[\cite{HooshKoh2025}]
A nontrivial finite group $G$ satisfies the strong CFS property if and only if $G$ is either
\begin{itemize}
  \item a cyclic group of prime order, or
  \item one of the four groups $C_4$, $C_2^2$, $C_2^3$, or $C_3^2$.
\end{itemize}

\label{section:Introduction}
Let $G$ be a finite group.  A subset $A \subseteq G$ is called a \textit{Lagrange subset} if $\lvert A\rvert$ divides $\lvert G\rvert$.  If there exists a subset $B \subseteq G$ such that every element $x\in G$ can be written \textit{uniquely} in the form
\[
x = ab,
\quad a\in A,\; b\in B,
\]
then $A$ is said to be a (left) \textit{factor} of $G$.  In this situation one immediately has
\[
\lvert G\rvert = \lvert A\rvert \,\lvert B\rvert,
\]
and both $A$ and $B$ are Lagrange subsets of $G$.

The problem of factoring a finite group $G$ into two (or more) Lagrange subsets goes back to Haj\'os’s
combinatorial proof of Minkowski’s conjecture on lattice tilings by unit cubes \cite{Hajos1941}.
In particular, Haj\'os\ showed that any lattice tiling of $\mathbb{R}^n$ by unit $n$-cubes must contain two
cubes sharing a full $(n-1)$-face, and he recast that argument in group‑theoretic terms to characterize exactly
which cyclic groups admit factorizations by arithmetic progressions \cite{Hajos1941}.
In the broader abelian setting, subsequent work of R\'edei \cite{Redei1952},
Sands \cite{Sands1974}, Szab\'o \cite{Szabo2004}, and others has connected these set‑factorizations with tiling problems, zero‑sum sequences, and
related combinatorial phenomena.  A good survey of these developments can be found in
\cite{Szabo-Sands2009}.

Bernstein \cite{Bernstein1968} extended Haj\'os’s theorem to the non‑abelian setting, showing that many of the
combinatorial tiling arguments carry over to arbitrary finite groups.  A related thread concerns the
existence of \textit{minimal logarithmic signatures} (MLS): conjecturally, every finite group $G$ admits
a factorization
\[
G = A_1 A_2 \cdots A_k
\]
where each $\lvert A_i\rvert$ is prime or equals $4$.
It is known that every finite solvable group has an
MLS, and that if a normal subgroup $K\lhd G$ and the quotient $G/K$ both admit MLS then so does $G$
\cite{Rahimipour2018}.  Consequently, any counterexample to the MLS conjecture must be a non-abelian
simple group, reducing the problem to the finite simple groups.

Hooshmand \cite[Problem 19.35]{Kourovka2018}
(see also \cite{Hooshmand2014,Banakh2018multifactorizable, Hooshmand2021})
asked whether for every factorization $\lvert G\rvert = m_1\cdots m_k$, there exist subsets
$A_i\subseteq G$ with $\lvert A_i\rvert=m_i$ and $G=A_1\cdots A_k$.
The case $k=2$ has been studied in \cite{Bildanov2020, Hooshmand2021}, where it is shown that a finite
group is $2$‑factorizable if and only if every finite simple group is $2$‑factorizable,
and all simple groups of order up to $10~000$ are $2$-factorizable \cite{Bildanov2020}.

For $k=3$, Bergman \cite{Bergman2020} proved that the alternating group $A_4$ is not $3$‑factorizable.
Brunault \cite{Banakh2018Factorizable} used computer methods to show that $A_5$ has no
$(2,3,5,2)$\nobreakdash-factorization (in fact, $A_5$ does not even admit
a $(2,15,2)$\nobreakdash-factorization; this was shown in \cite{Kab2021} without computer calculations).
More generally, it is proved in \cite{Kab2021} that for every integer $k\geq3$,
there exist integers $m_1,m_2,\ldots,m_k$ all greater than $1$
and a finite group of order $m_1m_2\ldots m_k$ that does not have any
$(m_1,m_2,\ldots,m_k)$\nobreakdash-factorization.
Among groups of order at most $100$, there are only eight groups that do not admit a $k$-factorization
for at least one $k\geq2$; all other groups are $k$-factorizable for every admissible $k$ \cite{Kab2021}.
It was also shown that the simple groups of orders $168$ and $360$ are $k$‑factorizable
for every admissible $k$ \cite{Kab2024}.

Recently, Chin, Wang, and Wong \cite{Chin-Wang-Wong} introduced the notion of \textit{complete
factorization}. Subsets $A_1, \ldots, A_k$ of $G$ form a complete factorization if they are pairwise
disjoint and every $g \in G$ is uniquely represented as $g = a_1 \cdots a_k$ with $a_i \in A_i$.
They proved that if $G$ is a finite abelian group and $|G|=m_1\ldots m_k$
where $m_1,\ldots,m_k$ are integers greater than $1$ and $k>2$, then
there exist subsets $A_1,\ldots,A_k$ of $G$ (with $|A_i|=m_i$ for all $i=1,2,\ldots,k$)
which form a complete factorization of the group $G$.
This result was extended to nilpotent groups in \cite{Kab2023}.

A finite group $G$ is said to have the \textit{strong CFS} (converse of factorization by subsets)
property if every Lagrange subset of $G$ is a factor of $G$. The notions of a Lagrange subset and a
strong CFS group were introduced in \cite{HooshKoh2025}, where the following was proved:

The problem of factoring a finite group $G$ into two (or more) Lagrange subsets goes back to Haj\'os’s
combinatorial proof of Minkowski’s conjecture on lattice tilings by unit cubes \cite{Hajos1941}.
In particular, Haj\'os\ showed that any lattice tiling of $\mathbb{R}^n$ by unit $n$-cubes must contain two
cubes sharing a full $(n-1)$-face, and he recast that argument in group‑theoretic terms to characterize exactly
which cyclic groups admit factorizations by arithmetic progressions \cite{Hajos1941}.
In the broader abelian setting, subsequent work of R\'edei \cite{Redei1952},
Sands \cite{Sands1974}, Szab\'o \cite{Szabo2004}, and others has connected these set‑factorizations with tiling problems, zero‑sum sequences, and
related combinatorial phenomena. A good survey of these developments can be found in
\cite{Szabo-Sands2009}.

Bernstein \cite{Bernstein1968} extended Haj\'os’s theorem to the non‑abelian setting, showing that many of the
combinatorial tiling arguments carry over to arbitrary finite groups. A related thread concerns the
existence of \textit{minimal logarithmic signatures} (MLS): conjecturally, every finite group $G$ admits
a factorization
\[
G = A_1 A_2 \cdots A_k
\]
where each $\lvert A_i\rvert$ is prime or equals $4$.
It is known that every finite solvable group has an
MLS, and that if a normal subgroup $K\lhd G$ and the quotient $G/K$ both admit MLS then so does $G$
\cite{Rahimipour2018}. Consequently, any counterexample to the MLS conjecture must be a non-abelian
simple group, reducing the problem to the finite simple groups.

Hooshmand \cite[Problem 19.35]{Kourovka2018}
(see also \cite{Hooshmand2014,Banakh2018multifactorizable, Hooshmand2021})
asked whether for every factorization $\lvert G\rvert = m_1\cdots m_k$, there exist subsets
$A_i\subseteq G$ with $\lvert A_i\rvert=m_i$ and $G=A_1\cdots A_k$.
The case $k=2$ has been studied in \cite{Bildanov2020, Hooshmand2021}, where it is shown that a finite
group is $2$‑factorizable if and only if every finite simple group is $2$‑factorizable,
and all simple groups of order up to $10~000$ are $2$-factorizable \cite{Bildanov2020}.

For $k=3$, Bergman \cite{Bergman2020} proved that the alternating group $A_4$ is not $3$‑factorizable.
Brunault \cite{Banakh2018Factorizable} used computer methods to show that $A_5$ has no
$(2,3,5,2)$\nobreakdash-factorization (in fact, $A_5$ does not even admit
a $(2,15,2)$\nobreakdash-factorization; this was shown in \cite{Kab2021} without computer calculations).
More generally, it is proved in \cite{Kab2021} that for every integer $k\geq3$,
there exist integers $m_1,m_2,\ldots,m_k$ all greater than $1$
and a finite group of order $m_1m_2\ldots m_k$ that does not have any
$(m_1,m_2,\ldots,m_k)$\nobreakdash-factorization.
Among groups of order at most $100$, there are only eight groups that do not admit a $k$-factorization
for at least one $k\geq2$; all other groups are $k$-factorizable for every admissible $k$ \cite{Kab2021}.
It was also shown that the simple groups of orders $168$ and $360$ are $k$‑factorizable
for every admissible $k$ \cite{Kab2024}.

Recently, Chin, Wang, and Wong \cite{Chin-Wang-Wong} introduced the notion of \textit{complete
factorization}. Subsets $A_1, \ldots, A_k$ of $G$ form a complete factorization if they are pairwise
disjoint and every $g \in G$ is uniquely represented as $g = a_1 \cdots a_k$ with $a_i \in A_i$.
They proved that if $G$ is a finite abelian group and $|G|=m_1\ldots m_k$
where $m_1,\ldots,m_k$ are integers greater than $1$ and $k>2$, then
there exist subsets $A_1,\ldots,A_k$ of $G$ (with $|A_i|=m_i$ for all $i=1,2,\ldots,k$)
which form a complete factorization of the group $G$.
This result was extended to nilpotent groups in \cite{Kab2023}.

