[cf. Theorem 2 in \cite{Maa96}]\label{thm:maa}
Let $X_1, X_2, X_3$  be i.i.d. random vectors with values in $\mathbb{R}^k$ and Lebesgue probability density $f_X$, let $Y_1, Y_2, Y_3$ be i.i.d. random vectors with values in $\mathbb{R}^k$ and  Lebesgue probability density $f_Y$ and let $X_1, X_2,X_3$ and $Y_1, Y_2, Y_3$ be independent. Let $d:\mathbb{R}^k\times \mathbb{R}^k\longrightarrow \mathbb{R}$ be a nonnegative, continuous function with
\begin{enumerate}
    \item[\textbf{(D1)}] $d(x,y)= 0$ if, and only if,  $x=y$,
    \item[\textbf{(D2)}] for all $a \in \mathbb{R}$ and $x,y,b \in \mathbb{R}^k$ $d(ax+b, ay+b) = |a|d(x,y)$.
\end{enumerate}
\noindent
Moreover, assume that
\begin{enumerate}[start=1,label={(\bfseries R\arabic*):}]
    \item[\textbf{(R1)}]  $\int_{\mathbb{R}^k}f^2_X(x) dx, \int_{\mathbb{R}^k}f^2_Y(y)dy<\infty$,
     \item[\textbf{(R2)}]  the zero vector is a Lebesgue point of the function $u(y)= \int f_Y(x+y) f_X(x) dx$, i.e. it holds that $\frac{1}{\lambda(B_r(0))} \int_{B_r(0)} |u(y) - u(0)| d y \underset{r \rightarrow 0}{\rightarrow} 0$, where $B_r(x)$ denotes the ball in $\mathbb{R}^k$ with radius $r$ around $x$ with respect to  the distance function $d$.
\end{enumerate}
\noindent
Then, it holds that
\begin{equation*}
    f_X = f_Y \textit{\quad if, and only if,  \quad } d(X_1, X_2) \overset{\mathcal{D}}{=} d(Y_1, Y_2) \overset{\mathcal{D}}{=}d(X_3, Y_3).
\end{equation*}

\begin{bsp} (Canberra distance) \label{bsp::Canberra}\\

\noindent
Let us consider the Canberra distance of $x=(x_1, \ldots, x_k)^{\top}$ and $y=(y_1, \ldots, y_k)^{\top}$ in $\R^k$ defined by
\begin{equation}\label{eqn::Canberra_k}
h(x, y) := \sum_{i=1}^k
\begin{cases}
\frac{|x_i - y_i|}{|x_i| + |y_i|}, & \text{if } |x_i| + |y_i| \neq 0, \\
0, & \text{if } x_i = y_i = 0.
\end{cases}
\end{equation}
\noindent
\textbf{Case $k = 1$:} \\
In this case, $h(x,y) = \frac{\lvert x - y \rvert}{\lvert x\rvert +  \lvert y\rvert}$. Consider for $x > 0$ and $t \in (0,1)$ the cases $y \geq x$ and $y  < x$ separately.
For $y \geq x$ and   $h(x,y)  < t$, we have
\begin{equation*}
    \frac{y - x}{x + y}  < t, \ \text{or, equivalently,} \  \   y  < \frac{x(t + 1)}{1 - t}.
\end{equation*}
For  $y  < x$ and   $h(x,y)  < t$, we have
\begin{equation*}
     \frac{x-y}{x + y}  < t, \ \text{or, equivalently,} \  \   y > \frac{x(1-t)}{1+t}.
\end{equation*}
It follows that
\begin{equation} \label{eqn::Canberra_1D_cube}
    B_t(x)= \left(x\frac{1-t}{1+t} , x\frac{1+t}{1-t} \right)
\end{equation}
and therefore $\mu(B_t(x)) = \frac{4tx}{1-t^2}$. Due to symmetry, for $x < 0$ we have $\mu(B_t(x)) = -\frac{4tx}{1-t^2}$ and we conclude
\begin{equation} \label{eqn::Phi_1D}
    \Phi(x, t) = \frac{4t \lvert x \rvert}{1-t^2} \textnormal{ \quad \quad for } t \in (0,1) \textnormal{ and } x \neq 0.
\end{equation}
Moreover, it holds that
\begin{equation*}
    B_t(0) =
\begin{cases}
\{ 0\} & t < 1 , \\
\R & t \geq 1,
\end{cases}
\end{equation*}
and therefore
\begin{equation*}
     \Phi(0,t) =
\begin{cases}
0 & t < 1 , \\
\infty & t \geq 1.
\end{cases}
\end{equation*}

\noindent
\textbf{Case $k > 1$:} \\
We firstly provide an upper and lower bounds on the volume of $B_t^{(h)}(x)$ for the case $x_i \neq 0$ for all $i \in \{1,...,k\}$. For the upper bound, we consider the rectangle in $\R^k$ with sides corresponding to  \eqref{eqn::Canberra_1D_cube}. Since $y_i \notin \left(x_i\frac{1-t}{1+t} , x_i\frac{1+t}{1-t} \right) $ implies $h(x,y) \geq t$, we have
\begin{equation*}
    B_t(x) \subseteq \prod\limits_{i=1}^k \left(x_i\frac{1-t}{1+t} , x_i\frac{1+t}{1-t} \right).
\end{equation*}
It follows that
\begin{equation} \label{eqn::Canberra_upper}
    \Phi(x,t) \leq \left(\frac{4t}{1-t^2} \right)^k \prod\limits_{i=1}^k \lvert x_i \rvert \textnormal{ \quad \quad for } t \in (0,1) \textnormal{ and } x_i \neq 0.
\end{equation}
For the lower bound consider the rectangle with sides
\begin{equation*}
   I_i = \left(x_i\frac{1-t/k}{1+t/k} , x_i\frac{1+t/k}{1-t/k} \right).
\end{equation*}
Since $y_i \in I_i$ for all $i \in \{1,...,k\}$ implies $y \in B_t(x)$, we conclude that
\begin{equation*}
     \prod\limits_{i=1}^k \left(x_i\frac{1-t/k}{1+t/k} , x_i\frac{1+t/k}{1-t/k} \right) \subseteq B_t(x)
\end{equation*}
and therefore
\begin{equation}\label{eqn::Canberra_lower}
    \left(\frac{4t/k}{1-(t/k)^2} \right)^k \prod\limits_{i=1}^k \lvert x_i \rvert \leq \Phi(x,t) \textnormal{ \quad \quad for } t \in (0,1) \textnormal{ and } x_i \neq 0.
\end{equation}
If for any $i \in \{1,...,k\}$ $x_i = 0$, then in that dimension the set of permissible $y_i$ is empty, and therefore $\Phi(0,t) = 0$ for $t \in (0,1)$. The set $\{x=(x_1, \ldots, x_k)^{\top} \mid x_i = 0 \ \text{for at least one $i \in \{1, \ldots, k\}$}\}$ has, however, Lebesgue measure $0$ in $\R^k$.
Accordingly,  for a.e. $x$
\begin{equation*}
    \Phi(x,t) \sim C(x) \left( \frac{t}{1-t^2} \right)^k,
\end{equation*}
where $0<C(x)\sim \prod_{i=1}^k \lvert x_i \rvert$. More precisely, with \eqref{eqn::Canberra_upper} and \eqref{eqn::Canberra_lower} it follows that $\delta_t(x,y) \leq C \delta(x,y)$ with $\delta(x,y) = \prod_{i= 1}^k \left|\frac{x_i}{y_i}\right|$ for some $C>0$. We conclude that the conditions \eqref{eqn::Phi_limit}, \eqref{eqn::Phi_positive} and \eqref{eqn::Phi_uniform} in the definition of volume-regular functions, i.e. Definition \ref{def:volume_reg},  are fulfilled.  Theorem \ref{thm::Main} then applies if its conditions on the densities $f$ and $g$ are fulfilled.\\

\noindent
As a concrete example, consider the density function $f$ of the standard normal distribution, i.e. $f(x) = \frac{1}{\sqrt{2 \pi}} e^{-\lVert x \rVert^2 /2}$, which is smooth on $\R^k$. Then, the Lebesgue differentiability condition is fulfilled since
\begin{align*}
      \frac{1}{\Phi(x,t)} \int_{B_t(x) } \lvert f(x) - f(y)\rvert d y &\leq \frac{1}{\Phi(x,t)} \sup_{y \in B_t(x)}\lvert f(x) - f(y)\rvert \Phi(x,t) \\
      &= \sup_{y \in B_t(x)}\lvert f(x) - f(y)\rvert \rightarrow 0
\end{align*}
for almost every $x \in \R^k$ and $t \rightarrow 0^{+}$. According to \eqref{eqn::Canberra_upper}, it further holds that
\begin{equation*}
    f^2(x)\Phi(x,t) \leq \frac{1}{2\pi}\left(\frac{4t}{1-t^2} \right)^k \prod\limits_{i=1}^k \lvert x_i \rvert  e^{-\lVert x \rVert^2 }
\end{equation*}
with the right-hand side of the above inequality being  integrable on $\R^k$. We conclude that $f \delta(\cdot,y), f^2 \delta(\cdot,y) \in L^1(\R^k)$ for any $t \in (0,1)$ and the integrability condition is fulfilled. In other words: Theorem \ref{thm::Main} holds for  normally distributed data and the Canberra distance.
\end{bsp}

\begin{Theorem}[cf. Theorem 2 in \cite{Maa96}]\label{thm:maa}
Let $X_1, X_2, X_3$  be i.i.d. random vectors with values in $\mathbb{R}^k$ and Lebesgue probability density $f_X$, let $Y_1, Y_2, Y_3$ be i.i.d. random vectors with values in $\mathbb{R}^k$ and  Lebesgue probability density $f_Y$ and let $X_1, X_2,X_3$ and $Y_1, Y_2, Y_3$ be independent. Let $d:\mathbb{R}^k\times \mathbb{R}^k\longrightarrow \mathbb{R}$ be a nonnegative, continuous function with
\begin{enumerate}
    \item[\textbf{(D1)}] $d(x,y)= 0$ if, and only if,  $x=y$,
    \item[\textbf{(D2)}] for all $a \in \R$ and $x,y,b \in \R^k$ $d(ax+b, ay+b) = |a|d(x,y)$.
\end{enumerate}
\noindent
Moreover, assume that
\begin{enumerate}[start=1,label={(\bfseries R\arabic*):}]
    \item[\textbf{(R1)}]  $\int_{\mathbb{R}^k}f^2_X(x) dx, \int_{\mathbb{R}^k}f^2_Y(y)dy<\infty$,
     \item[\textbf{(R2)}]  the zero vector is a Lebesgue point of the function $u(y)= \int f_Y(x+y) f_X(x) dx$, i.e. it holds that $\frac{1}{\lambda(B_r(0))} \int_{B_r(0)} |u(y) - u(0)| d y \underset{r \rightarrow 0}{\rightarrow} 0$, where $B_r(x)$ denotes the ball in $\mathbb{R}^k$ with radius $r$ around $x$ with respect to  the distance function $d$.
\end{enumerate}
\noindent
Then, it holds that
\begin{equation*}
    f_X = f_Y \textit{\quad if, and only if,  \quad } d(X_1, X_2) \overset{\mathcal{D}}{=} d(Y_1, Y_2) \overset{\mathcal{D}}{=}d(X_3, Y_3).
\end{equation*}
\end{Theorem}

\begin{equation} \label{eqn::Phi_uniform}
    c \delta(x,y) \leq  \delta_t (x,y) \leq C \delta(x,y)
\end{equation}
where $ \delta_t (x,y) := \frac{\Phi(x,t)}{ \Phi (y,t)}$ for some constants $c, C > 0$ and for all $0< t < \epsilon$.

\label{def:volume_reg}
\end{Def}
\noindent
We note that the volume function  $\Phi$ is well-defined for (on $\R^k \times \R^k$) measurable $h$. We further note that if condition \eqref{eqn::Phi_uniform} is fulfilled for at least one $y\in\mathbb{R}^k$, then almost any $\xi \in \R^k$ fulfills  \eqref{eqn::Phi_uniform}.  In particular, volume-regularity describes a space in which volumes of $h$-balls centered at different points are stably comparable as radii converge to zero and relative volume $\Phi(x,t)/\Phi(y,t)$ settles for small radii to a finite, positive $\delta (x,y)$ (and is dominated by $\delta (x,y)$) in almost every $x$, while allowing for a controlled divergence of $\Phi$ and $\delta_t$ for a fixed $t$ as $x \rightarrow \infty$. Classical Ahlfors-regular geometries, for example, satisfy this condition, and hence imply volume-regularity. We recall the definition of Ahlfors $\alpha$-regularity.
\begin{Def}(Ahlfors $\alpha$-regularity) \label{def::Ahlfors}\\
A metric measure space $(X, h, \mu)$ is called (locally) Ahlfors $\alpha$-regular, if there exists an $\alpha > 0$, $c, C > 0$ and $\epsilon > 0 $, such that
\begin{equation*}
    c t^\alpha \leq \mu (B^{(h)}_t(x)) \leq C t^\alpha
\end{equation*}
for any $0 < t < \epsilon$ and $x \in X$.
\end{Def}
\noindent
We further relax assumptions on $h$ to generalize Ahlfors $\alpha$-regularity for distance functions as in Definition \ref{def::gendist} and call the distance function $h$ itself Ahlfors $\alpha$-regular (with respect to $\R^k$ and Lebesgue measure).  If $h$ is Ahlfors $\alpha$-regular, then $\frac{c}{ C} \leq \delta_t(x,y) \leq \frac{C}{c}$, i.e. condition \eqref{eqn::Phi_uniform} is fulfilled. The $l_p$-induced distances are, for example, both volume- and Ahlfors-regular. The Canberra distance function on the other hand is volume-regular while not being Ahlfors-regular (see Example \ref{bsp::Canberra}).
We introduce Definition \ref{def:volume_reg}  as a generalization of Ahlfors regularity, accommodating for  distance functions for which the volume $\Phi$ may not have a polynomial dependence on the radius and may depend on the center of the ball.  \\
\noindent
One further analytic condition needed for the proof of Theorem \ref{thm:maa} is the Lebesgue differentiability relative to the homogeneous translation-invariant distance.  For a generalization of the theorem  we introduce the following concept of Lebesgue differentiability of a locally integrable function $f$ with respect to a function $h$:
\begin{Def} (Lebesgue differentiability with respect to $h$) \\
We call $f \in L^1_{loc} (\R^k)$ Lebesgue differentiable (with respect to $h$) in $x$ if
\begin{equation*}
    \lim_{t \rightarrow 0^{+}} \frac{\int_{B_t^{(h)}(x)} |f(y) - f(x)| d y}{ \Phi(x,t) } = 0.
\end{equation*}
\end{Def}
\noindent
It is important to note that Lebesgue differentiability with respect to a general distance function $h$ does not automatically follow from the assumption $f \in L^1_{\text{loc}}(\mathbb{R}^k)$. In the classical Lebesgue differentiation theorem, differentiability almost everywhere is guaranteed due to the regularity of Euclidean balls and the doubling property of the Lebesgue measure. Lebesgue differentiability with respect to $h$ requires for the underlying (metric) measure space $(\mathbb{R}^k, h, \mu)$ to satisfy conditions such as the doubling property, or more generally, the Vitali covering property. \\
A generalization of the classical differentiation theorem to such settings can be found e.g. in Theorem 1.8 of \cite{heinonen2001lectures}, which ensures differentiability of a locally integrable $f$ on a metric measure space with doubling measure, i.e. such that it holds
\begin{equation*}
    \mu (B_{2t}^{(h)} (x)) \leq C \mu(B_t^{(h)} (x))
\end{equation*}
for some $C \geq 1$ and any $x$ and $t$, and all balls have finite and positive measure.
\begin{Theorem}(Lebesgue differentiation theorem) \label{thm::Lebesgue_Diff}\\
Let $f \in L^1_{loc}(X)$ where $(X,h,\mu)$ is a doubling metric measure space. Then $f$ is Lebesgue differentiable with respect to $h$, i.e.
\begin{equation*}
    \lim_{t \rightarrow 0^{+}} \frac{1}{\mu(B_t^{(h)} (x))} \int_{B_t^{(h)} (x)} |f(x) - f(y)| d \mu(y) = 0
\end{equation*}
for almost every $x \in X$.
\end{Theorem}
\noindent
More generally, for a generalized distance $h$ and measure $\mu$, it is necessary that the family of $h$-balls $\{ B_t(x)^{(h)}\}$ forms a Vitali differentiation basis for locally doubling measure $\mu$ for an analogue statement to Theorem \ref{thm::Lebesgue_Diff} to hold  (cf. \cite{folland1999real}).
\noindent
Another assumption needed for generalization of Theorem \ref{thm:maa} guarantees  control of the magnitude of  fluctuations of densities within shrinking $h$-ball. This assumption requires the following definition of (uniformly bounded) centered oscillation.
\begin{Def} (Centered oscillation) \\
We call the functional
\begin{equation*}
    A_f^{(h)}(x, \epsilon) := \sup_{0< t < \epsilon} \frac{1}{\Phi(x,t)} \int_{B_t^{(h)}(x)} |f(y) - f(x)| d y
\end{equation*}
centered oscillation of $f$ in $x$ on the scale $\epsilon$ (with respect to $h$). We call $  A_f^{(h)}(x, \epsilon)$ uniformly bounded (on the scale $\epsilon$) if there exist a $C > 0$ such that
\begin{equation*}
     A_f^{(h)}(x, \epsilon) < C \textnormal{ \quad for almost every } x.
\end{equation*}

