusion derived from new reasoning.

Some discrimination is required because the options differ in subtle ways (range of uniformity, strength of error term, denominator scale, weaker true variant). However, solving it mainly tests recognition/memory of the precise statement, not substantial generative mathematical reasoning.

The distractors are plausible and mathematically targeted: one weakens the truth claim, one alters the uniformity range, one overstates uniformity of the error term, and one changes the key scale from tau to log n. These reflect realistic failure modes.

A technically strong multiple-choice item with good distractors and little answer leakage, but it is primarily a precise theorem-recall/restatement question rather than a genuine test of generative reasoning.