A finite group $G$ is said to have the \textit{strong CFS} (converse of factorization by subsets)
property if every Lagrange subset of $G$ is a factor of $G$. The notions of a Lagrange subset and a
strong CFS group were introduced in \cite{HooshKoh2025}, where the following was proved:

\begin{document}
\maketitle
\begin{abstract}
In a finite group, a subset is called a Lagrange subset if its size divides the group order, and a factor if it admits a complementary subset. We provide a new and comparatively direct proof of the classification of groups in which every Lagrange subset is a factor. We show that any nontrivial such group must be a cyclic group of prime order, the cyclic group of order 4, or an elementary abelian group of order 4, 8, or 9.
\end{abstract}

Hooshmand \cite[Problem 19.35]{Kourovka2018}
(see also \cite{Hooshmand2014,Banakh2018multifactorizable, Hooshmand2021})
asked whether for every factorization $\lvert G\rvert = m_1\cdots m_k$, there exist subsets
$A_i\subseteq G$ with $\lvert A_i\rvert=m_i$ and $G=A_1\cdots A_k$.
The case $k=2$ has been studied in \cite{Bildanov2020, Hooshmand2021}, where it is shown that a finite
group is $2$‑factorizable if and only if every finite simple group is $2$‑factorizable,
and all simple groups of order up to $10~000$ are $2$-factorizable \cite{Bildanov2020}.

For $k=3$, Bergman \cite{Bergman2020} proved that the alternating group $A_4$ is not $3$‑factorizable.
Brunault \cite{Banakh2018Factorizable} used computer methods to show that $A_5$ has no
$(2,3,5,2)$\nobreakdash-factorization (in fact, $A_5$ does not even admit
a $(2,15,2)$\nobreakdash-factorization; this was shown in \cite{Kab2021} without computer calculations).
More generally, it is proved in \cite{Kab2021} that for every integer $k\geq3$,
there exist integers $m_1,m_2,\ldots,m_k$ all greater than $1$
and a finite group of order $m_1m_2\ldots m_k$ that does not have any
$(m_1,m_2,\ldots,m_k)$\nobreakdash-factorization.
Among groups of order at most $100$, there are only eight groups that do not admit a $k$-factorization
for at least one $k\geq2$; all other groups are $k$-factorizable for every admissible $k$ \cite{Kab2021}.
It was also shown that the simple groups of orders $168$ and $360$ are $k$‑factorizable
for every admissible $k$ \cite{Kab2024}.

A finite group $G$ is said to have the \textit{strong CFS} (converse of factorization by subsets)
property if every Lagrange subset of $G$ is a factor of $G$. The notions of a Lagrange subset and a
strong CFS group were introduced in \cite{HooshKoh2025}, where the following was proved:

The authors of \cite{HooshKoh2025} used Cayley graph arguments to constrain the possible orders of $G$,
reducing the classification to an examination of about twenty concrete groups. In this paper, we offer a
streamlined proof of the "only if" direction, which breaks down into four independent steps:
\begin{enumerate}[label=(\arabic*)]
 \item Analysis of groups with "large" proper subgroups (Lemma \ref{lemma:main});
 \item Groups of even order containing elements of odd order (Lemma \ref{lemma:small subset});
 \item Investigation of groups of order $8$ (Lemma \ref{lemma:groups of order 8});
 \item A direct check for the three remaining groups: $D_4$, $C_8$, and $C_9$.
\end{enumerate}

\begin{lemma}
\label{lemma:small subset}
\begin{enumerate}[label=\textup{(\roman*)}]\!
 \item Let $G$ be a group of even order, and let $a\in G$, $a\neq e$.
 The subset $\{e,a\}$ is a factor of $G$ if and only if the order of $a$ is even.
 \item Let $G$ be an elementary abelian $2$-group.
 Every subset of size $4$ is a factor of $G$.
 \item\,Let $G$ be an elementary abelian $3$-group.
 Every subset of size $3$ is a factor of $G$.
\end{enumerate}
\end{lemma}
\begin{proof}[Proof of \textup{(i)}]
First suppose that the order of $a$ is even, say $|a| = 2m$ for some positive integer $m$.
Let $H = \langle a\rangle$.
Set
\[
B = \{e,a^2,\ldots,a^{2m-2}\}.
\]
Clearly $H = \{e,a\} B$, and it is straightforward to check that
$B \cap aB = \varnothing$.
Thus $\{e,a\}$ is a factor of $H$.
By Lemma~\ref{lemma:subset of subgroup}, $\{e,a\}$ is then a factor of $G$.

\begin{lemma}
\label{lemma:groups of order 8}
Let $G$ be a group of order $8$, and let $a,b\in G$ be two elements of order $4$ with $\langle a\rangle\neq\langle b\rangle$.
Then
\[
 A = \{a,a^2,a^3,b\}
\]
is not a left factor of $G$.
\end{lemma}
\begin{proof}
 Suppose to the contrary that there is an element $x\in G$ such that
 \[
 G=A\cup Ax \text{ and } A\cap Ax=\varnothing.
 \]
 Then $e\in Ax$ and hence $x=a^{-i}$ for some $i\in\{1,2,3\}$, or $x=b^{-1}$.
 If $x=a^{-i}$, then $a\in A\cap A x$, contradicting disjointness.
 If $x=b^{-1}=b^3$, then
 \[
 Ax=\{ab^3,a^2b^3,a^3b^3,e\}.
 \]
 Since $b^3\notin A$ and also $b^3\notin A x$,
 the element $b^3$ is not contained in $A\cup A x=G$, a clear contradiction.
\end{proof}

First, suppose that $|G|$ is even. By Lemma~\ref{lemma:small subset}\,(i),
every non-identity element of $G$ must have even order.
It follows that $G$ is a $2$-group.
Since $G$ cannot have a proper subgroup of order greater than $4$ (by Lemma~\ref{lemma:main}),
the order of $G$ must be $4$ or $8$.
If $|G|=4$, $G$ is either $C_4$ or $C_2^2$, both of which appear in the theorem's list.
If $|G|=8$, Lemma~\ref{lemma:groups of order 8} implies that $G$ cannot contain
two distinct cyclic subgroups of order $4$.
This eliminates the quaternion group $Q_8$ and the product $C_4 \times C_2$.
Since $C_2^3$ is a solution, the only remaining candidates to check are $D_4$ and $C_8$.

\begin{lemma}
\label{lemma:groups of order 8}
Let $G$ be a group of order $8$, and let $a,b\in G$ be two elements of order $4$ with $\langle a\rangle\neq\langle b\rangle$.
Then
\[
  A = \{a,a^2,a^3,b\}
\]
is not a left factor of $G$.
\end{lemma}

\begin{lemma}
\label{lemma:main}
Let $H$ be a proper subgroup of a finite group $G$ with $\lvert H\rvert \ge 5$.
Choose $h_0\in H$ with $h_0\neq e$ and choose $g\in G\setminus H$.
Let $h_1$ be an element of $H$ not in the set
$\{e,h_0,h_0^{-1}\}$,
and set
\[
H'= H\setminus\{e,h_0\}.
\]
Define
\begin{equation}\label{equ:subset-not-factor}
A= H'\cup\{g,h_1g\}.
\end{equation}
Then $\lvert A\rvert = \lvert H\rvert$ and $A$ is not a left factor of $G$.
\end{lemma}

\begin{lemma}
\label{lemma:small subset}
\begin{enumerate}[label=\textup{(\roman*)}]\!
    \item Let $G$ be a group of even order, and let $a\in G$, $a\neq e$.
    The subset $\{e,a\}$ is a factor of $G$ if and only if the order of $a$ is even.
    \item Let $G$ be an elementary abelian $2$-group.
    Every subset of size $4$ is a factor of $G$.
    \item\,Let $G$ be an elementary abelian $3$-group.
    Every subset of size $3$ is a factor of $G$.
\end{enumerate}
\end{lemma}

The authors of \cite{HooshKoh2025} used Cayley graph arguments to constrain the possible orders of $G$,
reducing the classification to an examination of about twenty concrete groups. In this paper, we offer a
streamlined proof of the "only if" direction, which breaks down into four independent steps:
\begin{enumerate}[label=(\arabic*)]
  \item Analysis of groups with "large" proper subgroups (Lemma \ref{lemma:main});
  \item Groups of even order containing elements of odd order (Lemma \ref{lemma:small subset});
  \item Investigation of groups of order $8$ (Lemma \ref{lemma:groups of order 8});
  \item A direct check for the three remaining groups: $D_4$, $C_8$, and $C_9$.
\end{enumerate}

\vspace{1ex}
\noindent\textbf{Notation.} Throughout, $G$ is a finite group with identity $e$. For any $A \subseteq G$ and $x \in G$, we denote $Ax = \{ax \mid a \in A\}$. We write $\langle g \rangle$ for the cyclic subgroup generated by $g$. A subset $A$ is a \textit{left factor} if $G = \bigsqcup_{b \in B} Ab$ for some $B \subseteq G$.

With these conventions in place, we turn in Section \ref{section:lemmas} to our lemmas,
and then in Section \ref{section:proof theorem} to the completion of the proof.

The post-theorem introduction explains that \cite{HooshKoh2025} used Cayley graph arguments to constrain the possible orders of $G$, reducing the classification to examining about twenty concrete groups. It then sketches a streamlined proof of the \emph{``only if''} direction, broken into four independent steps:
\begin{enumerate}[label=(\arabic*)]
  \item \emph{Analysis of groups with ``large'' proper subgroups} (Lemma \ref{lemma:main});
  \item \emph{Groups of even order containing elements of odd order} (Lemma \ref{lemma:small subset});
  \item \emph{Investigation of groups of order $8$} (Lemma \ref{lemma:groups of order 8});
  \item \emph{A direct check} for the three remaining groups: $D_4$, $C_8$, and $C_9$.
\end{enumerate}
It further indicates the paper proceeds by proving these lemmas in Section \ref{section:lemmas} and then completing the proof in Section \ref{section:proof theorem}.