\end{Def}
\noindent
Uniformly bounded centered oscillation is a mild regularity condition. In fact, many classes of densities, including bounded continuous functions, Lipschitz and Hölder functions,  satisfy it.\\
\noindent
The following theorem extends Theorem \ref{thm:maa} to volume-regular distance functions allowing for
Lebesgue differentiability of the data-generating densities with uniformly bounded, centered oscillations. More precisely, the theorem establishes that equality of interpoint distance distributions implies equality of the underlying distributions under these  assumptions.
\begin{Theorem} \label{thm::Main}
Let $h:\mathbb{R}^k\times\mathbb{R}^k\to[0,\infty)$ be a volume-regular generalized distance function. Let $X_1$, $X_2$ and $X_3$ be i.i.d. random variables  with density $f$, and $Y_1$, $Y_2$ and $Y_3$ i.i.d. random variables with density $g$, where $f,g$ are Lebesgue differentiable with uniformly bounded centered oscillations with respect to $h$ and $X_i$ and $Y_j$ are pairwise independent for all $i, j$.  Let further $f\cdot\delta(\cdot,\xi), g\cdot\delta(\cdot,\xi), f^2\cdot\delta(\cdot,\xi), g^2\cdot\delta(\cdot,\xi)\in L^1(\mathbb{R}^k)$ for a point $\xi\in\mathbb{R}^k$. Then it holds that
\begin{equation} \label{eqn::Maa_Thm}
    f = g \textnormal{ \quad \quad  } \Leftrightarrow \textnormal{ \quad \quad  }  h(X_1, X_2) \overset{\mathcal{D}}{=} h(Y_1, Y_2) \overset{\mathcal{D}}{=} h(X_3, Y_3).
\end{equation}

\label{sec::intro}
\noindent
Interpoint distance-based approaches to two-sample testing have been widely used due to their applicability in high-dimensional settings. A variety of methods rely on comparing interpoint distance distributions between and within samples. These include the energy distance \cite{szekely2004testing}, the maximum mean discrepancy (MMD) based test by Gretton et al. \cite{gretton2006kernel}, \cite{gretton2012kernel}, and graph-based statistics such as the minimum spanning tree test and nearest-neighbor methods \cite{Schilling01091986}, \cite{henze1988multivariate}, \cite{bhattacharya2021asymptoticdistributiondetectionthresholds}. Related distance-based procedures include the test of Rosenbaum \cite{rosenbaum2005exact}, which forms an optimal non-bipartite matching of the pooled sample under interpoint distances and tests homogeneity by counting cross-sample pairs. Montero-Manso and Vilar \cite{montero2019two} proposed a test based on comparing the distributions of interpoint distances directly by a Cramér–von Mises–type statistic. Rank-based statistics on the interpoint distance distributions have been developed by Liu et al. \cite{liu2022generalized}. More recently, Betken, Marjanovic and Proksch \cite{betken2024two} introduced a two-sample averaged Wilcoxon test, which applies rank-based statistics to all pairwise distances. The utility of comparing interpoint distance distributions has been further demonstrated in the field of generative modeling, where Jajeśniak et al. \cite{jajesniak2025deep} developed the Interpoint Inception Distance to evaluate models based on their feature representations. These methods share the structure of reducing the multivariate problem to a univariate comparison of distance-based quantities. \\
The basis for such approaches is provided by a theorem of Maa, Pearl and Bartoszynski \cite{Maa96}, which establishes that, under suitable regularity conditions, the equality of within-sample and between-sample interpoint distance distributions implies the equality of the underlying multivariate distributions. We compile the original statement with all its assumptions in the following theorem.

\end{Def}
\noindent
Uniformly bounded centered oscillation is a mild regularity condition. In fact, many classes of densities, including bounded continuous functions, Lipschitz and Hölder functions, satisfy it.\\
\noindent
The following theorem extends Theorem \ref{thm:maa} to volume-regular distance functions allowing for
Lebesgue differentiability of the data-generating densities with uniformly bounded, centered oscillations. More precisely, the theorem establishes that equality of interpoint distance distributions implies equality of the underlying distributions under these assumptions.
\begin{Theorem} \label{thm::Main}
Let $h:\mathbb{R}^k\times\mathbb{R}^k\to[0,\infty)$ be a volume-regular generalized distance function. Let $X_1$, $X_2$ and $X_3$ be i.i.d. random variables with density $f$, and $Y_1$, $Y_2$ and $Y_3$ i.i.d. random variables with density $g$, where $f,g$ are Lebesgue differentiable with uniformly bounded centered oscillations with respect to $h$ and $X_i$ and $Y_j$ are pairwise independent for all $i, j$. Let further $f\cdot\delta(\cdot,\xi), g\cdot\delta(\cdot,\xi), f^2\cdot\delta(\cdot,\xi), g^2\cdot\delta(\cdot,\xi)\in L^1(\mathbb{R}^k)$ for a point $\xi\in\mathbb{R}^k$. Then it holds that
\begin{equation} \label{eqn::Maa_Thm}
 f = g \textnormal{ \quad \quad } \Leftrightarrow \textnormal{ \quad \quad } h(X_1, X_2) \overset{\mathcal{D}}{=} h(Y_1, Y_2) \overset{\mathcal{D}}{=} h(X_3, Y_3).
\end{equation}
\end{Theorem}
\vspace{1em}
\noindent
\textbf{Proof. }
\noindent
The forward direction of the equivalence (\ref{eqn::Maa_Thm}) is straightforward, hence it remains to prove the converse. \\
Let $h(X_1, X_2) \overset{\mathcal{D}}{=} h(Y_1, Y_2) \overset{\mathcal{D}}{=} h(X_3, Y_3)$. In particular, then it is
\begin{equation} \label{eqn::Ps}
 \Prob (h(X_1,X_2) < t) = \Prob (h(Y_1,Y_2) < t) = \Prob (h(X_3,Y_3) < t)
\end{equation}
for any $t \in \R$.
Since by assumption (\ref{eqn::Phi_uniform}) for $t$ small enough it uniformly holds
\begin{equation*}
 f^j(x)\delta_t(x, \xi) \leq C f^j(x)\delta(x, \xi),
\end{equation*}
the integrability of $f^j \cdot \delta_t(\cdot, \xi)$ for $j = 1, 2$ follows from $ f^j\cdot\delta(\cdot,\xi) \in L^1(\R^k)$. Furthermore, since
\begin{equation*}
 \int_{\R^k} f^j(x) \Phi(x,t) dx = \Phi(\xi, t) \int_{\R^k} f^j(x) \delta_t(x, \xi) dx
\end{equation*}
the integrability of $f^j \cdot \Phi(\cdot, t)$ follows. The analog holds for $g^j \cdot \Phi(\cdot, t)$, $g^j \cdot \delta_t(\cdot, \xi)$. \\

\noindent
A natural question is whether our result extends to curved spaces, where translation invariance and homogeneity fail. Compact Riemannian manifolds provide a prototypical example: their geodesic balls have controlled volume growth and the Riemannian measure is doubling. Corollary \ref{cor::manifold} illustrates that our generalization continues to hold in this broader geometric setting.
\begin{cor} \label{cor::manifold}
 Let $\mathcal{M}$ be a smooth, compact $k$-dimensional Riemannian manifold equipped with Riemannian measure $\mu$ and geodesic distance $d_{\mathcal{M}}$. Let on $\mathcal{M}$ be given i.i.d. r.vs. $X_1$, $X_2$ and $X_3$ with density $f$ and i.i.d. $Y_1$, $Y_2$ and $Y_3$ with density $g$, where $f,g \in L^2(\mathcal{M})$ . Then it holds that
 \begin{equation*}
 f = g \textnormal{ \quad \quad } \Leftrightarrow \textnormal{ \quad \quad } d_{\mathcal{M}}(X_1, X_2) \overset{\mathcal{D}}{=} d_{\mathcal{M}}(Y_1, Y_2) \overset{\mathcal{D}}{=} d_{\mathcal{M}}(X_3, Y_3).
\end{equation*}
\end{cor}
\noindent
\textbf{Proof. }
The geodesic ball volume on $\mathcal{M}$ satisfies
\begin{equation*}
 \Phi(x,t)= \omega_{2,k}\, t^k\!\left[
1 - \frac{\mathrm{S}(x)}{6(k+2)}\,t^2
+ \frac{-3\|R\|^2(x) + 8\|\mathrm{Ric}\|^2(x) + 5\,\mathrm{S}(x)^2 - 12\,\Delta \mathrm{S}(x)}
{360\,(k+2)(k+4)}\,t^4
+ O(t^6)
\right]
\end{equation*}
where $\omega_{2,k}$ denotes the volume of the $k$-dimensional Euclidean unit ball, $S$ denotes scalar curvature, $R$ Riemannian curvature tensor and $\mathrm{Ric}$ Ricci tensor (cf. Theorem 3.1 in \cite{gray1974volume}). Conditions (\ref{eqn::Phi_limit}), (\ref{eqn::Phi_positive}) and (\ref{eqn::Phi_uniform}) follow. Since $\mathcal{M}$ is compact, it is also
\begin{equation} \label{eqn::Phi_bound_corollary}
 c t^k < \Phi(x,t) < Ct^k \textnormal{ \quad \quad in a.e. } x
\end{equation}
for some $c, C >0$ and integrability conditions of Theorem \ref{thm::Main} follow from $f, g \in L^2(\mathcal{M})$. Furthermore, $\mathcal{M}$ is with (\ref{eqn::Phi_bound_corollary}) a metric measurable space of globally doubling measure and by Lebesgue differentiation theorem, $f$ and $g$ are Lebesgue differentiable with respect to $d_{\mathcal{M}}$.\qed\\

\begin{Theorem} \label{thm::inequality}
Let $h$ be a symmetric generalized distance function, the conditions of Theorem \ref{thm::Main} be fulfilled and furthermore let for $\delta$ hold
\begin{equation} \label{eqn::delta_bonds}
 0 < \delta_* < \delta(x,\xi) < \delta^* < \infty \textnormal{\quad \quad} \textnormal{\quad for a.e. } x \in \R^k
\end{equation}
for some $\xi \in \R^k$ and $\delta_*, \delta^*$. Let $\Delta_K(t)$ denote Kolmogorov discrepancy, i.e.
\begin{equation*}
 \Delta_K(t) := \sup_{0 < u \leq t} \left| F_{XX} (u) - F_{XY} (u)\right| + \sup_{0 < u \leq t} \left| F_{YY} (u) - F_{XY} (u)\right|.
\end{equation*}
For some $\epsilon > 0 $ and $0 < t < \epsilon$ it holds then
\begin{equation} \label{eqn::ineq_L2}
 ||f-g||^2_{L^2} \leq \frac{1}{c \delta_*} \left[ \frac{\Delta_K(t)}{\Phi(\xi , t)} + r(\xi, t) \right]
\end{equation}
and
\begin{equation} \label{eqn::ineq_delta}
 \Delta_K (t) \leq C \delta^* \Phi(\xi, t) (||f||_{L^2} + ||g||_{L^2}) ||f-g||_{L^2},
\end{equation}
where $c$ and $C$ are as in Definition \ref{def:volume_reg}, with $r(\xi, t) \rightarrow 0$ as $t \rightarrow 0$.
\end{Theorem}
\noindent
\textbf{Proof.}
Following (\ref{eqn::pre_limit}), analogous integrals for $\Prob(h(X_3, Y_1) < t)$ and $\Prob(h(Y_2, Y_3) < t)$ are
\begin{align*}
 \frac{F_{XY}(t)}{\Phi(\xi, t)} &= \frac{1}{\Phi(\xi, t)} \int_{\R^k} f(x) \int_{B_t^{(h)}(x)}(g(y) - g(x) ) d y d x + \int_{\R^k} f(x) g(x) \delta_t(x, \xi) dx \\
 &= \frac{1}{\Phi(\xi, t)} \int_{\R^k} g(x) \int_{B_t^{(h)}(x)}(f(y) - f(x) ) d y d x + \int_{\R^k} g(x) f(x) \delta_t(x, \xi) dx
\end{align*}
due to the symmetry of $h$ and the first integral, and
\begin{equation*}
 \frac{F_{YY}(t)}{\Phi(\xi, t)} = \frac{1}{\Phi(\xi, t)} \int_{\R^k} g(x) \int_{B_t^{(h)}(x)}(g(y) - g(x) ) d y d x + \int_{\R^k} g^2(x) \delta_t(x, \xi) dx
\end{equation*}
and therefore
\begin{align*}
 \int_{\R^k} (f(x) - g(x))^2 \delta_t (x, \xi) d x &= \frac{1}{\Phi (\xi, t)} \left( F_{XX}(t) - F_{XY}(t)+F_{YY}(t) - F_{XY}(t) \right) \\
 &- \frac{1}{\Phi (\xi, t)} \int_{\R^k} (f - g)(x) \int_{B_t^{(h)} (x)} \left((f - g)(y) - (f -g)(x) \right) d y d x \\
 & \leq \frac{\Delta_K (t)}{\Phi (\xi, t)} + r(\xi, t)
\end{align*}
where
\begin{equation*}
 r(\xi, t):= \frac{1}{\Phi (\xi, t)} \int_{\R^k} |(f -g)(x)| \int_{B_t^{(h)} (x)} \left|(f - g)(y) - (f - g)(x) \right| d y d x .
\end{equation*}
In particular, since uniformly bounded centered oscillations of $f$ and $g$ imply uniformly bounded centered oscillation of $(f-g)$ (i.e. with $A_{|f-g|} \leq A_f + A_g$), by dominated convergence analogously to the proof of Theorem \ref{thm::Main} it follows that $r(\xi, t) \rightarrow 0$ as $t \rightarrow 0$.
Finally, by assumption there exist $\epsilon > 0 $ s.t. for $0 < t < \epsilon$ and a.e. $x$ it is $c \delta_* \leq c \delta(x, \xi) \leq \delta_t(x, \xi)$ and hence
\begin{equation*}
 c \delta_* ||f-g||^2_{L^2} \leq \int_{\R^k} (f(x) - g(x))^2 \delta_t (x, \xi) d x,
\end{equation*}
and the inequality (\ref{eqn::ineq_L2}) follows. \\