The post-theorem introduction explains that \cite{HooshKoh2025} used Cayley graph arguments to constrain the possible orders of $G$, reducing the classification to examining about twenty concrete groups. It then sketches a streamlined proof of the \emph{``only if''} direction, broken into four independent steps:
\begin{enumerate}[label=(\arabic*)]
  \item \emph{Analysis of groups with ``large'' proper subgroups}. We first prove the following lemma.

\begin{lemma}
\label{lemma:main}
Let $H$ be a proper subgroup of a finite group $G$ with $\lvert H\rvert \ge 5$.
Choose $h_0\in H$ with $h_0\neq e$ and choose $g\in G\setminus H$.
Let $h_1$ be an element of $H$ not in the set
$\{e,h_0,h_0^{-1}\}$,
and set
\[
H'= H\setminus\{e,h_0\}.
\]
Define
\begin{equation}\label{equ:subset-not-factor}
A= H'\cup\{g,h_1g\}.
\end{equation}
Then $\lvert A\rvert = \lvert H\rvert$ and $A$ is not a left factor of $G$.
\end{lemma}

  \item \emph{Groups of even order containing elements of odd order}. We next prove the following lemma.

\begin{lemma}
\label{lemma:small subset}
\begin{enumerate}[label=\textup{(\roman*)}]\!
    \item Let $G$ be a group of even order, and let $a\in G$, $a\neq e$.
    The subset $\{e,a\}$ is a factor of $G$ if and only if the order of $a$ is even.
    \item Let $G$ be an elementary abelian $2$-group.
    Every subset of size $4$ is a factor of $G$.
    \item\,Let $G$ be an elementary abelian $3$-group.
    Every subset of size $3$ is a factor of $G$.
\end{enumerate}
\end{lemma}

  \item \emph{Investigation of groups of order $8$}. We then prove the following lemma.

\begin{lemma}
\label{lemma:groups of order 8}
Let $G$ be a group of order $8$, and let $a,b\in G$ be two elements of order $4$ with $\langle a\rangle\neq\langle b\rangle$.
Then
\[
  A = \{a,a^2,a^3,b\}
\]
is not a left factor of $G$.
\end{lemma}

  \item \emph{A direct check} for the three remaining groups: $D_4$, $C_8$, and $C_9$.
\end{enumerate}
It further indicates the paper proceeds by proving these lemmas next and then completing the proof later.

Biconditional or Equivalence
Classification or Bijection

Question: Let $G$ be a nontrivial finite group. A subset $A\subseteq G$ is called a Lagrange subset if $|A|$ divides $|G|$. Call $A$ a factor of $G$ if there exists a subset $B\subseteq G$ such that the multiplication map $A\times B\to G$, $(a,b)\mapsto ab$, is bijective. The group $G$ is said to have the strong CFS property if every Lagrange subset of $G$ is a factor of $G$. Which finite groups $G$ have the strong CFS property?

Exactly the cyclic groups of prime order, together with the four groups $C_4$, $C_2^2$, $C_2^3$, and $C_3^2$.

Choices:
B. Exactly the cyclic groups of prime order, together with the five groups $C_4$, $C_2^2$, $C_2^3$, $C_3^2$, and $C_9$.
C. At least the cyclic groups of prime order and the four groups $C_4$, $C_2^2$, $C_2^3$, and $C_3^2$ have the strong CFS property.
D. Exactly the finite abelian groups in which every nontrivial proper divisor of $|G|$ is $2$, $3$, or a prime, namely the cyclic groups of prime order together with $C_4$, $C_2^2$, $C_2^3$, $C_3^2$, and $C_9$.
E. Exactly the cyclic groups of prime order, together with the groups $C_4$, $C_2^2$, $C_2^3$, $C_3^2$, and all groups of order $8$ having a unique cyclic subgroup of order $4$.

dropped the 'only if' classification/exhaustiveness

confuses the specific classified groups with a broader abelian/order-divisibility condition

misuses the order-8 lemma by turning elimination of groups with two distinct order-4 cyclic subgroups into a positive classification criterion

The stem defines the strong CFS property but does not reveal or strongly hint at the classification in the correct option. There is no explicit answer leakage.

This is essentially a direct classification-theorem recall question: it asks for the statement equivalent to the defined property, and the correct choice is just the exact classification. That makes it close to a restatement rather than a genuinely new inferential task.

The student must distinguish the exact classification from several near-miss alternatives, so some reasoning or careful theorem recall is needed. However, the task mainly tests recognition of the theorem rather than substantial generative mathematical reasoning.

The distractors are plausible and mathematically targeted: they include a weaker true overgeneralization, boundary-case insertions like C9, an overextension to all elementary abelian 2-groups, and a nonabelian candidate D4. These align well with common classification mistakes.

A solid MCQ in terms of answer concealment and distractor design, but it is largely a theorem-restatement/recognition item rather than one that strongly tests generative reasoning.


Question: Let $G$ be a nontrivial finite group. A subset $A\subseteq G$ is called a Lagrange subset if $|A|$ divides $|G|$. Call $A$ a factor of $G$ if there exists a subset $B\subseteq G$ such that the multiplication map $A\times B\to G$, $(a,b)\mapsto ab$, is bijective. The group $G$ is said to have the strong CFS property if every Lagrange subset of $G$ is a factor of $G$. Which finite groups $G$ have the strong CFS property?

Choices:
B. Exactly the cyclic groups of prime order, together with the five groups $C_4$, $C_2^2$, $C_2^3$, $C_3^2$, and $C_9$.
C. At least the cyclic groups of prime order and the four groups $C_4$, $C_2^2$, $C_2^3$, and $C_3^2$ have the strong CFS property.
D. Exactly the finite abelian groups in which every nontrivial proper divisor of $|G|$ is $2$, $3$, or a prime, namely the cyclic groups of prime order together with $C_4$, $C_2^2$, $C_2^3$, $C_3^2$, and $C_9$.
E. Exactly the cyclic groups of prime order, together with the groups $C_4$, $C_2^2$, $C_2^3$, $C_3^2$, and all groups of order $8$ having a unique cyclic subgroup of order $4$.

The stem only defines the strong CFS property and asks for a classification; it does not state or strongly hint at the listed exceptional groups.

The item is essentially asking for the statement of a classification theorem, so it is somewhat theorem-recall-based, though the answer choices are competing formulations rather than a verbatim restatement.

Selecting the correct option requires some reasoning about exactness versus overreach (e.g., excluding C_9 and rejecting weaker true statements), but it mainly tests recognition of the classification rather than substantial derivation.

The distractors are nuanced and plausible: one is weaker-but-true, others reflect natural overgeneralizations or boundary-case mistakes, and they are clearly distinct.

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


\label{Th-SynthesExistenceSW}
    Let $d\in\{1,2\}$, $\lambda,\omega\in\mathbb R$ and $\gamma>0$. Denote by $\mu_0$ the lowest eigenvalue of $-A$. In each of the following situation, there exists a positive solution of \cref{Eq-StationaryAGP} in $\mathrm{D}\left(\sqrt{|A|}\right)$,
    \begin{itemize}
        \item $\lambda>0$ and $\omega>-\mu_0$,
        \item $\lambda = 0$ and $\omega=\mu_0$,
        \item $\lambda<0$ and $\omega<-\mu_0$.
    \end{itemize}
    In every other cases, there is no non-trivial non-negative solution in $\mathrm{D}\left(\sqrt{|A|}\right)$.

\begin{cases}\complexI\p_t u+ H u +\lambda \left|u\right|^{2\gamma}u=0\\ u(0)=u_0,\end{cases} \label{Eq-GP}
\end{equation}
by a white noise potential. In the study of Bose-Einstein condensation, \cref{Eq-GP} appears as a mean-field limit of the many body Schrödinger equation, the nonlinear term appearing as a consequence of particle interactions \cite{BEC_Pitaevskii}. Adding a random potential in \cref{Eq-GP} can be seen as a way of modeling spatial inhomogeneities (see for example \cite{PhysRevLett.104.193901,PhysRevLett.109.243902} and references therein), thus one can see \cref{Eq-AGP} as a toy model of Bose-Einstein condensation in inhomogeneous environment with zero correlation length. In this context, at least formally, \cref{Eq:AGP} has two conserved quantities physically relevant, the mass
$$\mathcal{M}(u)=\frac{1}{2}\int_{\R^d}|u|^2\d x$$
and the energy
$$\mathcal{E}(u)=\frac{1}{2}\langle-(H+\xi)u,u\rangle - \frac{\lambda}{2\gamma+2}\int_{\R^d}|u|^{2\gamma+2}\d x.$$
Standing waves of \cref{Eq:AGP} then appear naturally as solutions of minimal energy at fixed mass. Formally, if $\phi$ minimize the energy at fixed mass, by a Lagrange multiplier argument, there exists $\omega\in\R$ such that
$$-(H+\xi)\phi-\lambda|\phi|^{2\gamma}\phi+\omega \phi = 0$$
and $u(t,x)=\e^{i\omega t}\phi(x)$ is solution of \cref{Eq:AGP}.\\

In dimension 2, it is known from \cite{Mackowiak_2025,MackowiakStrichartzConfAnderson} that \cref{Eq:AGP} should be interpreted in a renormalized sense. This renormalization has to be taken into account in the study of standing waves. The construction of the renormalized operator has been conducted in \cite{MackowiakStrichartzConfAnderson} and will be recall in \cref{Sec-AndersonOp2d}. The renormalized operator is defined through an exponential transform inspired by the one introduced in \cite{Haire_Labbe} to study the continuous parabolic Anderson model, and later used in \cite{debussche_weber_T2,Tzvetkov_2023,tzvetkov2020dimensional} to solve the Anderson nonlinear Schrödinger equation on the torus $\T^2$ and in \cite{debussche2017solution,debussche2023global} to solve the same equation but on the full space $\R^2$. Let $Y=(-H)^{-1}\xi$, it is almost surely of regularity $1-$, so one can define $\rho=\e^Y$ and search for a formula for $(H+\xi)u$ when $u$ is of the form $u=\rho v$. A formal computation gives
$$(H+\xi)u=\rho\left(Hv+2\nabla Y\cdot \nabla v + xY\cdot xv + |\nabla Y|^2v\right).$$
Unfortunately, $\nabla Y$ is of regularity $0-$, thus the product $|\nabla Y|^2$ barely fails to exist. In \cite{Mackowiak_2025}, the author proved that the wick product $\wick{|\nabla Y|^2}$ can be defined as the almost sure limit, in suitable distribution spaces of regularity $0-$, of
$$|\nabla Y_N|^2-\E[|\nabla Y_N|^2],$$
where $Y_N$ is a suitable regularisation of $Y_N$.  Thus, the renormalized operator can be formally defined as
$$(H+\xi)u=\rho\left(Hv+2\nabla Y\cdot \nabla v + xY\cdot xv + \wick{|\nabla Y|^2}v\right).$$
The rigorous construction of $H+\xi$ rely on the associated quadratic form.