\noindent
This theorem reduces a multivariate two-sample problem to a comparison of three induced univariate interpoint distance distributions and serves as a theoretical foundation for distance-based testing, feature selection procedures etc. The assumptions can be categorized into two types:  assumptions on the distance function, i.e., conditions on the topology and associated volume behavior induced by the distance ($\mathbf{(D1)}$-$\mathbf{(D2)}$) and the regularity assumptions on the densities relative to this distance based structure ($\mathbf{(R1)}$-$\mathbf{(R2)}$) .\\
\noindent
However, many metrics, distance functions and dissimilarities used in practice are neither homogeneous nor translation-invariant, and thus fail to satisfy condition $\mathbf{(D2)}$. Examples include the Canberra distance, whose usefulness has been demonstrated in two-sample testing and clustering methods on genomic data \cite{betken2024two}, \cite{proksch2023personalised}, the Bray-Curtis dissimilarity used in microbiology and ecology \cite{ricotta2017some}, and geodesic distances on Riemannian manifolds \cite{chu2024manifold}. These examples illustrate that many practically relevant distances fall outside of the scope of the original Theorem \ref{thm:maa}, which motivates our intention to generalize the theoretical basis to accommodate non-homogeneous, non-translation-invariant distances. \\

\noindent
The proof of the identifiability theorem in \cite{Maa96} leverages homogeneity of the space and polynomial convergence of ball volumes as $t \rightarrow 0$. Assuming translational invariance of $h$, i.e. assumption $\mathbf{(D2)}$ of Theorem \ref{thm:maa} for $a=1$, we have
    \begin{equation*}
        \mu(B_t^{(h)} (x)) = \mu(B_t^{(h)} (y)) \textnormal{ \quad \quad  } \forall x,y \in \mathbb{R}^k,
    \end{equation*}
    where $B_t^{(h)}(x)$ denotes the ball with center in $x$ and radius $t$ with respect to the distance $h$, while $\mu$ denotes the Lebesgue measure, and therefore $\Phi(x,t) = \Phi(y,t) = \Phi(t)$, where $\Phi(x, t):= \mu(B_t^{(h)} (x)) $. By homogeneity, i.e. assumption $\mathbf{(D2)}$ of Theorem \ref{thm:maa} for $b=0$, it furthermore holds that
        \begin{equation*}
        \Phi(t) =     \mu(B_t^{(h)} (0))=     \mu(|t|B_1^{(h)} (0))=|t|^k  \mu(B_1^{(h)} (0)).
            \end{equation*}
For the Canberra distance on $\mathbb{R}^k$, defined by
\begin{equation}\label{eqn::Canberra_k}
h(x, y) := \sum_{i=1}^k
\begin{cases}
\frac{|x_i - y_i|}{|x_i| + |y_i|}, & \text{if } |x_i| + |y_i| \neq 0 \\
0, & \text{if } x_i = y_i = 0
\end{cases}
\end{equation}
neither of the two conditions, translation invariance and homogeneity, of assumption $\mathbf{(D2)}$ in Theorem \ref{thm:maa} are fulfilled, but the volumes of balls $B_t^{(h)}(x)$ with center in $x$ and radius $t$  do converge to zero as $t$ approaches zero and have comparable convergence rates for different centers in space. To see this, consider the case $k=1$, for which it is $h(x,y) = \frac{|x - y|}{|x| + |y|}$ for $|x| + |y| \neq 0$. In this case, it can be directly calculated that \begin{equation} \label{eqn::Phi_Canberra_1D}
    \Phi(x, t) = \frac{4t \lvert x \rvert}{1-t^2} \textnormal{ \quad \quad for } t \in [ 0,1) \textnormal{ and } x \neq 0.
\end{equation}
For detailed calculations and expansion to case $k > 1$ see Example \ref{bsp::Canberra}.
Ball volumes in Canberra distance hence behave as rational functions in radius $t$ and are furthermore dependent on the positions of the balls. For visual comparison of one-dimensional Canberra ball volumes to ball volumes of the standard Euclidean distance and  an illustration of the shape of two-dimensional unit balls in Canberra distance see Figures \ref{fig:canberra_euclid} and \ref{fig:canberra_balls}. This motivates the definition of \textit{volume-regular distances} (see Definition \ref{def:volume_reg}), for which class we generalize Theorem \ref{thm:maa} under mild conditions on the densities $f$ and $g$.

\begin{figure}[H]
    \centering
    \includegraphics[width=0.9\textwidth]{Figure_2.jpg}
    \caption{Comparison of the volumes of one-dimensional balls in Canberra and Euclidean distances. The plots represent the volumes $\Phi(x,t)$ as level sets by colorscheme, where $x \in (0,10)$ and $t \in (0, 0.1)$. Left Canberra, right Euclidean balls. For a fixed radius $t$, the volume of an Euclidean ball deos not depend on its position in space, while  in Canberra distance balls become larger the further their center from the origin.  }
    \label{fig:canberra_euclid}
\end{figure}
\begin{figure}[H]
    \centering
    \includegraphics[width=0.9\textwidth]{Figure_1.jpg}
    \caption{Behavior of two-dimensional balls in Canberra distance. The figures are generated by computing Canberra distances on a dense grid of points and coloring those inside a ball of a given radius. Upper row: unit balls, with centers  $(10, 1)$,   $(10, 10)$,  $(5, 5)$ and  $(50, 10)$. Lower row: balls of radius $0.8$, $0.6$, $0.4$ and $0.2$ centered at $(10, 10)$. }
    \label{fig:canberra_balls}
\end{figure}

\noindent
In this article, we extend Theorem \ref{thm:maa} beyond homogeneous and translation invariant distance functions: under certain volume and oscillation controls of balls in the considered distance function and assuming Lebesgue differentiability of densities with respect to this distance function, equality of the within- and between-sample distance distributions ensures equality of laws. We complement this finding with computable $L^2$-error bounds expressed via Kolmogorov discrepancies of the three distance distributions. The key idea is to replace the geometric assumptions on the distance, which guarantee the well-behavedness of the balls uniformly in space as the radius shrinks, with more direct analytic conditions on the volumes of the balls.  We provide sufficient criteria under which the equivalence of interpoint distance distributions implies equality of the underlying distributions, and show that these conditions are satisfied for a wide range of distance functions of practical interest. Several corollaries and examples are provided to illustrate the scope of the generalization. The article is organized as follows. Section 2 introduces the standing assumptions (volume-regularity of $h$-balls, $h$-Lebesgue differentiability, bounded centered oscillation), states the generalization, and derives corollaries that recover the original theorem and extend the result to compact manifolds.  Furthermore, in Section 2 we complement the qualitative result with quantitative stability bounds and, under Ahlfors $\alpha$-volume growth and $\beta$-Hölder regularity of the densities, derive dimension-aware rates. In Section 3 we collect illustrative cases, i.e. domain-specific distances, and show that these generate volume-regular balls. Lastly, we outline open questions and future research directions in Section \ref{sec:conclusion}. \\

The post-theorem discussion gives a proof sketch for Theorem~\ref{thm:maa} as presented in \cite{Maa96}: it “leverages homogeneity of the space and polynomial convergence of ball volumes as $t \to 0$.” In particular, using translation invariance (assumption $\mathbf{(D2)}$ with $a=1$), one gets
\[
\mu(B_t^{(h)}(x))=\mu(B_t^{(h)}(y))\ \ \forall x,y\in\mathbb{R}^k,
\]
so “$\Phi(x,t)=\Phi(y,t)=\Phi(t)$,” where $\Phi(x,t):=\mu(B_t^{(h)}(x))$. Then, using homogeneity (assumption $\mathbf{(D2)}$ with $b=0$), the ball volumes satisfy the polynomial scaling
\[
\Phi(t)=\mu(B_t^{(h)}(0))=\mu(|t|B_1^{(h)}(0))=|t|^k\,\mu(B_1^{(h)}(0)).
\]
These uniform-in-space and polynomial volume properties (as $t\to 0$) are described as the geometric ingredients used in the identifiability proof.

Biconditional or Equivalence
Implication

Question: Let \(X_1,X_2,X_3\) be i.i.d. random vectors in \(\mathbb{R}^k\) with Lebesgue density \(f_X\), and let \(Y_1,Y_2,Y_3\) be i.i.d. random vectors in \(\mathbb{R}^k\) with Lebesgue density \(f_Y\). Assume that \(X_1,X_2,X_3,Y_1,Y_2,Y_3\) are mutually independent. Let \(d:\mathbb{R}^k\times\mathbb{R}^k\to\mathbb{R}\) be a nonnegative continuous function such that \(d(x,y)=0\) iff \(x=y\), and \(d(ax+b,ay+b)=|a|\,d(x,y)\) for all \(a\in\mathbb{R}\) and \(x,y,b\in\mathbb{R}^k\). Also assume \(\int_{\mathbb{R}^k} f_X^2(x)\,dx<\infty\), \(\int_{\mathbb{R}^k} f_Y^2(y)\,dy<\infty\), and that the origin is a Lebesgue point of \(u(y)=\int f_Y(x+y)f_X(x)\,dx\), meaning \(\frac{1}{\lambda(B_r(0))}\int_{B_r(0)} |u(y)-u(0)|\,dy\to 0\) as \(r\to 0\), where \(B_r(0)\) is the \(d\)-ball of radius \(r\) centered at \(0\). Which of the following statements is equivalent to \(f_X=f_Y\)?

\(d(X_1,X_2)\overset{\mathcal D}{=}d(Y_1,Y_2)\overset{\mathcal D}{=}d(X_3,Y_3)\); equivalently, the within-\(X\), within-\(Y\), and between-sample distances all have the same distribution.

Choices:
B. \(d(X_1,X_2)\overset{\mathcal D}{=}d(Y_1,Y_2)\); equivalently, it is enough that the two within-sample distance distributions coincide.
C. If \(f_X=f_Y\), then \(d(X_1,X_2)\overset{\mathcal D}{=}d(Y_1,Y_2)\overset{\mathcal D}{=}d(X_3,Y_3)\).
D. For some fixed indices \(i\neq j\) and \(m\neq n\), it suffices that \(d(X_i,X_j)\overset{\mathcal D}{=}d(X_m,Y_n)\); equivalently, equality in distribution of one within-sample distance and one between-sample distance already characterizes \(f_X=f_Y\).
E. \(d(X_1,X_2)\overset{\mathcal D}{=}d(X_3,Y_3)\) and \(d(Y_1,Y_2)\overset{\mathcal D}{=}d(X_3,Y_3)\) almost surely; equivalently, the within-\(X\), within-\(Y\), and between-sample distances can be coupled to be equal pointwise.

necessity of matching the between-sample distance law in addition to both within-sample laws

dropped the converse direction of the equivalence

full triple distributional equality replaced by a single pairwise distributional identity

distributional equality strengthened to almost-sure pointwise equality via coupling

The stem does not reveal the correct option explicitly or through obvious wording cues. It states technical assumptions and asks for an equivalent condition without signaling that the full triple distributional equality is the answer.

The item is essentially a direct theorem-recall question: under the theorem's hypotheses, it asks for the statement equivalent to \(f_X=f_Y\), and the correct choice restates that equivalence almost verbatim.

Some reasoning is needed to distinguish equivalence from one-way implication and to reject stronger or weaker variants, especially choices C, D, and E. However, the main task is still recognizing the exact theorem statement rather than generating a substantive mathematical conclusion.

The distractors are plausible and target meaningful failure modes: confusing necessity with sufficiency (B), weakening equivalence to implication (C), mishandling quantifiers/scope (D), and strengthening distributional equality to almost-sure coupling (E). They are distinct and mathematically aligned.

A solid recall-oriented MCQ with strong distractors and little answer leakage, but it is highly tautological and only moderately tests genuine reasoning.


\label{bc}

We note that
\begin{align}\label{bn}
\sum_{n=1}^\infty B(n)q^{n+1}&=q\sum_{n=1}^\infty\frac{q^{2n-1}}{(q;q^2)_n} \notag \\
&=\sum_{n=1}^\infty\frac{q^{2n}}{(q;q^2)_n},
\end{align}
here, an

\begin{theorem}\label{main theorem}
For $n>0$, we have
\begin{align}\label{main theorem eqn}
A(n)=B(n)=C(n+1)=\frac{1}{2}D(n+1),
\end{align}
where $C(n)$ is the number of partitions of $n$ with largest part even and parts not exceeding half of the largest part are distinct, and $D(n)$ is the number of partitions of $n$ into non-negative parts wherein the smallest part appear exactly twice and no other parts are repeated.
\end{theorem}

The grandfather of all partition identities is Euler's theorem \cite{andrews book}, namely,
\begin{align*}
A(n)=B(n),
\end{align*}
where $A(n)$ is the number of partitions of $n$ into distinct parts and $B(n)$ is the number of partitions of $n$ into odd parts.

This theorem contains all the elements that would suggest generalizations, and, over the centuries, generalizations have been found in profusion. The Rogers-Ramanujan identities \cite[p.~104]{andrews book} and Schur's 1926 theorem \cite[p.~116]{andrews book} kicked off the twentieth century's contributions, consult Henry Alder's survey \cite{alder} for an account of some of the results in the late 20th century.

However, little has ever been added to Euler's theorem itself. In this paper, we shall add two further partition functions.

\subjclass[2020]{Primary 11P84, 05A17}
 \keywords{Euler's theorem, Partitions, Glaisher's bijection}
\maketitle
\pagenumbering{arabic}
\pagestyle{headings}
\begin{abstract}
Euler's theorem asserts that $A(n)=B(n)$ where $A(n)$ is the number of partitions of $n$ into distinct parts and $B(n)$ is the number of partitions of $n$ into odd parts. In this paper, it is proved that for $n>0$,
\begin{align*}
A(n)=B(n)=C(n+1)=\frac{1}{2}D(n+1),
\end{align*}
where $C(n)$ is the number of partitions of $n$ with largest part even and parts not exceeding half of the largest part are distinct, and $D(n)$ is the number of partitions of $n$ into non-negative parts wherein the smallest part appear exactly twice and no other parts are repeated.