Even in dimension 1, the construction of $H+\xi$ is not straightforward as $\xi$ is not a function. On compact interval with no potential, $\p_x^2+\xi$ has first been constructed in \cite{FukushimaNakao} using its quadratic form and has been studied in \cite{DumazLabbeAndersonLoc1D} using a twisted Sturm-Liouville approach, allowing to construct $\p_x^2+\xi$ on unbounded interval too. In \cref{Sec-AndersonOp1d}, we propose a construction of $H+\xi$ as a self-adjoint unbounded operator through its quadratic form. \cref{Prop-IPP1d} shows $\mathrm{D}(H+\xi)=\mathrm{D}(\p_x^2+\xi)\cap \mathrm{D}(|x|^2)$, where $\p_x^2+\xi$ is the unbounded self-adjoint operator on $\L^2(\R)$ constructed in \cite{DumazLabbeAndersonLoc1D} and $\mathrm{D}(|x|^2)=\{u\in\L^2(\R),xu\in\L^2(\R)\}$.\\

We are then interested in standing waves for
\begin{equation}{~}
    \begin{cases}\complexI\p_t u+ Au+\lambda \left|u\right|^{2\gamma}u=0\\ u(0)=u_0,\end{cases}

\begin{cases}\complexI\p_t u+ H u + \xi u+\lambda \left|u\right|^{2\gamma}u=0\\ u(0)=u_0,\end{cases} \label{Eq:AGP}
\end{equation}
of the form $u(t,x)=\e^{i\omega t}\phi(x)$, where $H=\Delta-|x|^2$ is the Hermite operator, $\xi$ is a real spatial white noise, $\lambda\in\R$ and $\gamma>0$. Such a solution is called a standing wave of \cref{Eq:AGP} of frequency $\omega\in\R$ and profile $\phi$. The study of standing waves is motivated by the modeling of Bose-Einstein condensation.
In fact, \cref{Eq:AGP} can be seen as a perturbation of the Gross-Pitaevskii equation
\begin{equation}{~}
    \begin{cases}\complexI\p_t u+ H u +\lambda \left|u\right|^{2\gamma}u=0\\ u(0)=u_0,\end{cases}

\begin{theo}\label{Th-EnergyGS}

    Let $d\in\{1,2\}$, $\lambda,\omega\in\R$, $\gamma>0$ and $A$ be the self-adjoint operator constructed in \cref{Sec-AndersonOp1d,Sec-AndersonOp2d} with $\xi$ a spatial white noise, $\D^{1,2}=\mathrm{D}(\sqrt{|A|})$ and $\D^{-1,2}$ its dual. For $u\in\D^{1,2}$, we define the mass as
    $$\mathcal{M}(u)=\frac{1}{2}\int_{\R^d}|u|^2\d x$$
    and the energy as
    $$\mathcal{E}(u)=\frac{1}{2}\langle -Au,u\rangle_{\D^{-1,2},\D^{1,2}}-\frac{\lambda}{2\gamma+2}\int_{\R^d}|u|^{2\gamma+2}\d x.$$
    Then, it holds
    \begin{enumerate}
        \item For $\lambda\leqslant0$, for any $m>0$, there exists $\phi_m\in\D^{1,2}$ such that $\phi_m>0$ and
        $$\argmin_{\substack{u\in\D^{1,2}\\\mathcal{M}(u)=m}}\mathcal{E}(u) = \left\{\e^{\complexI\theta}\phi_m,\; \theta\in\R\right\}.$$
        \item For $\lambda>0$ and $\gamma<\frac{2}{d}$, for any $m>0$, there exists $\phi_m\in\displaystyle\argmin_{u\in\D^{1,2},\;\mathcal{M}(u)=m}\mathcal{E}(u)$ such that $\phi_m>0$.
        \item For $\lambda>0$ and $\gamma=\frac{2}{d}$, there exists $m_{*,\lambda}(\Xi)>0$ such that for all $m\in(0,m_{*,\lambda}(\Xi))$, there exists $\phi_m\in\displaystyle\argmin_{u\in\D^{1,2},\;\mathcal{M}(u)=m}\mathcal{E}(u)$ such that $\phi_m>0$. Moreover, there exists $m_{*,\lambda}(\Xi)\leqslant m^*_\lambda(\Xi)\leqslant m^*_\lambda$ such that
        $$\forall m>m^*_\lambda(\Xi), \inf_{\substack{u\in\D^{1,2}\\\mathcal{M}(u)=m}}\mathcal{E}(u)=-\infty,$$
        where $m^*_\lambda$ is the critical mass of the classical nonlinear Schrödinger equation (i.e.\ without any potential).
        \item For $\lambda>0$ and $\gamma>\frac{2}{d}$, for any $m>0$,
        $$\inf_{\substack{u\in\D^{1,2}\\\mathcal{M}(u)=m}}\mathcal{E}(u)=-\infty.$$
    \end{enumerate}

\end{theo}

\label{Eq-AGP}
\end{equation}
in dimension $d\in\{1,2\}$, where $A=H+\xi$. As $A$ is independent of time, standing waves $u(t,x)=\e^{\complexI\omega t}\phi(x)$ solves \cref{Eq-AGP} if and only if $\p

In this paper, we are interested in particular solutions of the Anderson-Gross-Pitaevskii
\begin{equation}{~}
    \begin{cases}\mathrm{i}\p_t u+ H u + \xi u+\lambda \left|u\right|^{2\gamma}u=0\\ u(0)=u_0,\end{cases} \label{Eq:AGP}
\end{equation}
of the form $u(t,x)=\mathrm{e}^{i\omega t}\phi(x)$, where $H=\Delta-|x|^2$ is the Hermite operator, $\xi$ is a real spatial white noise, $\lambda\in\mathbb R$ and $\gamma>0$. Such a solution is called a standing wave of \cref{Eq:AGP} of frequency $\omega\in\mathbb R$ and profile $\phi$. The study of standing waves is motivated by the modeling of Bose-Einstein condensation.
In fact, \cref{Eq:AGP} can be seen as a perturbation of the Gross-Pitaevskii equation
\begin{equation}{~}
    \begin{cases}\mathrm{i}\p_t u+ H u +\lambda \left|u\right|^{2\gamma}u=0\\ u(0)=u_0,\end{cases} \label{Eq-GP}
\end{equation}
by a white noise potential. In the study of Bose-Einstein condensation, \cref{Eq-GP} appears as a mean-field limit of the many body Schrödinger equation, the nonlinear term appearing as a consequence of particle interactions \cite{BEC_Pitaevskii}. Adding a random potential in \cref{Eq-GP} can be seen as a way of modeling spatial inhomogeneities (see for example \cite{PhysRevLett.104.193901,PhysRevLett.109.243902} and references therein), thus one can see \cref{Eq-AGP} as a toy model of Bose-Einstein condensation in inhomogeneous environment with zero correlation length. In this context, at least formally, \cref{Eq:AGP} has two conserved quantities physically relevant, the mass
$$\mathcal{M}(u)=\frac{1}{2}\int_{\mathbb R^d}|u|^2\mathrm{d} x$$
and the energy
$$\mathcal{E}(u)=\frac{1}{2}\langle-(H+\xi)u,u\rangle - \frac{\lambda}{2\gamma+2}\int_{\mathbb R^d}|u|^{2\gamma+2}\mathrm{d} x.$$
Standing waves of \cref{Eq:AGP} then appear naturally as solutions of minimal energy at fixed mass. Formally, if $\phi$ minimize the energy at fixed mass, by a Lagrange multiplier argument, there exists $\omega\in\mathbb R$ such that
$$-(H+\xi)\phi-\lambda|\phi|^{2\gamma}\phi+\omega \phi = 0$$
and $u(t,x)=\mathrm{e}^{i\omega t}\phi(x)$ is solution of \cref{Eq:AGP}.\\

In dimension 2, it is known from \cite{Mackowiak_2025,MackowiakStrichartzConfAnderson} that \cref{Eq:AGP} should be interpreted in a renormalized sense. This renormalization has to be taken into account in the study of standing waves. The construction of the renormalized operator has been conducted in \cite{MackowiakStrichartzConfAnderson} and will be recall in \cref{Sec-AndersonOp2d}. The renormalized operator is defined through an exponential transform inspired by the one introduced in \cite{Haire_Labbe} to study the continuous parabolic Anderson model, and later used in \cite{debussche_weber_T2,Tzvetkov_2023,tzvetkov2020dimensional} to solve the Anderson nonlinear Schrödinger equation on the torus $\mathbb T^2$ and in \cite{debussche2017solution,debussche2023global} to solve the same equation but on the full space $\mathbb R^2$. Let $Y=(-H)^{-1}\xi$, it is almost surely of regularity $1-$, so one can define $\rho=\mathrm{e}^Y$ and search for a formula for $(H+\xi)u$ when $u$ is of the form $u=\rho v$. A formal computation gives
$$(H+\xi)u=\rho\left(Hv+2\nabla Y\cdot \nabla v + xY\cdot xv + |\nabla Y|^2v\right).$$
Unfortunately, $\nabla Y$ is of regularity $0-$, thus the product $|\nabla Y|^2$ barely fails to exist. In \cite{Mackowiak_2025}, the author proved that the wick product $\wick{|\nabla Y|^2}$ can be defined as the almost sure limit, in suitable distribution spaces of regularity $0-$, of
$$|\nabla Y_N|^2-\mathbb E[|\nabla Y_N|^2],$$
where $Y_N$ is a suitable regularisation of $Y_N$.  Thus, the renormalized operator can be formally defined as
$$(H+\xi)u=\rho\left(Hv+2\nabla Y\cdot \nabla v + xY\cdot xv + \wick{|\nabla Y|^2}v\right).$$
The rigorous construction of $H+\xi$ rely on the associated quadratic form.