The grandfather of all partition identities is Euler's theorem \cite{andrews book}, namely,
\begin{align*}
A(n)=B(n),
\end{align*}
where $A(n)$ is the number of partitions of $n$ into distinct parts and $B(n)$ is the number of partitions of $n$ into odd parts.

However, little has ever been added to Euler's theorem itself. In this paper, we shall add two further partition functions.

For example, if $n=6$, the four sets of partitions are following:
\begin{center}
\begin{tabular}{ p{2cm} p{3.5cm} p{3cm} p{4.1cm} p{4.1cm}}

On the other hand, we have
\begin{align}
\sum_{n=1}^\infty C(n)q^{n}&=\sum_{n=1}^\infty\frac{(-q;q)_nq^{2n}}{(q^{n+1};q)_n}\nonumber\\
&=\sum_{n=1}^\infty\frac{(q^2;q^2)_nq^{2n}}{(q;q)_{2n}} \label{c1n}\\
&=\sum_{n=1}^\infty\frac{q^{2n}}{(q;q^2)_n}.\label{cn}
\end{align}

Let us fix $C(0)=1$. From \eqref{c1n}, we have
\begin{align}\label{before p19}
\sum_{n=0}^\infty C(n)q^{n}&=\sum_{n=0}^\infty\frac{(q^2;q^2)_nq^{2n}}{(q;q)_{2n}}\nonumber\\
&=(q^2;q^2)_\infty\sum_{n=0}^\infty\frac{q^{2n}}{(q;q)_{2n}(q^{2n+2};q^2)_\infty}.
\end{align}
We now employ the following Euler's identity \cite[p.~19, (2.2.5)]{andrews book}
\begin{align}\label{euler identity}
\frac{1}{(t;q)_\infty}=\sum_{m=0}^\infty\frac{t^m}{(q,q)_m},
\end{align}
(with replacing $q$ by $q^2$ and then letting $t=q^{2n+2}$) in \eqref{before p19} so as to obtain
\begin{align}\label{series}
\sum_{n=0}^\infty C(n)q^{n}&=(q^2;q^2)_\infty\sum_{n=0}^\infty\frac{q^{2n}}{(q;q)_{2n}}\sum_{m=0}^\infty\frac{q^{2nm+2m}}{(q^2;q^2)_m}\nonumber\\
&=(q^2;q^2)_\infty\sum_{m,n=0}^\infty\frac{q^{2n+2nm+2m}}{(q;q)_{2n}(q^2;q^2)_m}\nonumber\\
&=(q^2;q^2)_\infty\sum_{m,n=0}^\infty\frac{1}{2}\left(1+(-1)^n\right)\frac{q^{n+nm+2m}}{(q;q)_{n}(q^2;q^2)_m}\nonumber\\
&=\frac{1}{2}(q^2;q^2)_\infty\sum_{m=0}^\infty\frac{q^{2m}}{(q^2;q^2)_m}\left\{\sum_{n=0}^\infty\frac{q^{n(m+1)}}{(q;q)_n}+\sum_{n=0}^\infty\frac{(-1)^nq^{n(m+1)}}{(q;q)_n}\right\}.
\end{align}
Upon invoking \eqref{euler identity} twice, once with letting $t=q^{m+1}$ and once with letting $t=-q^{m+1}$, and then substituting both resulting expressions in \eqref{series}, we conclude that
\begin{align}\label{3.4}
\sum_{n=0}^\infty C(n)q^{n}&=\frac{1}{2}(q^2;q^2)_\infty\sum_{m=0}^\infty\frac{q^{2m}}{(q^2;q^2)_m}\left\{\frac{1}{(q^{m+1};q)_\infty}+\frac{1}{(-q^{m+1};q)_\infty}\right\}\nonumber\\
&=\frac{1}{2}(-q;q)_\infty\sum_{m=0}^\infty\frac{q^{2m}}{(-q;q)_m}+\frac{1}{2}(q;q)_\infty\sum_{m=0}^\infty\frac{q^{2m}}{(q;q)_m}\nonumber\\
&=\frac{1}{2}\sum_{m=0}^\infty q^{2m}(-q^{m+1};q)_\infty+\frac{1}{2}(1-q),
\end{align}
where the last step follows upon again using \eqref{euler identity} with letting $t=q^2$.

\subsection{A Bijective proof of $A(n-1)=\frac{1}{2}D(n)$}
We can first remove the non-negative part condition for $D(n)$ as follows:
\begin{quote}
For $n\ge 2$, $D(n)$ counts the number of partitions of $n$ where only the smallest part can repeat at most twice and all other parts are distinct.
\end{quote}
In other words, a partition $\lambda=(\lambda_1,\ldots, \lambda_{\ell})$ counted by $D(n)$ satisfies
$$
\lambda_1>\lambda_2> \cdots > \lambda_{\ell-1}\ge \lambda_{\ell} \text{ if $\ell>1$}
$$
or
$\lambda=(\lambda_1)$ has only one part.

Let $M \le 2N$ be an even number, which can be written as
$$
M=2^k a \text{ for some odd integer $a$ and some $ k\ge 1$}.
$$
We apply Glaisher's bijection $\phi$ to $M$ and obtain $2^k$ parts of size $a$. Note that since
$
2^k a \le 2N,
$
$$
a \le N.
$$
Also, if $M\le N$, then $M$ can appear only once, so we get all distinct positive powers $2^k$ such that
\begin{equation}
2^k\le N/a. \label{power1}
\end{equation}
On the other hand, if $N< M\le 2N$, then
\begin{equation}
2^k> N/a, \label{power2}
\end{equation}
and $M$ can repeat.
Suppose $M$ appears $f$ many times. Upon applying Glaisher's bijection $\phi$ to $f$ copies of $M$, we obtain
$$
 f 2^{k} \text{ copies of $a$}.
$$
By writing $f$ as a binary expansion,
$$
f 2^k =(f_0 \,2^{0}+f_1 \, 2^{1} +\cdots ) 2^k,
$$
where $f_j$ is either $0$ or $1$ for $j\ge 0$. By \eqref{power2}, we see that each summand in the above expression represents a distinct power of $2$ greater than $N/a$, i.e.,
\begin{equation}
f_j \, 2^{j+k} >N/a. \label{power3}
\end{equation}
It follows from \eqref{power1} and \eqref{power3} that $a$ can appear with any multiplicity greater than $1$. This proves that the resulting partition is counted by $B(n)$.

For example, if $n=6$, the four sets of partitions are following:
\begin{center}
\begin{tabular}{ p{2cm} p{3.5cm} p{3cm} p{4.1cm} p{4.1cm}}

 \vspace{2mm}$A(6)$\newline  &  \vspace{2mm} $B(6)$ & \vspace{2mm} $C(7)$& \vspace{2mm} $D(7)$ \\

 $6$ \newline 5+1 \newline 4+2\newline 3+2+1   & 5+1 \newline 3+3 \newline 3+1+1+1 \newline 1+1+1+1+1+1  & 6+1\newline 4+3 \newline 4+2+1 \newline 2+2+2+1  & 0+0+7\newline 0+0+6+1\newline 0+0+5+2\newline 0+0+4+3\newline 0+0+4+2+1\newline 1+1+5\newline 1+1+2+3\newline 2+2+3
\end{tabular}
\end{center}

We conclude the introduction with the following remark.
\begin{remark}
In their recent paper \cite{ba}, M. El Bachraoui and the first author considered partitions with multiple appearances by the first part.  All parts were assumed to be positive. It would be a simple matter to extend the results of that paper to the case of non-negative parts in that this would add $(-q;q)_\infty$ to the generating functions in question.
\end{remark}

This paper is organised as follows. In section \ref{bc}, we provide the brief proof that $B(n)=C(n+1)$. In Section \ref{cd}, we prove that $C(n)=\frac{1}{2}D(n)$. We also provide bijective proofs our assertions in Theorem \ref{main theorem} in Section \ref{bijective proofs}.

The post-theorem introduction does not give a proof sketch beyond an outline of the paper: in Section~\ref{bc} they "provide the brief proof that $B(n)=C(n+1)$"; in Section~\ref{cd} they "prove that $C(n)=\frac{1}{2}D(n)$"; and in Section~\ref{bijective proofs} they "provide bijective proofs" of the assertions in Theorem~\ref{main theorem}.

The post-theorem introduction does not give a proof sketch beyond an outline of the paper: next they note that
\begin{align}\label{bn}
\sum_{n=1}^\infty B(n)q^{n+1}&=q\sum_{n=1}^\infty\frac{q^{2n-1}}{(q;q^2)_n} \notag \\
&=\sum_{n=1}^\infty\frac{q^{2n}}{(q;q^2)_n},
\end{align}
here, an
and also that
\begin{align}
\sum_{n=1}^\infty C(n)q^{n}&=\sum_{n=1}^\infty\frac{(-q;q)_nq^{2n}}{(q^{n+1};q)_n}\nonumber\\
&=\sum_{n=1}^\infty\frac{(q^2;q^2)_nq^{2n}}{(q;q)_{2n}} \label{c1n}\\
&=\sum_{n=1}^\infty\frac{q^{2n}}{(q;q^2)_n}.\label{cn}
\end{align}

Comparing \eqref{bn} and \eqref{cn}, we see that $B(n)=C(n+1)$.

\section{Proof that $C(n)=\frac{1}{2}D(n)$}\label{cd}

Let us fix $C(0)=1$. From \eqref{c1n}, we have
\begin{align}\label{before p19}
\sum_{n=0}^\infty C(n)q^{n}&=\sum_{n=0}^\infty\frac{(q^2;q^2)_nq^{2n}}{(q;q)_{2n}}\nonumber\\
&=(q^2;q^2)_\infty\sum_{n=0}^\infty\frac{q^{2n}}{(q;q)_{2n}(q^{2n+2};q^2)_\infty}.
\end{align}
They then prove that $C(n)=\frac{1}{2}D(n)$, and finally they provide bijective proofs as follows:
\label{bijective proofs}

\subsection{A Bijective proof of $A(n-1)=\frac{1}{2}D(n)$}
We can first remove the non-negative part condition for $D(n)$ as follows:
\begin{quote}
For $n\ge 2$, $D(n)$ count

\label{main theorem}
For $n>0$, we have
\begin{align}\label{main theorem eqn}
A(n)=B(n)=C(n+1)=\frac{1}{2}D(n+1),
\end{align}
where $C(n)$ is the number of partitions of $n$ with largest part even and parts not exceeding half of the largest part are distinct, and $D(n)$ is the number of partitions of $n$ into non-negative parts wherein the smallest part appear exactly twice and no other parts are repeated.,

Question: For each integer \(n>0\), let \(A(n)\) be the number of partitions of \(n\) into distinct parts, and let \(B(n)\) be the number of partitions of \(n\) into odd parts. Also, for an integer \(m\), let \(C(m)\) be the number of partitions of \(m\) whose largest part is even and whose parts not exceeding half of the largest part are distinct, and let \(D(m)\) be the number of partitions of \(m\) into non-negative parts such that the smallest part appears exactly twice and no other part is repeated. Which statement is equivalent to the equality \(A(n)=B(n)\) for all \(n>0\)?

For every \(n>0\), \(A(n)=B(n)=C(n+1)=\tfrac12 D(n+1)\).

Choices:
B. For every \(n>0\), \(A(n)=B(n)=C(n)=\tfrac12 D(n)\).
C. For every \(n>0\), \(A(n)=B(n)=C(n+1)\).
D. For every \(n>0\), \(A(n)=B(n)=C(n+1)=D(n+1)\).
E. For every \(n>0\), \(A(n)=B(n)=C(n+1)=\tfrac12 D(n)\).

The stem defines the partition functions and asks for an equivalent statement, but it does not explicitly reveal the needed index shift or the factor of 1/2. The correct answer is not stated or strongly hinted at in the wording.

The item is close to asking for the exact theorem-level equivalence rather than a genuinely new conclusion. It is not a pure restatement of A(n)=B(n), since the C and D relations must be matched correctly, but it still mainly tests recognition of the precise formulation.

Some reasoning is needed to distinguish subtle variants involving shifted indices, a missing factor, and a weaker true statement. However, the task is still largely theorem recognition/verification rather than substantial generative mathematical reasoning.

The distractors are plausible and well targeted: off-by-one shifts, omission of the 1/2 factor, and inclusion of a weaker true statement. These reflect realistic mathematical errors and are meaningfully distinct.

A solid MCQ with no real answer leakage and strong distractors, but it leans toward recalling the exact theorem statement rather than forcing deep generative reasoning.


Let $\mathbb{F}_{2^n}$ be the finite field with $2^n$ elements. A function $f$ from $\mathbb{F}_{2^n}$ into itself is called almost perfect nonlinear (APN) if for any non zero $a\in \mathbb{F}_{2^n}$ and any $b\in \mathbb{F}_{2^n}$, the equation
$$
f(x+a)+f(x)=b
$$
admits at most two solutions.

APN functions play a central role in modern cryptography since they provide optimal resistance against differential cryptanalysis (\cite{diffBS}) when used as substitution boxes in block ciphers. Beyond cryptography, they also appear as optimal objects in coding theory, combinatorics, and projective geometry \cite{semibi,DO,hypDE}.

Despite their importance, only a few infinite families of APN functions are currently known, and their classification up to CCZ- or EA-equivalence remains an open problem (see \cite{LK} for a list of known APN families and for the definition of these equivalence relations).

Several of the known families that can be defined in even dimension have been obtained using the so called bivariate construction introduced by Carlet in \cite{carletbiv}. In particular, let $n=2m$, we can decompose $\mathbb{F}_{2^n}$ as $\mathbb{F}_{2^m}\times\mathbb{F}_{2^m}$, and a function from $\mathbb{F}_{2^m}\times\mathbb{F}_{2^m}$ into itself can be represented using bivariate polynomials, that is, $f(x,y)=(f_1(x,y),f_2(x,y))$ with $f_1,f_2\in\mathbb{F}_{2^m}[x,y]$.

In \cite{carletbiv}, the author considered functions $f(x,y)$ where $f_1(x,y)$ was given by the Maiorana-McFarland function $xy$, and provided some necessary and sufficient conditions for the APN property of $f(x,y)$. He also introduced a class of APN function in bivariate form which was later proved (see \cite{APN}) to be equivalent to the hexanomial family constructed in \cite{BChex}.
The bivariate construction was later used for obtaining other classes of APN functions (\cite{APN,tan,ZP}). Recently, in \cite{Golbiproj}, G\"olo\u{g}lu
proposed a generalization of the bivariate construction based on the so-called biprojective polynomials. Bi-projective polynomials has been used for constructing several classes of APN functions lately \cite{CLVbiproj,GKbiproj,LZLQ}.

Within specific families, the APN property is intrinsically connected to the existence of polynomials with well-defined structural properties. Accordingly, a fundamental problem is to determine whether APN functions derived from these constructions exist in infinitely many dimensions or whether they are restricted to finitely many instances \cite{BCPZ,Bhex,BTThex,Golhex}.

In particular, the existence of several classes of bivariate APN families constructed to date relies on the fact that a certain projective polynomial, that is a polynomial of type $x^{2^r+1}+x+a$, admits no roots over $\mathbb{F}_{2^m}$ \cite{CLVbiproj,carletbiv,tan}.