Even in dimension 1, the construction of $H+\xi$ is not straightforward as $\xi$ is not a function. On compact interval with no potential, $\p_x^2+\xi$ has first been constructed in \cite{FukushimaNakao} using its quadratic form and has been studied in \cite{DumazLabbeAndersonLoc1D} using a twisted Sturm-Liouville approach, allowing to construct $\p_x^2+\xi$ on unbounded interval too. In \cref{Sec-AndersonOp1d}, we propose a construction of $H+\xi$ as a self-adjoint unbounded operator through its quadratic form. \cref{Prop-IPP1d} shows $\mathrm{D}(H+\xi)=\mathrm{D}(\p_x^2+\xi)\cap \mathrm{D}(|x|^2)$, where $\p_x^2+\xi$ is the unbounded self-adjoint operator on $\mathrm L^2(\mathbb R)$ constructed in \cite{DumazLabbeAndersonLoc1D} and $\mathrm{D}(|x|^2)=\{u\in\mathrm L^2(\mathbb R),xu\in\mathrm L^2(\mathbb R)\}$.\\

We are then interested in standing waves for
\begin{equation}{~}
    \begin{cases}\mathrm{i}\p_t u+ Au+\lambda \left|u\right|^{2\gamma}u=0\\ u(0)=u_0,\end{cases} \label{Eq-AGP}
\end{equation}
in dimension $d\in\{1,2\}$, where $A=H+\xi$. As $A$ is independent of time, standing waves $u(t,x)=\mathrm{e}^{\mathrm{i}\omega t}\phi(x)$ solves \cref{Eq-AGP} if and only if $\phi$ is solution of
\begin{equation}
    -A\phi-\lambda|\phi|^{2\gamma}\phi+\omega \phi = 0. \label{Eq-StationaryAGP}
\end{equation}
This paper aim to construct standing waves in presence of a white noise potential and present some of their properties.

When $\xi=0$ and $H=\Delta$, standing waves are well-known object. See for example \cite{NehariExistence,BerestyckiLions,StraussExistenceSolitary} for existence, \cite{BerestyckiLions} for non-existence, \cite{GidasSymmetry,KwongUniqueness} for uniqueness and \cite{BerestyckiInstabilité,CazenaveLionsStability,CazenaveStabilityInstability,WeinsteinSharpInterpolation,GrillakisStability,GrillakisStabilityII} for results on stability. Let us stress out many of these results relies on the particular structure of \cref{Eq-StationaryAGP} in this case, which is not preserved in presence of potentials. When $\xi=0$ and $H=\Delta-V$ with $V$ a regular and potentially unbounded potential, we refer the reader to \cite{KavianSelfSimilar,FukuizumiStabilityInstability,HiroseStructure,HiroseUniqueness,FukuizumiExponentialDecay,MaedaSymmetry,SelemBifurcation,GuoSeringerMassConcentration,PelinovskyGSEnergyCriticalGP} and references therein for many results on existence, uniqueness, localization and stability.

Nonlinear elliptic equations in presence of white noise potential on compact manifold have been studied in \cite{EulryVariational} using mountain pass methods and in \cite{ZhangConstraintMinAnderson} using constraint minimization. In both \cite{EulryVariational,ZhangConstraintMinAnderson}, the Anderson operator $\Delta+\xi$ is constructed using paracontrolled calculus. We will present here the construction of solution of \cref{Eq-StationaryAGP} through constraint minimization, namely energy minimization at fixed mass and action minimization. This allow to construct a positive solution of \cref{Eq-StationaryAGP} whenever such a solution exists.

In dimension 2, it is known from \cite{Mackowiak_2025,MackowiakStrichartzConfAnderson} that \cref{Eq:AGP} should be interpreted in a renormalized sense. This renormalization has to be taken into account in the study of standing waves. The construction of the renormalized operator has been conducted in \cite{MackowiakStrichartzConfAnderson} and will be recall in \cref{Sec-AndersonOp2d}. The renormalized operator is defined through an exponential transform inspired by the one introduced in \cite{Haire_Labbe} to study the continuous parabolic Anderson model, and later used in \cite{debussche_weber_T2,Tzvetkov_2023,tzvetkov2020dimensional} to solve the Anderson nonlinear Schrödinger equation on the torus $\mathbb T^2$ and in \cite{debussche2017solution,debussche2023global} to solve the same equation but on the full space $\mathbb R^2$. Let $Y=(-H)^{-1}\xi$, it is almost surely of regularity $1-$, so one can define $\rho=\mathrm{e}^Y$ and search for a formula for $(H+\xi)u$ when $u$ is of the form $u=\rho v$. A formal computation gives
$$(H+\xi)u=\rho\left(Hv+2\nabla Y\cdot \nabla v + xY\cdot xv + |\nabla Y|^2v\right).$$
Unfortunately, $\nabla Y$ is of regularity $0-$, thus the product $|\nabla Y|^2$ barely fails to exist. In \cite{Mackowiak_2025}, the author proved that the wick product $\wick{|\nabla Y|^2}$ can be defined as the almost sure limit, in suitable distribution spaces of regularity $0-$, of
$$|\nabla Y_N|^2-\mathbb E[|\nabla Y_N|^2],$$
where $Y_N$ is a suitable regularisation of $Y_N$. Thus, the renormalized operator can be formally defined as
$$(H+\xi)u=\rho\left(Hv+2\nabla Y\cdot \nabla v + xY\cdot xv + \wick{|\nabla Y|^2}v\right).$$
The rigorous construction of $H+\xi$ rely on the associated quadratic form.

Even in dimension 1, the construction of $H+\xi$ is not straightforward as $\xi$ is not a function. On compact interval with no potential, $\p_x^2+\xi$ has first been constructed in \cite{FukushimaNakao} using its quadratic form and has been studied in \cite{DumazLabbeAndersonLoc1D} using a twisted Sturm-Liouville approach, allowing to construct $\p_x^2+\xi$ on unbounded interval too. In \cref{Sec-AndersonOp1d}, we propose a construction of $H+\xi$ as a self-adjoint unbounded operator through its quadratic form. \cref{Prop-IPP1d} shows $\mathrm{D}(H+\xi)=\mathrm{D}(\p_x^2+\xi)\cap \mathrm{D}(|x|^2)$, where $\p_x^2+\xi$ is the unbounded self-adjoint operator on $\mathrm L^2(\mathbb R)$ constructed in \cite{DumazLabbeAndersonLoc1D} and $\mathrm{D}(|x|^2)=\{u\in\mathrm L^2(\mathbb R),xu\in\mathrm L^2(\mathbb R)\}$.\\

We are then interested in standing waves for
\begin{equation}{~}
 \begin{cases}\mathrm{i}\p_t u+ Au+\lambda \left|u\right|^{2\gamma}u=0\\ u(0)=u_0,\end{cases} \label{Eq-AGP}
\end{equation}
in dimension $d\in\{1,2\}$, where $A=H+\xi$. As $A$ is independent of time, standing waves $u(t,x)=\mathrm{e}^{\mathrm{i}\omega t}\phi(x)$ solves \cref{Eq-AGP} if and only if $\phi$ is solution of
\begin{equation}
 -A\phi-\lambda|\phi|^{2\gamma}\phi+\omega \phi = 0. \label{Eq-StationaryAGP}
\end{equation}
This paper aim to construct standing waves in presence of a white noise potential and present some of their properties.

When $\xi=0$ and $H=\Delta$, standing waves are well-known object. See for example \cite{NehariExistence,BerestyckiLions,StraussExistenceSolitary} for existence, \cite{BerestyckiLions} for non-existence, \cite{GidasSymmetry,KwongUniqueness} for uniqueness and \cite{BerestyckiInstabilité,CazenaveLionsStability,CazenaveStabilityInstability,WeinsteinSharpInterpolation,GrillakisStability,GrillakisStabilityII} for results on stability. Let us stress out many of these results relies on the particular structure of \cref{Eq-StationaryAGP} in this case, which is not preserved in presence of potentials. When $\xi=0$ and $H=\Delta-V$ with $V$ a regular and potentially unbounded potential, we refer the reader to \cite{KavianSelfSimilar,FukuizumiStabilityInstability,HiroseStructure,HiroseUniqueness,FukuizumiExponentialDecay,MaedaSymmetry,SelemBifurcation,GuoSeringerMassConcentration,PelinovskyGSEnergyCriticalGP} and references therein for many results on existence, uniqueness, localization and stability.

Nonlinear elliptic equations in presence of white noise potential on compact manifold have been studied in \cite{EulryVariational} using mountain pass methods and in \cite{ZhangConstraintMinAnderson} using constraint minimization. In both \cite{EulryVariational,ZhangConstraintMinAnderson}, the Anderson operator $\Delta+\xi$ is constructed using paracontrolled calculus. We will present here the construction of solution of \cref{Eq-StationaryAGP} through constraint minimization, namely energy minimization at fixed mass and action minimization. This allow to construct a positive solution of \cref{Eq-StationaryAGP} whenever such a solution exists.