Projective polynomials and their roots have been studied in several works, such as \cite{Bluher,BTThex,HKproj1,HKproj2}.
So, applying Bluher’s results (\cite{Bluher}), one obtains that these constructions yield an APN function in every dimension in which they are defined.

 For the case of the APN class introduced in \cite{APN}, the existence of instances coming from this construction is related to the roots of a certain polynomial, which is not projective.

In particular, the APN class given in \cite{APN} is the following:

Let $\mathbb{F}_{2^n}$ be the finite field with $2^n$ elements. A function $f$ from $\mathbb{F}_{2^n}$ into itself is called almost perfect nonlinear (APN) if for any non zero $a\in \mathbb{F}_{2^n}$ and any $b\in \mathbb{F}_{2^n}$, the equation
$$
f(x+a)+f(x)=b
$$
admits at most two solutions.

Several of the known families that can be defined in even dimension have been obtained using the so called bivariate construction introduced by Carlet in \cite{carletbiv}. In particular, let $n=2m$, we can decompose $\mathbb{F}_{2^n}$ as $\mathbb{F}_{2^m}\times\mathbb{F}_{2^m}$, and a function from $\mathbb{F}_{2^m}\times\mathbb{F}_{2^m}$ into itself can be represented using bivariate polynomials, that is, $f(x,y)=(f_1(x,y),f_2(x,y))$ with $f_1,f_2\in\mathbb{F}_{2^m}[x,y]$.

In particular, the existence of several classes of bivariate APN families constructed to date relies on the fact that a certain projective polynomial, that is a polynomial of type $x^{2^r+1}+x+a$, admits no roots over $\mathbb{F}_{2^m}$ \cite{CLVbiproj,carletbiv,tan}.

Projective polynomials and their roots have been studied in several works, such as \cite{Bluher,BTThex,HKproj1,HKproj2}.
So, applying Bluher’s results (\cite{Bluher}), one obtains that these constructions yield an APN function in every dimension in which they are defined.

For the case of the APN class introduced in \cite{APN}, the existence of instances coming from this construction is related to the roots of a certain polynomial, which is not projective.

In particular, the APN class given in \cite{APN} is the following:

For the case of the APN class introduced in \cite{APN}, the existence of instances coming from this construction is related to the roots of a certain polynomial, which is not projective.

The authors showed that for $n\le 12$ (so $m\le 6$) it was possible to produce new APN functions (up to CCZ-equivalence). However, if such functions exist also for higher dimensions is an open problem.

Denote $q=2^{m/2}$.
First observe that $P_{c,b}(X)$ has a zero in $\mathbb{F}_{q^2}$ if and only if there exists $x\in \mathbb{F}_{q^2}^*$ such that
\begin{align}
\frac{cx^{2^r+1} + bx^{2^r} + 1}{x} \label{eq0}
\end{align}
is a $(q+1)$-root of unity. This is equivalent to ask that
\begin{align*}
\frac{cx^{2^r+1} + bx^{2^r} + 1}{x}=\frac{x^q}{c^qx^{(2^r+1)q} + b^qx^{2^rq} + 1}.
\end{align*}
Let $x=x_0+\xi x_1$, where $\{1,\xi\}$ is an $\mathbb{F}_q$ basis of $\mathbb{F}_{q^2}$ and $x_0,x_1 \in \mathbb{F}_q$. The previous condition (since $x\neq 0$) can be equivalently rewritten as
\begin{align*}
\left(c(x_0+\xi x_1)^{2^r+1} + b(x_0+\xi x_1)^{2^r} + 1\right)\left(c^q(x_0+\xi^q x_1)^{2^r+1} + b^q(x_0+\xi^q x_1)^{2^r} + 1\right)+(x_0+\xi x_1)(x_0+\xi^q x_1)=0.
\end{align*}
In order to prove that for each $b,c \in \mathbb{F}_{q^2}$ there is at least a solution $(\overline{x_0},\overline{x_1}) \in \mathbb{F}_q^2$ to the above equation, we consider the algebraic curve $\mathcal{D}_{b,c,r}$ defined by
\begin{align*}
\left(c(X+\xi Y)^{2^r+1} + b(X+\xi Y)^{2^r} + 1\right)\left(c^q(X+\xi^q Y)^{2^r+1} + b^q(X+\xi^q Y)^{2^r} + 1\right)+(X+\xi Y)(X+\xi^q Y)=0.
\end{align*}
Via the change of variables $(X+\xi Y,X+\xi^q Y)\mapsto (X,Y)$, $\mathcal{D}_{b,c,r}$ is affinely equivalent to the plane curve
$\mathcal{C}_{b,c,r}$ defined by
\begin{align*}
\left(cX^{2^r+1} + bX^{2^r} + 1\right)\left(c^qY^{2^r+1} + b^qY^{2^r} + 1\right)+XY=0.
\end{align*}
Our strategy consists in proving that $\mathcal{C}_{b,c,r}$, $b, c \in \F_{q^2}$, $c \ne 0$, $r \ge 1$, is absolutely irreducible and so is $\mathcal{D}_{b,c,r}$. Hence, by the Hasse-Weil bound we obtain the existence of at least one point $(\overline{x_0},\overline{x_1}) \in \mathbb{F}_q^2$ in $\mathcal{D}_{b,c,r}$. The case $c = 0$ is treated separataly. Therefore, by Theorem \ref{Th:APN} the function $f_{b,c,r}(x,y)$ is not APN.

\begin{lemma}\label{lm:H90}
Let $\alpha\in\mathbb{F}_{2^m}$ and let $j$ be such that $\gcd(j,m)=1$. Then, $Tr_{2}^{2^m}(\alpha)=0$ if and only if there exists $\beta\in\mathbb{F}_{2^m}$ such that $\alpha=\beta^{2^j}-\beta$.
Here $Tr_{2}^{2^m}$ is the trace map from $\mathbb{F}_{2^m}$ onto $\mathbb{F}_2$.
\end{lemma}

\begin{lemma}
Let $b\in \mathbb{F}_{q^2}^*$, and $r$ be such that $\gcd(r,m)=1$. Then, the polynomial
 $\big(bX^{2^r} + 1\big)^{q+1} + X^{q+1}$ has a zero in $\mathbb{F}_{q^2}$.
\end{lemma}
\begin{proof}
 We note that $\big(bX^{2^r} + 1\big)^{q+1} + X^{q+1}$ has a zero in $\mathbb{F}_{q^2}$ if and only if there exist $x\in\mathbb{F}_{q^2}$ and $u\in\mu_{q+1}$ such that
 \begin{equation}\label{eq:c0}
 bx^{2^r} + ux+1=0.
 \end{equation}
 Now, $b=u't$ for some $t\in\mathbb{F}_{q}^*$ and $u' \in \mu_{q+1}$. Therefore, performing the substitution $x\mapsto b^{-2^{-r}}x$ and considering $u=u'^{2^{-r}}\in\mu_{q+1}$, Equation \eqref{eq:c0} becomes
 \begin{equation}\label{eq:c02}
 x^{2^r}+t'x+1=0,
 \end{equation}
 where $t'=t^{-2^{-r}}$.

As a consequences we get the following:
\begin{theorem}\label{th:b0}
Let $n = 2m$ with $m$ even, and let $r < m/2$ be such that $\gcd(r,m) = 1$. Then, for any $b\in \mathbb{F}_{2^{m}}^*$, the function $f_{b,0,r}(x,y)$ defined as in Theorem \ref{Th:APN} is not APN.
\end{theorem}

In \cite{APN}, the authors show that for $m\le 6$, we have instance of APN functions coming from Theorem \ref{Th:APN} for $r=1$. To check that $f_{b,c,1}$ cannot be APN for $8 \le m\le 16$ we need the following proposition which allows us to reduce the number of pairs $(b,c)$.
\begin{prop}\label{prop:red}
 Let $k\ge 0$ be an integer, and $u\in\mu_{q+1}$. Then, for any $b,c\in\mathbb{F}_{q^2}$ the equation
 \begin{equation}\label{eq:sol1}
 \big(cx^{2^r+1} + bx^{2^r} + 1\big)^{q+1} + x^{q+1}=0
 \end{equation}

By \eqref{eq0}, the existence of a root of the polynomial
\[
P_{c,b}(X) = \bigl(cX^{2^r+1} + bX^{2^r} + 1\bigr)^{q+1} + X^{q+1}
\]
is equivalent to the existence of an element $u \in \mu_{q+1}$ such that the equation
\[
cx^{2^r+1} + bx^{2^r} + ux + 1 = 0
\]
admits a root in $\mathbb{F}_{q^2}$. This equation can be transformed into
\begin{equation}\label{eq:projpol}
x^{2^r+1} + x + A = 0,
\end{equation}
where
\[
 A = \frac{(ub+c)c^{2^r-1}}{\bigl(uc^{2^r-1} + b^{2^r}\bigr)^{2^{-r}+1}},
\]
under the assumption that $uc^{2^r-1}+b^{2^r} \neq 0$, see for instance \cite{Bluher}. In \cite[Theorem 2.1]{BTThex} it has been proved that equation \eqref{eq:projpol} admits no solution over $\mathbb{F}_{q^2}$ if and only if
\begin{equation}\label{eq:condition}
A = \frac{a(a+1)^{2^r+2^{-r}}}{(a+a^{2^{-r}})^{2^r+1}},
\end{equation}
for some non-cube $a$.
For the case $r=1$, the previous request is equivalent to ask that
\begin{align*}
A=a+\frac{1}{a},
\end{align*}
for some non-cube $a$.
So, for $r=1$, using MAGMA \cite{Magma} it is possible to check that one can always find some $u \in \mu_{q+1}$ such that {$uc+b^{2} \neq 0$ and} the associated value of $A$ does \emph{not} belong to the set
\[
\Biggl\{ a+\frac{1}{a} : a \text{ not a cube} \Biggr\},
\]
for any choice of $b,c\in\mathbb{F}_{2^m}$ and $8\le m\le 16$. Therefore, the function $f_{b,c,1}$ cannot be APN. So, we get the following result.
\begin{cor}
 Let $m\ge 8$ be an even integer. Then, for any choice $b,c \in \mathbb{F}_{2^m}$ the function $f_{b,c,1}$ as in Theorem \ref{Th:APN} is not APN.
\end{cor}

\begin{theorem}\label{Th:APN}\cite[Theorem 6.2]{APN}
Let $n = 2m$ with $m$ even, and let $r < m/2$ be such that $\gcd(r,m) = 1$.  Consider $b,c \in \mathbb{F}_{2^{m}}$
Then
\[
f_{b,c,r}(x,y) = \big(xy,\, x^{2^r+1} + x^{2^{r+m/2}} y^{2^{m/2}} + bxy^{2^r} + cy^{2^r+1}\big)
\]
is APN if and only if
\[
P_{c,b}(X)=\big(cX^{2^r+1} + bX^{2^r} + 1\big)^{2^{m/2}+1} + X^{2^{m/2}+1}
\]
has no zero in $\mathbb{F}_{2^m}$.
\end{theorem}

The authors showed that for $n\le 12$ (so $m\le 6$) it was possible to produce new APN functions (up to CCZ-equivalence). However, if such functions exist also for higher dimensions is an open problem.

The aim of this work is to investigate such an open question. In particular, we prove that for each $r <m/8-1$ there are no instances of $b,c \in \mathbb{F}_{2^{m}}$ for which $f_{b,c,r}(x,y)$ is APN.

The main tool is given by application of techniques from algebraic varieties over finite fields.

Denote $q=2^{m/2}$.
First observe that $P_{c,b}(X)$ has a zero in $\mathbb{F}_{q^2}$ if and only if there exists $x\in \mathbb{F}_{q^2}^*$ such that
\begin{align}
\frac{cx^{2^r+1} + bx^{2^r} + 1}{x}  \label{eq0}
\end{align}
is a $(q+1)$-root of unity. This is equivalent to ask that
\begin{align*}
\frac{cx^{2^r+1} + bx^{2^r} + 1}{x}=\frac{x^q}{c^qx^{(2^r+1)q} + b^qx^{2^rq} + 1}.
\end{align*}
Let $x=x_0+\xi x_1$, where $\{1,\xi\}$ is an $\mathbb{F}_q$ basis of $\mathbb{F}_{q^2}$ and $x_0,x_1 \in \mathbb{F}_q$. The previous condition (since $x\neq 0$) can be equivalently rewritten as
\begin{align*}
\left(c(x_0+\xi x_1)^{2^r+1} + b(x_0+\xi x_1)^{2^r} + 1\right)\left(c^q(x_0+\xi^q x_1)^{2^r+1} + b^q(x_0+\xi^q x_1)^{2^r} + 1\right)+(x_0+\xi x_1)(x_0+\xi^q x_1)=0.
\end{align*}
In order to prove that for each $b,c \in \mathbb{F}_{q^2}$ there is at least a solution $(\overline{x_0},\overline{x_1}) \in \mathbb{F}_q^2$ to the above equation, we consider the algebraic curve $\mathcal{D}_{b,c,r}$ defined by
\begin{align*}
\left(c(X+\xi Y)^{2^r+1} + b(X+\xi Y)^{2^r} + 1\right)\left(c^q(X+\xi^q Y)^{2^r+1} + b^q(X+\xi^q Y)^{2^r} + 1\right)+(X+\xi Y)(X+\xi^q Y)=0.
\end{align*}
Via the change of variables $(X+\xi Y,X+\xi^q Y)\mapsto (X,Y)$, $\mathcal{D}_{b,c,r}$ is affinely equivalent to the plane curve
$\mathcal{C}_{b,c,r}$ defined by
\begin{align*}
\left(cX^{2^r+1} + bX^{2^r} + 1\right)\left(c^qY^{2^r+1} + b^qY^{2^r} + 1\right)+XY=0.
\end{align*}
Our strategy consists in proving that $\mathcal{C}_{b,c,r}$, $b, c \in \F_{q^2}$, $c \ne 0$, $r \ge 1$, is absolutely irreducible and so is $\mathcal{D}_{b,c,r}$. Hence, by the Hasse-Weil bound we obtain the existence of at least one point $(\overline{x_0},\overline{x_1}) \in \mathbb{F}_q^2$ in $\mathcal{D}_{b,c,r}$. The case $c = 0$ is treated separataly. Therefore, by Theorem \ref{Th:APN} the function $f_{b,c,r}(x,y)$ is not APN.

Let $q=2^{m/2}$. Using Theorem~\ref{Th:APN}, it suffices to show that $P_{c,b}(X)$ has a zero in $\mathbb{F}_{q^2}$ (hence in $\mathbb{F}_{2^m}$) for the relevant parameters, so that $f_{b,c,r}(x,y)$ is not APN.

First, observe that “$P_{c,b}(X)$ has a zero in $\mathbb{F}_{q^2}$ if and only if there exists $x\in \mathbb{F}_{q^2}^*$ such that
\[
\frac{cx^{2^r+1} + bx^{2^r} + 1}{x}
\]
is a $(q+1)$-root of unity,” which is equivalently rewritten as
\[
\frac{cx^{2^r+1} + bx^{2^r} + 1}{x}=\frac{x^q}{c^qx^{(2^r+1)q} + b^qx^{2^rq} + 1}.
\]
Write $x=x_0+\xi x_1$ with $\{1,\xi\}$ an $\mathbb{F}_q$-basis of $\mathbb{F}_{q^2}$ and $x_0,x_1\in\mathbb{F}_q$. Then (since $x\neq 0$) the condition becomes a plane equation in $(x_0,x_1)$.