Nonlinear elliptic equations in presence of white noise potential on compact manifold have been studied in \cite{EulryVariational} using mountain pass methods and in \cite{ZhangConstraintMinAnderson} using constraint minimization. In both \cite{EulryVariational,ZhangConstraintMinAnderson}, the Anderson operator $\Delta+\xi$ is constructed using paracontrolled calculus. We will present here the construction of solution of \cref{Eq-StationaryAGP} through constraint minimization, namely energy minimization at fixed mass and action minimization. This allow to construct a positive solution of \cref{Eq-StationaryAGP} whenever such a solution exists.

In order to do so, we use constrained minimization methods. In \cref{Sec-EnergyGS}, we study the minimization of energy at fixed mass and prove the following result.

Let $d\in\{1,2\}$, $\lambda,\omega\in\R$, $\gamma>0$ and $A$ the self-adjoint operator constructed in \cref{Sec-AndersonOp1d,Sec-AndersonOp2d} with $\xi$ a spatial white noise. Let $\D^{1,2}=\mathrm{D}(\sqrt{|A|})$ and $\D^{-1,2}$ its dual. Denote by $\mu_0$ the smallest eigenvalue of $-A$. Let $\omega>-\mu_0$ and define $$\mathcal{S}_\omega = \mathcal{E} + \omega \mathcal{M}$$ and $$\mathcal{N}_\omega =\left\{u\in\D^{1,2}\backslash\{0\},\; \langle \nabla\mathcal{S}_\omega u,u \rangle_{\D^{-1,2},\D^{1,2}}=0\right\}.$$
 Then, there exists $\psi_\omega\in\displaystyle\argmin_{\mathcal{N}_\omega}\mathcal{S}_\omega$ such that $\psi_\omega>0$.

\begin{itemize}
 \item Assume $\gamma>\frac{2}{d}$, let $v\in\W^{1,2}\backslash\{0\}$ and $u=\rho v$, then for any $\alpha>0$, \cref{Eq-EnergyVside} implies
 \begin{align*}
 \mathcal{E}(u_\alpha) &= - \frac{\lambda\alpha^{d\gamma}}{2\gamma+2}\int_{\R^d}|v(x)|^{2\gamma+2}\rho(x/\alpha)^{2\gamma+2}\d x &(I)\\
 &+ \frac{\alpha^2}{2}\int_{\R^d}\rho(x/\alpha)^2|\nabla v(x)|^2\d x &(II)\\
 &+ \frac{1}{2\alpha^2}\int_{\R^d} (1-Y(x/\alpha))\rho(x/\alpha)^2|xv(x)|^2\d x &(III)\\
 &-\frac{1}{2}\langle Z,|u_\alpha|^2\rangle. &(IV)
 \end{align*}
 \cref{Lem-EstimScalTransf,Lem-ProductRule} imply $(IV)=O(\alpha^s)$ as $\alpha$ goes to infinity, for any $s\in(0,2)$, as $Z\in\W^{0-,\infty}$. By dominated convergence, $(III)=O(\alpha^{-2})$ as $\alpha$ goes to infinity. Finally, $(II)=O(\alpha^2)$ and
 $$(I)\sim - \frac{\lambda\alpha^{d\gamma}\rho(0)^{2\gamma+2}}{2\gamma+2}\int_{\R^d}|v(x)|^{2\gamma+2}\d x.$$
 Chose $v = \sqrt{2m}\rho(0)^{-1}h_0$ ($m>0$), then $\mathcal{M}(u_\alpha)$ converges to $m$ whereas $\mathcal{E}(u_\alpha)$ goes to $-\infty$, because $d\gamma>2$. Thus, \cref{Lem-AsympEnergy} implies
 $$\forall m>0,\; \inf_{\Atop{u\in\D^{1,2}}{\mathcal{M}(u)=m}}\mathcal{E}(u)=-\infty.$$
 \item Assume $\gamma=\frac{2}{d}$ and let $Q$ be the unique positive and radial solution of
 $$\Delta Q + \lambda Q^{2\gamma+1} = Q.$$
 Recall $\mathcal{M}(Q)=m_\lambda^*=\frac{1}{2}\left(\frac{(\gamma+1)J}{\lambda}\right)^{\frac{1}{\gamma}}$. For $c>0$, let $u=c\rho Q$. For any $\alpha>0$, it holds
 \begin{align*}
 \mathcal{E}(u_\alpha) &= \frac{\alpha^2}{2}\int_{\R^d}\rho(x/\alpha)^2|c\nabla Q(x)|^2\d x - \frac{\lambda\alpha^{2}}{2\gamma+2}\int_{\R^d}|cQ(x)|^{2\gamma+2}\rho(x/\alpha)^{2\gamma+2}\d x &(I)\\
 &+ \frac{c^2}{2\alpha^2}\int_{\R^d} (1-Y(x/\alpha))\rho(x/\alpha)^2|xQ(x)|^2\d x -\frac{1}{2}\langle Z,|u_\alpha|^2\rangle. &(II)
 \end{align*}
 As previously $(II)=O(\alpha^s)$ as $\alpha$ goes to infinity, for any $s\in(0,2)$, and for $c\rho(0)\neq 1$,
 $$(I)\sim\frac{\alpha^2\lambda(c\rho(0))^2}{2\gamma+2}\left(1-(c\rho(0))^{2\gamma}\right)\int_{\R^d}|Q(x)|^{2\gamma+2}\d x$$
 as $\alpha$ goes to infinity, using \cref{Eq-PohozaevQ}. Thus, for $c\rho(0)>1$, \cref{Lem-AsympEnergy} implies
 $$\inf_{\Atop{u\in\D^{1,2}}{\mathcal{M}(u)=c\rho(0)m^*_\lambda}}\mathcal{E}(u)=-\infty,$$
 that is
 $$\forall m>m^*_\lambda,\; \inf_{\Atop{u\in\D^{1,2}}{\mathcal{M}(u)=m}}\mathcal{E}(u)=-\infty.$$
 Let $m'>m>0$ and $\phi\in\D^{1,2}$ such that $\mathcal{M}(\phi)=m$. Let $\psi=\sqrt{\frac{m'}{m}}\phi$. Then,
 \begin{align*}
 \mathcal{E}(\phi) &= \frac{m}{m'}\left[\frac{1}{2}\langle -A\psi,\psi\rangle - \frac{\lambda}{2\gamma+2}\left(\frac{m}{m'}\right)^{\gamma}\int_{\R^d}|\psi|^{2\gamma+2}\d x\right]\\
 &=\frac{m}{m'}\mathcal{E}(\psi) + \frac{m\lambda}{m'(2\gamma+2)}\left[1 - \left(\frac{m}{m'}\right)^{\gamma}\right]\int_{\R^d}|\phi|^{2\gamma+2}\d x\\
 &\geqslant \frac{m}{m'}\mathcal{E}(\psi).
 \end{align*}
 Thus,
 $$m'\inf_{\Atop{u\in\D^{1,2}}{\mathcal{M}(u)=m}}\mathcal{E}(u) \geqslant m \inf_{\Atop{u\in\D^{1,2}}{\mathcal{M}(u)=m}'}\mathcal{E}(u).$$
 In particular, if $\displaystyle\inf_{u\in\D^{1,2},\; \mathcal{M}(u)=m}\mathcal{E}(u)=-\infty$, then $\displaystyle\inf_{u\in\D^{1,2},\; \mathcal{M}(u)=m'}\mathcal{E}(u)=-\infty$. Similarly, if $\displaystyle\inf_{u\in\D^{1,2},\; \mathcal{M}(u)=m'}\mathcal{E}(u)>-\infty$, then $\displaystyle\inf_{u\in\D^{1,2},\; \mathcal{M}(u)=m}\mathcal{E}(u)>-\infty$. Hence, we can define
 \begin{align*}
 m^*_\lambda(\Xi) &= \inf\left\{m>0,\; \inf_{\Atop{u\in\D^{1,2}}{\mathcal{M}(u)=m}}\mathcal{E}(u)=-\infty\right\}\\
 &= \sup\left\{m>0,\; \inf_{\Atop{u\in\D^{1,2}}{\mathcal{M}(u)=m}}\mathcal{E}(u)>-\infty\right\} \in [0,m^*_\lambda]
 \end{align*}
 To conclude, we only have to prove a lower bound for $m^*_\lambda(\Xi)$.

Our aim is now to prove \cref{Th-ACtionGS}. Showing directly that the minimization problem
\begin{equation}
 \inf_{\mathcal{N}_{\omega,\lambda}}\mathcal{S}_{\omega,\lambda}\label{Eq-MinAction}
\end{equation}
has non-negative solutions may be a bit technical as it is not straightforward that
$$u\in\argmin_{\mathcal{N}_{\omega,\lambda}}\mathcal{S}_{\omega,\lambda}\Rightarrow |u|\in\argmin_{\mathcal{N}_{\omega,\lambda}}\mathcal{S}_{\omega,\lambda}.$$
Thus, we will introduce two other, but equivalent, minimization problems which are easier to study. Remark if $u$ solves \cref{Eq-StationaryAGP} for some $\lambda>0$, then $\alpha u$ solves \cref{Eq-StationaryAGP} for $\lambda'=\frac{\lambda}{|\alpha|^{2\gamma}}$. Thus, the parameter $\lambda$ can be seen as a Lagrange multiplier, and solving \cref{Eq-StationaryAGP} for some $\lambda>0$ is equivalent to solving \cref{Eq-StationaryAGP} for all $\lambda>0$. Following an idea of \cite{ROSE1988207}, we define for $u\in\D^{1,2}$
\begin{equation}
 \mathcal{P}_\omega(u)=\langle (-A+\omega) u, u\rangle,\label{Eq-POmega}
\end{equation}
\begin{equation}
 \mathcal{Q}(u) = \int_\R |u|^{2(\gamma+1)}\d x\label{Eq-Q(u)}
\end{equation}
and if $u\neq0,$
\begin{equation}
 \mathcal{J}_\omega(u) = \frac{\mathcal{P}_\omega(u)}{\mathcal{Q}(u)^{1/(\gamma+1)}}.\label{Eq-JOmega}
\end{equation}
Let us stress out $\mathcal{J}_\omega(\alpha u)=\mathcal{J}_\omega(u)$ for any $\alpha\in\C^*$, thus if $u$ minimizes $\mathcal{J}_\omega$, $\alpha u$ minimizes $\mathcal{J}_\omega$ too. Using standard argument of variational calculus, one can show minimizers of both
\begin{equation}
 \inf_{\Atop{u\in\D^{1,2}}{\mathcal{Q}(u)=q}}\mathcal{P}_\omega(u)\label{Eq-MinP/Q=q}
\end{equation}
and
\begin{equation}
 \inf_{\D^{1,2}\backslash\{0\}}\mathcal{J}_\omega\label{Eq-MinJ}
\end{equation}
are solutions of \cref{Eq-StationaryAGP} for some $\lambda>0$. For $\lambda>0$, one can rewrite
$$\mathcal{S}_{\omega,\lambda}(u)=\frac{1}{2}\mathcal{P}_\omega(u)-\frac{\lambda}{2(\gamma+1)}\mathcal{Q}(u)$$
and
\begin{equation}
 \mathcal{N}_{\omega,\lambda} = \left\{u\in\D^{1,2}\backslash\{0\},\; \mathcal{J}_\omega(u) = \lambda \mathcal{Q}(u)^{\gamma/(\gamma+1)} \right\}.\label{Eq-NehariJQ}
\end{equation}
As remarked in \cite{ROSE1988207}, minimizers of \cref{Eq-MinP/Q=q,Eq-MinJ} are essentially the same, that is up to a multiplicative factor, due to the the homogeneity of $\mathcal{J}_\omega$. The following proposition shows minimization problems \cref{Eq-MinAction,Eq-MinP/Q=q,Eq-MinJ} are equivalent, up to a multiplicative factor (see also \cite{LiuRotatingActionGS}).

In order to do so, we use constrained minimization methods. In \cref{Sec-EnergyGS}, we study the minimization of energy at fixed mass and prove the following result.

\begin{theo}\label{Th-EnergyGS}

    Let $d\in\{1,2\}$, $\lambda,\omega\in\mathbb R$, $\gamma>0$ and $A$ be the self-adjoint operator constructed in \cref{Sec-AndersonOp1d,Sec-AndersonOp2d} with $\xi$ a spatial white noise, $\mathcal D^{1,2}=\mathrm{D}(\sqrt{|A|})$ and $\mathcal D^{-1,2}$ its dual. For $u\in\mathcal D^{1,2}$, we define the mass as
    $$\mathcal{M}(u)=\frac{1}{2}\int_{\mathbb R^d}|u|^2\mathrm{d} x$$
    and the energy as
    $$\mathcal{E}(u)=\frac{1}{2}\langle -Au,u\rangle_{\mathcal D^{-1,2},\mathcal D^{1,2}}-\frac{\lambda}{2\gamma+2}\int_{\mathbb R^d}|u|^{2\gamma+2}\mathrm{d} x.$$
    Then, it holds
    \begin{enumerate}
        \item For $\lambda\leqslant0$, for any $m>0$, there exists $\phi_m\in\mathcal D^{1,2}$ such that $\phi_m>0$ and
        $$\argmin_{\substack{u\in\mathcal D^{1,2}\\\mathcal{M}(u)=m}}\mathcal{E}(u) = \left\{\mathrm{e}^{\mathrm{i}\theta}\phi_m,\; \theta\in\mathbb R\right\}.$$
        \item For $\lambda>0$ and $\gamma<\frac{2}{d}$, for any $m>0$, there exists $\phi_m\in\displaystyle\argmin_{u\in\mathcal D^{1,2},\;\mathcal{M}(u)=m}\mathcal{E}(u)$ such that $\phi_m>0$.
        \item For $\lambda>0$ and $\gamma=\frac{2}{d}$, there exists $m_{*,\lambda}(\Xi)>0$ such that for all $m\in(0,m_{*,\lambda}(\Xi))$, there exists $\phi_m\in\displaystyle\argmin_{u\in\mathcal D^{1,2},\;\mathcal{M}(u)=m}\mathcal{E}(u)$ such that $\phi_m>0$. Moreover, there exists $m_{*,\lambda}(\Xi)\leqslant m^*_\lambda(\Xi)\leqslant m^*_\lambda$ such that
        $$\forall m>m^*_\lambda(\Xi), \inf_{\substack{u\in\mathcal D^{1,2}\\\mathcal{M}(u)=m}}\mathcal{E}(u)=-\infty,$$
        where $m^*_\lambda$ is the critical mass of the classical nonlinear Schrödinger equation (i.e.\ without any potential).
        \item For $\lambda>0$ and $\gamma>\frac{2}{d}$, for any $m>0$,
        $$\inf_{\substack{u\in\mathcal D^{1,2}\\\mathcal{M}(u)=m}}\mathcal{E}(u)=-\infty.$$
    \end{enumerate}

\end{theo}

It is classical that the set of minimal energy states of fixed mass is stable in $\mathcal D^{1,2}$ for \cref{Eq-AGP}. In particular, \cref{Th-EnergyGS} implies the orbital stability of the ground-state $\phi_m$ in the defocusing case ($\lambda<0$). \cref{Prop-ExistenceEnergyGS,Prop-FrequencyEnergyGS} show that, for $\lambda\leqslant 0$ or $\gamma<\frac{2}{d}$, energy minimization at fixed mass provide positive solutions of \cref{Eq-StationaryAGP} for any possible value of $\omega$. We also prove the small mass asymptotic of $\phi_m$ and its frequency $\omega_m$.\\

In \cref{Sec-ActionGS}, we investigate the minimization of action $\mathcal{S}_\omega = \mathcal{E} + \omega \mathcal{M}$. In the focusing case ($\lambda> 0$), one easily verifies the action is not lowerbounded. We thus introduce the Nehari manifold and show the following result.

\begin{theo}\label{Th-ACtionGS}

    Let $d\in\{1,2\}$, $\lambda,\omega\in\mathbb R$, $\gamma>0$ and $A$ the self-adjoint operator constructed in \cref{Sec-AndersonOp1d,Sec-AndersonOp2d} with $\xi$ a spatial white noise. Let $\mathcal D^{1,2}=\mathrm{D}(\sqrt{|A|})$ and $\mathcal D^{-1,2}$ its dual. Denote by $\mu_0$ the smallest eigenvalue of $-A$. Let $\omega>-\mu_0$ and define $$\mathcal{S}_\omega = \mathcal{E} + \omega \mathcal{M}$$ and $$\mathcal{N}_\omega =\left\{u\in\mathcal D^{1,2}\backslash\{0\},\; \langle \nabla\mathcal{S}_\omega u,u \rangle_{\mathcal D^{-1,2},\mathcal D^{1,2}}=0\right\}.$$
    Then, there exists $\psi_\omega\in\displaystyle\argmin_{\mathcal{N}_\omega}\mathcal{S}_\omega$ such that $\psi_\omega>0$.

\end{theo}

Let us stress out an important property of solution of \cref{Eq-StationaryAGP} compared to the noiseless case. In presence of white noise, standing waves have a limited Sobolev regularity.
\begin{theo}\label{Th-MaxRegSW}
    Let $d\in\{1,2\}$, $\lambda,\omega\in\mathbb R$ and $\gamma>0$. Almost surely any solution $u$ of \cref{Eq-StationaryAGP} belongs to $\mathcal W^{2-\frac{d}{2}-,q}$ for all $q\in[2,+\infty)$, where $\mathcal W^{s,q}$ spaces are defined in \cref{Sec-ConfiningSobolev}. Moreover, almost surely any solution $u$ of \cref{Eq-StationaryAGP} which belongs to $\mathrm{H}^{2-\frac{d}{2},q}(U)$ for some open set $U\subset\mathbb R^d$ and $q\in[2,+\infty]$ vanishes everywhere in $U$.
\end{theo}

In order to prove Theorem~\ref{Th-SynthesExistenceSW}, the text says they "use constrained minimization methods".

- In \cref{Sec-EnergyGS}, they "study the minimization of energy at fixed mass" (Theorem~\ref{Th-EnergyGS}). It is then stated that \cref{Prop-ExistenceEnergyGS,Prop-FrequencyEnergyGS} "show that, for $\lambda\leqslant 0$ or $\gamma<\frac{2}{d}$, energy minimization at fixed mass provide positive solutions of \cref{Eq-StationaryAGP} for any possible value of $\omega$".

- In \cref{Sec-ActionGS}, they instead "investigate the minimization of action $\mathcal{S}_\omega=\mathcal{E}+\omega\mathcal{M}$". In the focusing case ($\lambda>0$), they note that "one easily verifies the action is not lowerbounded", so they "introduce the Nehari manifold" and obtain an action minimizer on it (Theorem~\ref{Th-ACtionGS}) for $\omega>-\mu_0$.

- They also "stress out" that, in the presence of white noise, standing waves have "limited Sobolev regularity" (Theorem~\ref{Th-MaxRegSW}), giving the maximal regularity and a unique-continuation-type statement (if a solution lies in $\mathrm{H}^{2-\frac{d}{2},q}(U)$ on some open set $U$, then it "vanishes everywhere in $U$").

In order to prove the main theorem, the text says they "use constrained minimization methods".