To prove that “for each $b,c\in\mathbb{F}_{q^2}$ there is at least a solution $(\overline{x_0},\overline{x_1})\in\mathbb{F}_q^2$,” introduce the algebraic curve $\mathcal{D}_{b,c,r}$ over $\mathbb{F}_q$ defined by that equation, and via the change of variables $(X+\xi Y, X+\xi^q Y)\mapsto (X,Y)$ pass to an affinely equivalent plane curve $\mathcal{C}_{b,c,r}$:
\[
\left(cX^{2^r+1} + bX^{2^r} + 1\right)\left(c^qY^{2^r+1} + b^qY^{2^r} + 1\right)+XY=0.
\]
The stated strategy is to prove $\mathcal{C}_{b,c,r}$ (hence $\mathcal{D}_{b,c,r}$) is “absolutely irreducible” for $b,c\in\F_{q^2}$, $c\neq 0$, $r\ge 1$ (with “the case $c=0$ treated separately”). Then “by the Hasse-Weil bound we obtain the existence of at least one point $(\overline{x_0},\overline{x_1})\in\mathbb{F}_q^2$ in $\mathcal{D}_{b,c,r}$,” yielding a zero of $P_{c,b}$ in $\mathbb{F}_{q^2}$. “Therefore, by Theorem~\ref{Th:APN} the function $f_{b,c,r}(x,y)$ is not APN.”

Let $q=2^{m/2}$. In establishing the main theorem, it suffices to show that $P_{c,b}(X)$ has a zero in $\mathbb{F}_{q^2}$ (hence in $\mathbb{F}_{2^m}$) for the relevant parameters, so that $f_{b,c,r}(x,y)$ is not APN.

First, observe that “$P_{c,b}(X)$ has a zero in $\mathbb{F}_{q^2}$ if and only if there exists $x\in \mathbb{F}_{q^2}^*$ such that
\[
\frac{cx^{2^r+1} + bx^{2^r} + 1}{x}
\]
is a $(q+1)$-root of unity,” which is equivalently rewritten as
\[
\frac{cx^{2^r+1} + bx^{2^r} + 1}{x}=\frac{x^q}{c^qx^{(2^r+1)q} + b^qx^{2^rq} + 1}.
\]
Write $x=x_0+\xi x_1$ with $\{1,\xi\}$ an $\mathbb{F}_q$-basis of $\mathbb{F}_{q^2}$ and $x_0,x_1\in\mathbb{F}_q$. Then (since $x\neq 0$) the condition becomes a plane equation in $(x_0,x_1)$.

To prove that “for each $b,c\in\mathbb{F}_{q^2}$ there is at least a solution $(\overline{x_0},\overline{x_1})\in\mathbb{F}_q^2$,” introduce the algebraic curve $\mathcal{D}_{b,c,r}$ over $\mathbb{F}_q$ defined by that equation, and via the change of variables $(X+\xi Y, X+\xi^q Y)\mapsto (X,Y)$ pass to an affinely equivalent plane curve $\mathcal{C}_{b,c,r}$:
\[
\left(cX^{2^r+1} + bX^{2^r} + 1\right)\left(c^qY^{2^r+1} + b^qY^{2^r} + 1\right)+XY=0.
\]
The stated strategy is to prove $\mathcal{C}_{b,c,r}$ (hence $\mathcal{D}_{b,c,r}$) is “absolutely irreducible” for $b,c\in\F_{q^2}$, $c\neq 0$, $r\ge 1$ (with “the case $c=0$ treated separately”). Then “by the Hasse-Weil bound we obtain the existence of at least one point $(\overline{x_0},\overline{x_1})\in\mathbb{F}_q^2$ in $\mathcal{D}_{b,c,r}$,” yielding a zero of $P_{c,b}$ in $\mathbb{F}_{q^2}$. This completes the proof of the main theorem.

\label{Th:APN}To prove the main theorem, we use Theorem 6.2 of \cite[Theorem 6.2]{APN}.
Let $n = 2m$ with $m$ even, and let $r < m/2$ be such that $\gcd(r,m) = 1$. Consider $b,c \in \mathbb{F}_{2^{m}}$.
Then
\[
f_{b,c,r}(x,y) = \big(xy,\, x^{2^r+1} + x^{2^{r+m/2}} y^{2^{m/2}} + bxy^{2^r} + cy^{2^r+1}\big)
\]
is APN if and only if
\[
P_{c,b}(X)=\big(cX^{2^r+1} + bX^{2^r} + 1\big)^{2^{m/2}+1} + X^{2^{m/2}+1}
\]
has no zero in $\mathbb{F}_{2^m}$.

Biconditional or Equivalence
Nonexistence

Question:
Let $n=2m$ with $m$ even, let $r<m/2$ satisfy $\gcd(r,m)=1$, and let $b,c\in\mathbb F_{2^m}$. Identify $\mathbb F_{2^n}$ with $\mathbb F_{2^m}\times\mathbb F_{2^m}$, and define
\[
f_{b,c,r}(x,y)=\big(xy,\,x^{2^r+1}+x^{2^{r+m/2}}y^{2^{m/2}}+bxy^{2^r}+cy^{2^r+1}\big).
\]
A function $F:\mathbb F_{2^n}\to\mathbb F_{2^n}$ is called almost perfect nonlinear (APN) if for every nonzero $a\in\mathbb F_{2^n}$ and every $d\in\mathbb F_{2^n}$, the equation $F(z+a)+F(z)=d$ has at most two solutions $z$. Which statement is equivalent to saying that $f_{b,c,r}$ is APN?

The polynomial
\[
P_{c,b}(X)=\big(cX^{2^r+1}+bX^{2^r}+1\big)^{2^{m/2}+1}+X^{2^{m/2}+1}
\]
has no zero in $\mathbb F_{2^m}$.

Choices:
B.
The polynomial
\[
P_{c,b}(X)=\big(cX^{2^r+1}+bX^{2^r}+1\big)^{2^{m/2}+1}+X^{2^{m/2}+1}
\]
has no zero in $\mathbb F_{2^n}$.
C.
The polynomial
\[
P_{c,b}(X)=\big(cX^{2^r+1}+bX^{2^r}+1\big)^{2^{m/2}+1}+X^{2^{m/2}+1}
\]
has no zero in $\mathbb F_{2^m}^*$.
D.
There exists a $(2^{m/2}+1)$-st root of unity $u\in\mathbb F_{2^m}$ such that the polynomial
\[
cX^{2^r+1}+bX^{2^r}+uX+1
\]
has no zero in $\mathbb F_{2^m}$.
E.
The polynomial
\[
P_{c,b}(X)=\big(cX^{2^r+1}+bX^{2^r}+1\big)^{2^{m/2}+1}+X^{2^{m/2}+1}
\]
has at most one zero in $\mathbb F_{2^m}$.

allowing the obvious possibility of a zero at $X=0$

replace zero-free condition by uniqueness of zero

The stem does not explicitly reveal the correct option. It gives the APN definition and asks for an equivalent condition, but the exact polynomial criterion is not stated in the question itself.

This is essentially a theorem-recall item: it asks which statement is equivalent to the APN property for the specific family, and the correct choice is the theorem statement itself.

Some reasoning is needed because the distractors are close variants involving field size, exclusion of zero, weakened uniqueness, and an auxiliary root-of-unity condition. However, the item mainly rewards recognition of the exact equivalence rather than substantial derivation.

The distractors are mathematically plausible and target realistic failure modes: strengthening the field of root exclusion, weakening to nonzero roots only, altering quantifiers, and replacing zero-freeness with uniqueness.

A technically well-constructed recall/discrimination question with strong distractors, but it is largely tautological because it directly asks for the known equivalent theorem statement rather than eliciting deeper generative reasoning.


\begin{theorem}
		\label{t:SplittingResult}
		Let $v, w, x, y$ be weight matrices satisfying \emph{(S)}.  If $v, y$ satisfy $(\DN)$ and $w, x$ satisfy $(\Omega)$, then
		$$
		\ExtPLS^1(\lambda^1(v, w), \lambda^\infty(x, y)) = 0.
		$$
	\end{theorem}

\begin{theorem}\label{main-intro}
Let $\mu, \tau > 1/2$. Set $\alpha_\mu = (j^{1/\mu})_{j \in \N}$ and  $\alpha_\tau = (j^{1/\tau})_{j \in \N}$. Then,
$$
\mathcal{Z}^{(\mu)}_{(\tau)} \cong \Lambda(\alpha_\mu, \alpha_\tau).
$$
\end{theorem}

\subsection{A Pe\l czy\'nski-Vogt decomposition result for (PLS)-spaces} Given two locally convex spaces $X$ and $Y$, we write $X \cong Y$  if $X$ and $Y$ are isomorphic, and $X \compl Y$ if $X$ is isomorphic to a complemented subspace of $Y$.

In his study of the complemented subspaces of the classical Banach sequence spaces $\ell_p$  and $c_0$, Pe\l czy\'nski showed the following decomposition result \cite[Proposition 4]{Pelczynski} (see also \cite[Theorem 2.2.4]{Albiac-Kalton-TopicsBanach}): \emph{Let $Y$ = $\ell_p$ $(1 \leq p < \infty)$ or $c_0$}.
\begin{equation}
\label{PVstat}
\mbox{\emph{ Let $X$ be a locally convex space with $X \compl Y$ and $Y \compl X$. Then, $X \cong Y$.}}
\end{equation}
The proof of this result is elementary and relies on the identities $\ell_p(\ell_p) \cong \ell_p$ and $c_0(c_0) \cong c_0$.

In the context of sequence space representations,  Vogt later obtained the following variant of Pe\l czy\'nski's result \cite[Proposition 1.2]{V-SeqSpRepTestFuncDist}:  \emph{Let $E$ be a locally convex space and set $Y = E^{\N}, E^{(\N)}$, or $s(E)$. Then, \eqref{PVstat} holds.} The proof of this result is very similar to the one of Pe\l czy\'nski, now using that $Y \varepsilon Y \cong Y$. Applying the splitting theory for Fr\'echet spaces, \cite{firstsplittingVogt} (see also  \cite{V-Ext1Frechet}), Vogt \cite[Satz 1.4]{V-IsomorphSatzPotRaum} showed the much deeper result that \eqref{PVstat} holds if $Y = \Lambda_{\infty}(\alpha)$ is a stable nuclear power series space of infinite type. Moreover, \eqref{PVstat} is also true if $Y = \Lambda_{0}(\alpha)$ is a nuclear power series space of finite type \cite[Satz 1.4]{V-IsomorphSatzPotRaum}, as follows from a theorem of Mityagin, which states that every locally convex space $X$ with $X \compl \Lambda_0(\alpha)$ is isomorphic to a power series of finite type \cite[Theorem 1.1]{MH} (see also \cite[Corollary 29.20]{M-V-IntroFunctAnal}), together with basic properties of the diametral dimension.

 Results of this form are often called \emph{Pe\l czy\'nski-Vogt decomposition results}. They were used by  Valdivia and Vogt in the 1980s to establish sequence space representations of most of the function and distribution spaces arising in Schwartz’s theory of distributions \cite{V-TopLCS, V-SeqSpRepTestFuncDist}; see \cite{B-D-N-SeqSpRepWilson,D-SeqSpRepEntFunc, D-N-SeqSpRepTMIB, L-BasesGerms, L-DiamDimWeighSpGerms}  for more recent, related works.

 $(\PLS)$-spaces are locally convex spaces that can be written as the countable projective limit of $(\LS)$-spaces. Important examples include the space of distributions, the space of real analytic functions, and the space $\mathcal{O}_M$ of slowly increasing smooth functions.
The class of $(\PLS)$-spaces plays an important role in the modern theory of locally convex spaces, particularly in connection with the derived projective limit functor \cite{W-DerivFuncFunctAnal}. We refer to the survey article \cite{Domanski-PLS} of Doma\'nski for more information on  $(\PLS)$-spaces.

 In the first part of this article, we prove a Pe\l czy\'nski-Vogt decomposition result for $(\PLS)$-type power series spaces of infinite type. Let  $\Lambda_{\infty}(\alpha)$ and  $\Lambda_{\infty}(\beta)$ be nuclear and assume that $\Lambda_{\infty}(\alpha)$ is stable. Consider the $(\PLS)$-space
 \begin{align*}
 	\Lambda_\infty(\alpha, \beta)
 	&= \Lambda_{\infty}(\alpha) \varepsilon \Lambda'_{\infty}(\beta) \\
	&= \{ c = (c_{i,j})_{(i,j) \in \N^2} \mid \forall n \in \N \, \exists N \in \N :  \| c\|_{n,N} = \sup_{(i,j) \in \N^2} |c_{i,j}|e^{n\alpha_i -N \beta_j} < \infty \}.
 \end{align*}
 In Theorem \ref{t:PelczynskiPLS}, we show that \eqref{PVstat} holds for $Y =  \Lambda_\infty(\alpha, \beta)$. To this end, we extend the technique of Vogt's proof \cite[Satz 1.4]{V-IsomorphSatzPotRaum} of the decomposition result for power series of infinite type to the setting of $(\PLS)$-spaces.

 One of the main ingredients in our proof is a new splitting result for  $(\PLS)$-type sequence spaces, namely, that every topologically exact sequence of $(\PLS)$-spaces
 $$
 \SES{ \Lambda_\infty(\alpha, \beta)}{X}{ \Lambda_\infty(\gamma, \delta)}{}{}
$$
with $\Lambda_{\infty}(\gamma)$ and $\Lambda_{\infty}(\delta)$ nuclear, splits (see Theorem \ref{t:SplittingResult} for a more general result). In his PhD thesis \cite[Theorem 5.14]{K-SplitPowerSeriesSpPLS}, Kunkle showed a similar result, but for a different type of sequence spaces. We closely follow Kunkle’s approach to show the above result. For the reader’s convenience, and since the results from \cite{K-SplitPowerSeriesSpPLS} have never been published, we include here a full proof. See \cite{Domanski-splitting} and the references therein for other works related to the splitting theory for  $(\PLS)$-spaces.

We expect that an analogous decomposition should also hold in the finite-type case, namely, that \eqref{PVstat} is true for $ Y = \Lambda_{0}(\alpha) \varepsilon \Lambda'_{0}(\beta)$, with $\Lambda_{0}(\alpha)$ and $\Lambda_{0}(\beta)$ nuclear, however, we have not yet been able been able to prove this.

\subsection{Sequence space representations for multiplier spaces of Gelfand-Shilov spaces of Beurling type} In his classical book \cite[Chapitre VII, $\S $5]{Schwartzbook}, Schwartz introduced the space
$$
 \mathcal{O}_M = \{ f \in C^\infty(\R) \mid \forall n \in \N \, \exists N \in \N :  \| f\|_{n,N} = \max_{p \leq n}  |f^{(p)}(x)|(1+|x|)^{-N}< \infty \}.
$$
and showed that it was equal to the multiplier space of the space $\mathcal{S}$ of rapidly decreasing smooth functions, that is,
$$
 \mathcal{O}_M = \{ f \in \mathcal{S}' \mid \varphi \cdot f \in \mathcal{S} \mbox{ for all $\varphi \in \mathcal{S}$}\}.
$$
Moreover, the natural $(\PLS)$-space topology on  $\mathcal{O}_M$ coincides with the operator topology induced by the embedding
	\[ \mathcal{O}_{M} \rightarrow L_{b}(\mathcal{S}, \mathcal{S}), \quad f \mapsto ( \varphi \mapsto \varphi \cdot f) . \]

In his doctoral thesis \cite[Chapitre II, Theor\`eme 16, p.\ 131]{G-ProdTensTopEspNucl}, Grothendieck proved that the space  $\mathcal{O}_M$ is ultrabornological. He achieved this by showing that $\mathcal{O}_M \compl s(s')$ and verifying directly that  $s(s')$ is ultrabornological. Later, Valdivia \cite{V-RepOM} showed that in fact $\mathcal{O}_{M} \cong s(s')$. To this end, he showed, in addition, that  $s(s') \compl \mathcal{O}_M$  and used the Pe\l czy\'nski-Vogt decomposition result for $s(s')$. We refer to \cite{B-D-N-SeqSpRepWilson} for a constructive proof of the isomorphism $\mathcal{O}_{M} \cong s(s')$.

In  \cite{D-N-WeighPLBUltradiffFuncMultSp, D-N-BarrelWeighPLBUltradiffFunc}, we studied the structural and linear topological properties of the multiplier space of Gelfand-Shilov spaces; see \cite{D-P-V-MultConvTempUltraDist, S-InclThMoyalMultAlgGenGSSp} for related works. In the second part of this article, we apply the Pe\l czy\'nski-Vogt decomposition result for the spaces $\Lambda_{\infty}(\alpha,\beta)$  to obtain sequence space representations for multiplier spaces of Gelfand-Shilov spaces of Beurling type. We now state a sample of this result.

Fix $\mu,\tau >0$. For $h >0$ and $\lambda \in \R$ we define the Banach space
$$
\Sigma^{\mu,h}_{\tau,\lambda} = \{ \varphi \in C^\infty(\R) \mid  \| f\| =  \sup_{p \in \N} \sup_{x \in \R} \frac{|\varphi^{(p)}(x)|e^{-\frac{1}{\lambda}|x|^{1/\tau}}}{h^{p}p!^\mu} < \infty \}.
 $$
Consider the Fr\'echet space
$$
\Sigma^{\mu}_{\tau}  = \varprojlim_{h \to 0^+} \Sigma^{\mu,h}_{\tau,h}  .
$$
The spaces $\Sigma^\tau_\mu$ are the projective analogues of the classical Gelfand-Shilov spaces $\mathcal{S}^\mu_\tau$ \cite{G-S-GenFunc2} and have been considered in e.g.\ \cite{C-G-P-R-AnistropicShubinOpExpGSSp, D-SeqSpRepEntFunc, D-N-SeqSpRepBBSp, Petersson}. We mention that $\Sigma^\tau_\mu$ is non-trivial if and only if $\mu + \tau >1$ (cf.\ \cite[Section 8]{G-S-GenFunc2}).

Define the $(\PLS)$-space
$$
\mathcal{Z}^{(\mu)}_{(\tau)} = \varprojlim_{h \to 0^+}  \varinjlim_{\lambda \to 0^-}  \Sigma^{\mu,h}_{\tau,\lambda}.
$$
In \cite[Theorem 5.7(i)]{D-N-WeighPLBUltradiffFuncMultSp} we showed that $\mathcal{Z}^{(\mu)}_{(\tau)}$ is the multiplier space  of $\Sigma^{\mu}_{\tau}$, i.e.,
$$
\mathcal{Z}^{(\mu)}_{(\tau)} = \{ f \in (\Sigma^\tau_\mu)' \mid \varphi \cdot f \in\Sigma^{\mu}_{\tau}  \mbox{ for all $\varphi \in\Sigma^{\mu}_{\tau} $}\},
$$
and that the natural $(\PLS)$-space topology on  $\mathcal{Z}^{(\mu)}_{(\tau)}$ coincides with the topology induced by the embedding
	\[ \mathcal{Z}^{(\mu)}_{(\tau)} \rightarrow L_{b}(\Sigma^{\mu}_{\tau} ,\Sigma^{\mu}_{\tau} ), \quad f \mapsto ( \varphi \mapsto \varphi \cdot f) . \]
Moreover, $\mathcal{Z}^{(\mu)}_{(\tau)}$ is ultrabornological  \cite[Theorem 5.7(ii)]{D-N-WeighPLBUltradiffFuncMultSp}.

Here, we obtain a  sequence space representation for $\mathcal{Z}^{(\mu)}_{(\tau)}$:

In his study of the complemented subspaces of the classical Banach sequence spaces $\ell_p$ and $c_0$, Pe\l czy\'nski showed the following decomposition result \cite[Proposition 4]{Pelczynski} (see also \cite[Theorem 2.2.4]{Albiac-Kalton-TopicsBanach}): \emph{Let $Y$ = $\ell_p$ $(1 \leq p < \infty)$ or $c_0$}.
\begin{equation}
\label{PVstat}
\mbox{\emph{ Let $X$ be a locally convex space with $X \compl Y$ and $Y \compl X$. Then, $X \cong Y$.}}
\end{equation}
The proof of this result is elementary and relies on the identities $\ell_p(\ell_p) \cong \ell_p$ and $c_0(c_0) \cong c_0$.

In the context of sequence space representations, Vogt later obtained the following variant of Pe\l czy\'nski's result \cite[Proposition 1.2]{V-SeqSpRepTestFuncDist}: \emph{Let $E$ be a locally convex space and set $Y = E^{\N}, E^{(\N)}$, or $s(E)$. Then, \eqref{PVstat} holds.} The proof of this result is very similar to the one of Pe\l czy\'nski, now using that $Y \varepsilon Y \cong Y$. Applying the splitting theory for Fr\'echet spaces, \cite{firstsplittingVogt} (see also \cite{V-Ext1Frechet}), Vogt \cite[Satz 1.4]{V-IsomorphSatzPotRaum} showed the much deeper result that \eqref{PVstat} holds if $Y = \Lambda_{\infty}(\alpha)$ is a stable nuclear power series space of infinite type. Moreover, \eqref{PVstat} is also true if $Y = \Lambda_{0}(\alpha)$ is a nuclear power series space of finite type \cite[Satz 1.4]{V-IsomorphSatzPotRaum}, as follows from a theorem of Mityagin, which states that every locally convex space $X$ with $X \compl \Lambda_0(\alpha)$ is isomorphic to a power series of finite type \cite[Theorem 1.1]{MH} (see also \cite[Corollary 29.20]{M-V-IntroFunctAnal}), together with basic properties of the diametral dimension.

In the first part of this article, we prove a Pe\l czy\'nski-Vogt decomposition result for $(\PLS)$-type power series spaces of infinite type. Let $\Lambda_{\infty}(\alpha)$ and $\Lambda_{\infty}(\beta)$ be nuclear and assume that $\Lambda_{\infty}(\alpha)$ is stable. Consider the $(\PLS)$-space
 \begin{align*}
 \Lambda_\infty(\alpha, \beta)
 &= \Lambda_{\infty}(\alpha) \varepsilon \Lambda'_{\infty}(\beta) \\
 &= \{ c = (c_{i,j})_{(i,j) \in \N^2} \mid \forall n \in \N \, \exists N \in \N : \| c\|_{n,N} = \sup_{(i,j) \in \N^2} |c_{i,j}|e^{n\alpha_i -N \beta_j} < \infty \}.
 \end{align*}
 In Theorem \ref{t:PelczynskiPLS}, we show that \eqref{PVstat} holds for $Y = \Lambda_\infty(\alpha, \beta)$. To this end, we extend the technique of Vogt's proof \cite[Satz 1.4]{V-IsomorphSatzPotRaum} of the decomposition result for power series of infinite type to the setting of $(\PLS)$-spaces.

Fix $\mu,\tau >0$. For $h >0$ and $\lambda \in \R$ we define the Banach space
$$
\Sigma^{\mu,h}_{\tau,\lambda} = \{ \varphi \in C^\infty(\R) \mid \| f\| = \sup_{p \in \N} \sup_{x \in \R} \frac{|\varphi^{(p)}(x)|e^{-\frac{1}{\lambda}|x|^{1/\tau}}}{h^{p}p!^\mu} < \infty \}.
 $$
Consider the Fr\'echet space
$$
\Sigma^{\mu}_{\tau} = \varprojlim_{h \to 0^+} \Sigma^{\mu,h}_{\tau,h} .
$$
The spaces $\Sigma^\tau_\mu$ are the projective analogues of the classical Gelfand-Shilov spaces $\mathcal{S}^\mu_\tau$ \cite{G-S-GenFunc2} and have been considered in e.g.\ \cite{C-G-P-R-AnistropicShubinOpExpGSSp, D-SeqSpRepEntFunc, D-N-SeqSpRepBBSp, Petersson}. We mention that $\Sigma^\tau_\mu$ is non-trivial if and only if $\mu + \tau >1$ (cf.\ \cite[Section 8]{G-S-GenFunc2}).

Define the $(\PLS)$-space
$$
\mathcal{Z}^{(\mu)}_{(\tau)} = \varprojlim_{h \to 0^+} \varinjlim_{\lambda \to 0^-} \Sigma^{\mu,h}_{\tau,\lambda}.
$$
In \cite[Theorem 5.7(i)]{D-N-WeighPLBUltradiffFuncMultSp} we showed that $\mathcal{Z}^{(\mu)}_{(\tau)}$ is the multiplier space of $\Sigma^{\mu}_{\tau}$, i.e.,
$$
\mathcal{Z}^{(\mu)}_{(\tau)} = \{ f \in (\Sigma^\tau_\mu)' \mid \varphi \cdot f \in\Sigma^{\mu}_{\tau} \mbox{ for all $\varphi \in\Sigma^{\mu}_{\tau} $}\},
$$
and that the natural $(\PLS)$-space topology on $\mathcal{Z}^{(\mu)}_{(\tau)}$ coincides with the topology induced by the embedding
 \[ \mathcal{Z}^{(\mu)}_{(\tau)} \rightarrow L_{b}(\Sigma^{\mu}_{\tau} ,\Sigma^{\mu}_{\tau} ), \quad f \mapsto ( \varphi \mapsto \varphi \cdot f) . \]
Moreover, $\mathcal{Z}^{(\mu)}_{(\tau)}$ is ultrabornological \cite[Theorem 5.7(ii)]{D-N-WeighPLBUltradiffFuncMultSp}.

Here, we obtain a sequence space representation for $\mathcal{Z}^{(\mu)}_{(\tau)}$:

\begin{theorem}
		\label{t:PelczynskiPLS}
		Let $\alpha, \beta$ be exponent sequences satisfying \emph{(N)} with $\alpha$ stable.
		Let $X$ be a lcHs.
		If $X \compl \Lambda_{\infty}(\alpha, \beta)$ and $\Lambda_{\infty}(\alpha, \beta) \compl X$, then $X \cong \Lambda_{\infty}(\alpha, \beta)$.
	\end{theorem}

Here, we obtain a sequence space representation for $\mathcal{Z}^{(\mu)}_{(\tau)}$:

\begin{lemma}
 \label{l:CharBoundedSp}
 Let $\mathcal{A} = (a_n)_{n \in \N}$ be a weight matrix system on an index set $\mathcal{I}$, let $b$ be a dual weight matrix on an index set $\mathcal{J}$, and let $\alpha \in \mathfrak{C}^\infty$.
 \begin{enumerate}
 \item Let $T_{i, j} \in \C$, $(i, j) \in \mathcal{I} \times \mathcal{J}$, be numbers satisfying
 \begin{equation}
\label{coochar2} |T_{i, j}| \leq \inf_{M \in \N} \alpha_3(M) \frac{a_{\alpha_1, M, i}}{b_{\alpha_2(M), j}}, \qquad \forall (i, j) \in \mathcal{I} \times \mathcal{J} .
\end{equation}
 Then,
 $$
T: \lambda^1(\mathcal{A}) \to k^\infty(b), \quad c \mapsto \left(\sum_{i \in \mathcal{I}} c_i T_{i, j}\right)_{j \in \mathcal{J}},
 $$
 is a well-defined mapping that belongs to $A_\alpha(\mathcal{A},b)$.
 \item If $T \in A_\alpha(\mathcal{A},b)$, there exist unique numbers $T_{i, j} \in \C$, $(i, j) \in \mathcal{I} \times \mathcal{J}$, satisfying \eqref{coochar2} such that
 $$
 T(c) = \left(\sum_{i \in \mathcal{I}} T_{i, j}c_i\right)_{j \in \mathcal{J}} , \qquad \forall c \in \lambda^1(\mathcal{A}) .
$$
 \end{enumerate}
 \end{lemma}
 \begin{proof}
(1) For all $M \in \N$ and $c \in \ell^1( a_{\alpha_1,M})$, it holds that
$$
\sup_{j \in \mathcal{J}} b_{\alpha_2(M), j} \sum_{i \in \mathcal{I}} |T_{i, j}||c_i| \leq \alpha_3(M)\|c\|_{1, a_{\alpha_1,M}}.
$$
This implies the result. \\
(2) Let $c_{00}(\mathcal{I})$ denote the space consisting of all $c \in \C^\mathcal{I}$ with only finitely many non-zero entries. For $i \in \mathcal{I}$ we set $e_i = (\delta_{i,i'})_{i' \in \mathcal{I}}$, where $\delta_{i,i'}$ is the Kronecker delta symbol.
Put $T(e_i) = (T_{i,j})_{j \in \mathcal{J}}$. Then,
\begin{equation}
\label{reprfins}
T(c) = \left(\sum_{i \in \mathcal{I}} T_{i, j}c_i\right)_{j \in \mathcal{J}}, \qquad \forall c \in c_{00}(I).
\end{equation}
Since $T \in A_\alpha(\mathcal{A},b)$, we find that for all $M \in \N$ and $(i, j) \in \mathcal{I} \times \mathcal{J}$
$$
|T_{i,j}| \leq \frac{ \|T(e_i)\|_{\infty,b_{\alpha_2(M)}}}{b_{\alpha_2(M), j}} \leq \alpha_3(M) \frac{\|e_i\|_{1, a_{\alpha_1,M}}}{b_{\alpha_2(M), j}} = \alpha_3(M) \frac{a_{\alpha_1,M}}{b_{\alpha_2(M), j}}.
$$
The result now follows from \eqref{reprfins}, part (1), and the fact that $c_{00}(\mathcal{I})$ is dense in $\lambda^1(\mathcal{A})$.
\end{proof}

Let $\omega$ be a BMT-weight function and let $\eta$ be a weight function. For $h >0$ and $\lambda \in \R$ we write $\mathcal{S}_{\eta,\lambda}^{\omega,h}$ for the Banach space consisting of all $\varphi \in C^\infty(\R)$ such that
$$
\sup_{p \in \N} \sup_{x \in \R} |\varphi^{(p)}(x)| \exp \left( \frac{1}{\lambda} \eta(x) - \frac{1}{h} \phi^{*}(h p) ) \right) < \infty.
$$
We define the \emph{Gelfand-Shilov spaces (of Beurling and Roumieu type)} as
\[
\mathcal{S}_{(\eta)}^{(\omega)}=\varprojlim_{h \to0^+} \mathcal{S}_{\eta,h}^{\omega,h} \qquad \mbox{and}
\qquad \mathcal{S}_{\{\eta\}}^{\{\omega\}}=\varinjlim_{h\to +\infty} \mathcal{S}_{\eta,h}^{\omega,h}.
\]
We use $\mathcal{S}_{[\eta]}^{[\omega]}$ as a common notation for $\mathcal{S}_{(\eta)}^{(\omega)}$ and $\mathcal{S}_{\{\eta\}}^{\{\omega\}}$; a similar convention is used for other spaces as well.
 \begin{example}\label{CGS}
Let $\mu, \tau >0$. For $h>0$ and $\lambda \in \R$ we define $\mathcal{S}_{\tau,\lambda}^{\mu,h}$ as the Banach space consisting of all $\varphi \in C^\infty(\R)$ such that
$$
 \sup_{p \in \N} \sup_{x \in \R} \frac{|\varphi^{(p)}(x)|e^{\frac{1}{\lambda}|x|^{1/\tau}}}{h^{p}p!^\mu} < \infty.
$$
We set
\[
\Sigma^{\mu}_{\tau}=\varprojlim_{h \to0^+} \mathcal{S}_{\tau,h}^{\mu,h} \qquad \mbox{and}
\qquad \mathcal{S}^{\mu}_{\tau}=\varinjlim_{h\to +\infty} \mathcal{S}_{\tau,h}^{\mu,h}.
\]
We already considered the space $\Sigma^{\mu}_{\tau}$ in the introduction. The spaces $\mathcal{S}^{\mu}_\tau$ were introduced by Gelfand and Shilov \cite{G-S-GenFunc2}. Using the same notation as in Example \ref{GevreyW}, it holds that $\Sigma^\mu_\tau = \mathcal{S}^{(\omega_\mu)}_{(\omega_\tau)}$ and $\mathcal{S}^\mu_\tau = \mathcal{S}^{\{\omega_\mu\}}_{\{\omega_\tau\}}$.
\end{example}

\begin{theorem}
 \label{t:SeqSpRep}
 Let $\omega$ be a BMT-weight function and let $\eta$ be a weight function such that $\mathcal{S}^{(\omega)}_{(\eta)}$ is Gabor accessible.
 Then,
 \begin{equation}
 \label{SSR}
 \mathcal{Z}^{(\omega)}_{(\eta)} \cong \Lambda_\infty(\alpha(\omega), \alpha(\eta)).
 \end{equation}
 \end{theorem}
 \begin{proof}
 As $\mathcal{S}_{(\eta)}^{(\omega)}$ is Gabor accessible, there are $\psi, \gamma \in \mathcal{S}_{(\eta)}^{(\omega)}$ and $a,b >0$ such that $(\psi,\gamma)$ is a pair of dual windows on $a \Z \times b \Z$.
Condition $(\alpha)$ implies that
$$
\Lambda_\infty( \alpha(\omega), \alpha(\eta)) \cong \lambda^\infty(\mathcal{A}^{a,b}_{(\omega), (\eta)}) \cong \lambda^\infty(\mathcal{A}^{1/b,1/a}_{(\omega), (\eta)}).
$$
Hence, by Theorem \ref{t:PelczynskiPLS}, it suffices to verify that $\mathcal{Z}_{(\eta)}^{(\omega)} \compl \lambda^\infty(\mathcal{A}^{a,b}_{(\omega), (\eta)})$ and $\lambda^\infty(\mathcal{A}^{1/b,1/a}_{(\omega), (\eta)}) \compl \mathcal{Z}_{(\eta)}^{(\omega)}$. Proposition \ref{p:GaborCont} shows that the mappings
$$
 C^{a,b}_\psi: \mathcal{Z}_{(\eta)}^{(\omega)} \rightarrow \lambda^\infty(\mathcal{A}^{a,b}_{(\omega), (\eta)}) \qquad \mbox{and} \qquad
D^{a,b}_\gamma: \lambda^\infty(\mathcal{A}^{a,b}_{(\omega), (\eta)}) \to \mathcal{Z}_{(\eta)}^{(\omega)}
$$
are continuous, and in view of \eqref{comp11}, it holds that $D^{a,b}_\gamma \circ C^{a,b}_\psi = \operatorname{id}_{ \mathcal{Z}_{(\eta)}^{(\omega)}}$. This shows that $\mathcal{Z}_{(\eta)}^{(\omega)} \compl \lambda^\infty(\mathcal{A}^{a,b}_{(\omega), (\eta)})$ . Another application of Proposition \ref{p:GaborCont} gives that the mappings
$$
 C^{\frac{1}{b}, \frac{1}{a}}_\psi: \mathcal{Z}_{(\eta)}^{(\omega)} \rightarrow \lambda^\infty(\mathcal{A}^{1/b,1/a}_{(\omega), (\eta)}) \qquad \mbox{and} \qquad
D^{\frac{1}{b}, \frac{1}{a}}_\gamma: \lambda^\infty(\mathcal{A}^{1/b,1/a}_{(\omega), (\eta)}) \to \mathcal{Z}_{(\eta)}^{(\omega)}
$$
are continuous, and by \eqref{WR}, it holds that $(ab)^{-1}C^{\frac{1}{b}, \frac{1}{a}}_\psi \circ D^{\frac{1}{b},\frac{1}{a}}_\gamma = \operatorname{id}_{ \lambda^\infty(\mathcal{A}^{1/b,1/a}_{(\omega), (\eta)})}$.
Therefore $\lambda^\infty(\mathcal{A}^{1/b,1/a}_{(\omega), (\eta)}) \compl \mathcal{Z}_{(\eta)}^{(\omega)}$.
\end{proof}

We end this article with several remarks.
\begin{remark} Let $\omega$ be a BMT-weight function and let $\eta$ be a weight function
\begin{enumerate}
\item We believe that the sequence space representation \eqref{SSR} holds under the assumption that $\mathcal{S}^{(\omega)}_{(\eta)} \neq \{0\}$, but are unable to show this. This would follow from Theorem \ref{t:SeqSpRep} if it holds that every non-trivial $\mathcal{S}^{(\omega)}_{(\eta)}$ is Gabor accessible (see Remark \ref{GAR}).
\item Similarly to the sequence space representation \eqref{SSR}, we believe that in the Roumieu case
\begin{equation}
\label{SSRR}
\mathcal{Z}^{\{\omega\}}_{\{\eta\}} \cong \Lambda_0(\alpha(\eta), \alpha(\omega)).
\end{equation}
In the case that $\mathcal{S}^{\{\omega\}}_{\{\eta\}}$ is Gabor accessible, by using the same argument as in Theorem \ref{t:SeqSpRep}, this would be implied by a positive solution to the question posed in Remark \ref{openqabst}.
\item Let $\psi \in L^2(\R)$. We define
 \begin{align*}
 \psi_{0,l} &= T_{l} \psi, \qquad l \in \Z, \\
 \psi_{k,l} &= \frac{1}{\sqrt{2}} T_{\frac{l}{2}}(M_k+ (-1)^{k+l} M_{-k})\psi, \qquad (k,l)\in\N_{>0} \times \Z ,
 \end{align*}
and set $\mathcal{W}(\psi) = \{ \psi_{k,n} \mid (k,n)\in\N\times\Z \}$. We call $\mathcal{W}(\psi)$ a \emph{Wilson basis} if it is an orthonormal basis in $L^2(\R)$. See \cite[Section 8.5]{G-FoundTFAnalysis} for more information.

The proof of Theorem \ref{main-intro} combines the Pe\l czy\'nski-Vogt decomposition result for the $(\PLS)$-type power series spaces $\Lambda_{\infty}(\alpha,\beta)$ of infinite type with properties of Gabor frames---a fundamental tool in time-frequency analysis \cite{G-FoundTFAnalysis}. In \cite{D-N-SeqSpRepBBSp}, we used the same technique to obtain sequence space representations for Gelfand-Shilov spaces, in particular $\Sigma^{\mu}_{\tau}$. The restriction $\mu, \tau > 1/2$ is imposed because we rely on a result on dual windows of Gabor frames from \cite{Janssen}.
Finally, we note that in  \cite{D-N-WeighPLBUltradiffFuncMultSp, D-N-BarrelWeighPLBUltradiffFunc} we also made use of time-frequency analysis, namely, the short-time Fourier transform \cite{G-FoundTFAnalysis}, which can be seen as a continuous version of Gabor frames.

\subsection{Outline of the paper} Section \ref{sect-prelim} collects notation and preliminary notions. In Section \ref{sect-seqsp}, we define and discuss various sequence spaces. We prove a splitting result for $(\PLS)$-type sequence spaces in Section \ref{sect-splitting}. This section is based on \cite{K-SplitPowerSeriesSpPLS}. Section \ref{sect-PV} is devoted to the proof of the  Pe\l czy\'nski-Vogt decomposition result for $(\PLS)$-type power series spaces. In Section \ref{sect-appl}, we apply this result to obtain sequence space representations for multiplier spaces of Gelfand-Shilov spaces of Beurling type. In particular, we show Theorem \ref{main-intro} there.

The post-theorem introduction says that the proof of Theorem \ref{main-intro} \emph{combines} (i) the Pe\l czy\'nski--Vogt decomposition result for the $(\PLS)$-type power series spaces $\Lambda_{\infty}(\alpha,\beta)$ of infinite type with (ii) properties of Gabor frames. It notes the restriction $\mu,\tau>1/2$ is imposed because the argument \emph{relies on} a result on dual windows of Gabor frames from \cite{Janssen}. The outline further indicates the structure: one proves a splitting result for $(\PLS)$-type sequence spaces (Section \ref{sect-splitting}), then proves the Pe\l czy\'nski--Vogt decomposition for $(\PLS)$-type power series spaces (Section \ref{sect-PV}), and finally \emph{applies this result} to obtain sequence space representations for multiplier spaces of Gelfand--Shilov spaces of Beurling type, where 'in particular' Theorem \ref{main-intro} is shown (Section \ref{sect-appl}).

The post-theorem introduction says that, to prove the main theorem, the proof \emph{combines} (i) the Pe\l czy\'nski--Vogt decomposition result for the $(\PLS)$-type power series spaces $\Lambda_{\infty}(\alpha,\beta)$ of infinite type with (ii) properties of Gabor frames. It notes the restriction $\mu,\tau>1/2$ is imposed because the argument \emph{relies on} a result on dual windows of Gabor frames from Janssen, \cite{Janssen}. The outline further indicates the structure: one proves a splitting result for $(\PLS)$-type sequence spaces (Section \ref{sect-splitting}), then proves the Pe\l czy\'nski--Vogt decomposition for $(\PLS)$-type power series spaces (Section \ref{sect-PV}), and finally \emph{applies this result} to obtain sequence space representations for multiplier spaces of Gelfand--Shilov spaces of Beurling type, where, in particular, the main theorem is shown (Section \ref{sect-appl}).

Question:
Let $\mu,\tau>1/2$, and define
\[
\Sigma^{\mu,h}_{\tau,\lambda}=\left\{\varphi\in C^{\infty}(\mathbb R)\;\middle|\; \sup_{p\in\mathbb N}\sup_{x\in\mathbb R}\frac{|\varphi^{(p)}(x)|e^{-\frac1\lambda |x|^{1/\tau}}}{h^p p!^{\mu}}<\infty\right\},
\qquad
\mathcal Z^{(\mu)}_{(\tau)}:=\varprojlim_{h\to0^+}\varinjlim_{\lambda\to0^-}\Sigma^{\mu,h}_{\tau,\lambda}.
\]
If $\alpha_{\mu}=(j^{1/\mu})_{j\in\mathbb N}$ and $\alpha_{\tau}=(j^{1/\tau})_{j\in\mathbb N}$, which sequence space is $\mathcal Z^{(\mu)}_{(\tau)}$ isomorphic to?

$\Lambda(\alpha_{\mu},\alpha_{\tau})$, i.e. the Köthe-type space of double sequences $c=(c_{i,j})_{(i,j)\in\mathbb N^2}$ such that for every $n\in\mathbb N$ there exists $N\in\mathbb N$ with \[\sup_{(i,j)\in\mathbb N^2}|c_{i,j}|\,e^{n i^{1/\mu}-N j^{1/\tau}}<\infty.\]

Choices:
B. $\Lambda(\alpha_{\tau},\alpha_{\mu})$, i.e. the Köthe-type space of double sequences $c=(c_{i,j})_{(i,j)\in\mathbb N^2}$ such that for every $n\in\mathbb N$ there exists $N\in\mathbb N$ with \[\sup_{(i,j)\in\mathbb N^2}|c_{i,j}|\,e^{n i^{1/\tau}-N j^{1/\mu}}<\infty.\]
C. A complemented subspace of $\Lambda(\alpha_{\mu},\alpha_{\tau})$; in particular, $\mathcal Z^{(\mu)}_{(\tau)} \compl \Lambda(\alpha_{\mu},\alpha_{\tau})$ and $\Lambda(\alpha_{\mu},\alpha_{\tau}) \compl \mathcal Z^{(\mu)}_{(\tau)}$.
D. $\Lambda(\alpha_{\mu},\alpha_{\tau})$, i.e. the Köthe-type space of double sequences $c=(c_{i,j})_{(i,j)\in\mathbb N^2}$ such that there exists $N\in\mathbb N$ for every $n\in\mathbb N$ with \[\sup_{(i,j)\in\mathbb N^2}|c_{i,j}|\,e^{n i^{1/\mu}-N j^{1/\tau}}<\infty.\]
E. $\Lambda(\alpha_{\mu},\alpha_{\tau})$ for all $\mu,\tau>0$, i.e. the Köthe-type space of double sequences $c=(c_{i,j})_{(i,j)\in\mathbb N^2}$ such that for every $n\in\mathbb N$ there exists $N\in\mathbb N$ with \[\sup_{(i,j)\in\mathbb N^2}|c_{i,j}|\,e^{n i^{1/\mu}-N j^{1/\tau}}<\infty.\]

assignment of $\alpha_\mu$ and $\alpha_\tau$ to the two sequence directions

dropped the conclusion of isomorphism and kept only mutual complemented-embedding consequence

quantifier order in the defining growth condition ($\forall n\,\exists N$ replaced by $\exists N\,\forall n$)

range restriction $\mu,\tau>1/2$ required by the Gabor-dual-window input

The stem does not reveal the correct option explicitly. It defines the function space and auxiliary sequences, then asks for the corresponding sequence-space identification without giving away the exact target.

This is essentially a direct theorem-recall item: it asks which sequence space the given space is isomorphic to, which closely matches a standard statement of the result rather than probing a derived consequence.

Some reasoning is needed to distinguish the precise isomorphism from nearby variants (swapped parameters, altered quantifiers, weaker embedding statement), but the main task is still recognition/recall of the theorem rather than substantial generative mathematical reasoning.

The distractors are strong and mathematically meaningful: one swaps the roles of the parameters, one weakens isomorphism to complemented embeddings, one alters quantifier order, and one overgeneralizes the parameter range. These reflect realistic failure modes.

A mathematically well-constructed MCQ with strong distractors and no answer leakage, but it is mostly a direct theorem-identification question rather than a non-tautological or strongly generative reasoning task.