Next, they "study the minimization of energy at fixed mass" and use the following result.

\begin{theo}\label{Th-EnergyGS}

    Let $d\in\{1,2\}$, $\lambda,\omega\in\R$, $\gamma>0$ and $A$ be the self-adjoint operator constructed in \cref{Sec-AndersonOp1d,Sec-AndersonOp2d} with $\xi$ a spatial white noise, $\D^{1,2}=\mathrm{D}(\sqrt{|A|})$ and $\D^{-1,2}$ its dual. For $u\in\D^{1,2}$, we define the mass as
    $$\mathcal{M}(u)=\frac{1}{2}\int_{\R^d}|u|^2\d x$$
    and the energy as
    $$\mathcal{E}(u)=\frac{1}{2}\langle -Au,u\rangle_{\D^{-1,2},\D^{1,2}}-\frac{\lambda}{2\gamma+2}\int_{\R^d}|u|^{2\gamma+2}\d x.$$
    Then, it holds
    \begin{enumerate}
        \item For $\lambda\leqslant0$, for any $m>0$, there exists $\phi_m\in\D^{1,2}$ such that $\phi_m>0$ and
        $$\argmin_{\substack{u\in\D^{1,2}\\\mathcal{M}(u)=m}}\mathcal{E}(u) = \left\{\e^{\complexI\theta}\phi_m,\; \theta\in\R\right\}.$$
        \item For $\lambda>0$ and $\gamma<\frac{2}{d}$, for any $m>0$, there exists $\phi_m\in\displaystyle\argmin_{u\in\D^{1,2},\;\mathcal{M}(u)=m}\mathcal{E}(u)$ such that $\phi_m>0$.
        \item For $\lambda>0$ and $\gamma=\frac{2}{d}$, there exists $m_{*,\lambda}(\Xi)>0$ such that for all $m\in(0,m_{*,\lambda}(\Xi))$, there exists $\phi_m\in\displaystyle\argmin_{u\in\D^{1,2},\;\mathcal{M}(u)=m}\mathcal{E}(u)$ such that $\phi_m>0$. Moreover, there exists $m_{*,\lambda}(\Xi)\leqslant m^*_\lambda(\Xi)\leqslant m^*_\lambda$ such that
        $$\forall m>m^*_\lambda(\Xi), \inf_{\substack{u\in\D^{1,2}\\\mathcal{M}(u)=m}}\mathcal{E}(u)=-\infty,$$
        where $m^*_\lambda$ is the critical mass of the classical nonlinear Schrdinger equation (i.e.\ without any potential).
        \item For $\lambda>0$ and $\gamma>\frac{2}{d}$, for any $m>0$,
        $$\inf_{\substack{u\in\D^{1,2}\\\mathcal{M}(u)=m}}\mathcal{E}(u)=-\infty.$$
    \end{enumerate}

\end{theo}

It is then stated that \cref{Prop-ExistenceEnergyGS,Prop-FrequencyEnergyGS} show that, for $\lambda\leqslant 0$ or $\gamma<\frac{2}{d}$, energy minimization at fixed mass provides positive solutions of
\begin{equation}
    -A\phi-\lambda|\phi|^{2\gamma}\phi+\omega \phi = 0. \label{Eq-StationaryAGP}
\end{equation}
for any possible value of $\omega$.

Later, they instead "investigate the minimization of action $\mathcal{S}_\omega=\mathcal{E}+\omega\mathcal{M}$". In the focusing case ($\lambda>0$), they note that "one easily verifies the action is not lowerbounded", so they "introduce the Nehari manifold" and obtain an action minimizer on it (Theorem~\ref{Th-ACtionGS}) for $\omega>-\mu_0$.

They also "stress out" that, in the presence of white noise, standing waves have "limited Sobolev regularity" (Theorem~\ref{Th-MaxRegSW}), giving the maximal regularity and a unique-continuation-type statement (if a solution lies in $\mathrm{H}^{2-\frac{d}{2},q}(U)$ on some open set $U$, then it "vanishes everywhere in $U$").

\label{Th-SynthesExistenceSW}
    Let $d\in\{1,2\}$, $\lambda,\omega\in\mathbb R$ and $\gamma>0$. Denote by $\mu_0$ the lowest eigenvalue of $-A$. In each of the following situation, there exists a positive solution of
\begin{equation}
    -A\phi-\lambda|\phi|^{2\gamma}\phi+\omega \phi = 0. \label{Eq-StationaryAGP}
\end{equation}
 in $\mathrm{D}\left(\sqrt{|A|}\right)$,
    \begin{itemize}
        \item $\lambda>0$ and $\omega>-\mu_0$,
        \item $\lambda = 0$ and $\omega=\mu_0$,
        \item $\lambda<0$ and $\omega<-\mu_0$.
    \end{itemize}
    In every other cases, there is no non-trivial non-negative solution in $\mathrm{D}\left(\sqrt{|A|}\right)$.,

Question:
Let \(d\in\{1,2\}\), let \(A=H+\xi\) be the Anderson Hamiltonian in dimension \(d\), and let \(\mu_0\) denote the lowest eigenvalue of \(-A\). For \(\lambda,\omega\in\mathbb R\) and \(\gamma>0\), consider the stationary equation
\[
-A\phi-\lambda|\phi|^{2\gamma}\phi+\omega\phi=0.
\]
Which statement about positive or nonnegative solutions \(\phi\in \mathrm D(\sqrt{|A|})\) is valid?

A positive solution \(\phi\in \mathrm D(\sqrt{|A|})\) exists in each of the following three cases: \(\lambda>0\) and \(\omega>-\mu_0\); \(\lambda=0\) and \(\omega=\mu_0\); \(\lambda<0\) and \(\omega<-\mu_0\). In every other case, there is no non-trivial non-negative solution in \(\mathrm D(\sqrt{|A|})\).

Choices:
B. A positive solution \(\phi\in \mathrm D(\sqrt{|A|})\) exists in each of the following three cases: \(\lambda>0\) and \(\omega\ge -\mu_0\); \(\lambda=0\) and \(\omega=\mu_0\); \(\lambda<0\) and \(\omega\le -\mu_0\). In every other case, there is no non-trivial non-negative solution in \(\mathrm D(\sqrt{|A|})\).
C. If \(\lambda>0\) and \(\omega>-\mu_0\), or if \(\lambda=0\) and \(\omega=\mu_0\), or if \(\lambda<0\) and \(\omega<-\mu_0\), then there exists a non-trivial non-negative solution \(\phi\in \mathrm D(\sqrt{|A|})\).
D. A positive solution \(\phi\in \mathrm D(\sqrt{|A|})\) exists whenever \(\lambda>0\) and \(\omega\in\mathbb R\), whenever \(\lambda=0\) and \(\omega=\mu_0\), and whenever \(\lambda<0\) and \(\omega< -\mu_0\). In every other case, there is no non-trivial non-negative solution in \(\mathrm D(\sqrt{|A|})\).
E. A positive solution \(\phi\in \mathrm D(\sqrt{|A|})\) exists in each of the following three cases: \(\lambda>0\) and \(\omega>-\mu_0\); \(\lambda=0\) and \(\omega\ge \mu_0\); \(\lambda<0\) and \(\omega<-\mu_0\). In every other case, there is no non-trivial non-negative solution in \(\mathrm D(\sqrt{|A|})\).

The stem gives the operator, equation, and parameter names, but it does not reveal the correct parameter regime or the full existence/nonexistence classification. There is no explicit answer leakage.

The item is close to asking for the theorem’s exact conclusion, so it is partly a theorem-recall question. However, the options differ in logical strength, boundary cases, and converse claims, so it is not a pure verbatim restatement.

Some reasoning is needed to distinguish strict vs. non-strict thresholds, sign-dependent regimes, and the stronger full classification from a merely weaker true statement. Still, the task is mainly recognition/recall of the correct theorem statement rather than substantial derivation.

The distractors are plausible and mathematically targeted: one mishandles boundary cases, one is a weaker true statement, one flips the sign regime, and one overgeneralizes the classification. These align well with common failure modes.

A solid MCQ with strong distractors and no answer leakage, but it mainly tests precise theorem recall/comparison rather than deep generative reasoning.


\label{thm:main}
    Let $\Lambda_\alpha$ be a Delone set associated with an $\alpha$-Kakutani tiling of $\R$. Then $\Lambda_\alpha$ is uniformly spread if and only if
    \begin{equation}\label{eq:special values}
            r_\alpha:=\frac{\log(\min\{\alpha,1-\alpha\})}{\log(1-\min\{\alpha,1-\alpha\})}\in\left\{1,\frac{3}{2},2,3,4  \right\}.
    \end{equation}

A set $\Lambda\subset\R^d$ is \emph{Delone} if it is \emph{uniformly discrete} and \emph{relatively dense}, that is, if there exist constants $0<r,R<\infty$ so that $B(x,r)\cap\Lambda=x$ and $B(y,R)\cap\Lambda\neq\emptyset$ for all $x\in\Lambda$ and $y\in \R^d$, respectively. clearly lattices and their translations are Delone sets in $\R^d$, and there are numerous non-lattice examples that are not periodic in any way. Given such a non-lattice Delone set, a natural question is to ask how ``close'' it is to a lattice in $\R^d$. One natural interpretation of such a question involves the concept of \emph{bounded displacement (BD) equivalence}, where two Delone sets $\Lambda,\Gamma\subset\R^d$ are BD equivalent if there exists a bijection $\varphi:\Lambda\rightarrow \Gamma$ satisfying $\sup_{x\in\Lambda}\|x-\varphi(x)\|<\infty$.
A Delone set $\Lambda\subset\R^d$ is \emph{uniformly spread} if it is BD equivalent to some lattice in $\R^d$, or equivalently, by a simple application of Hall's marriage theorem, if there is some $c>0$ for which $\Lambda$ and $c\cdot\Z^d$ are BD equivalent. See \cite{SmilanskySolomon-2025} for proofs and a comprehensive discussion.

The Delone sets considered in this manuscript are those associated with $\alpha$-Kakutani tilings for $\alpha\in(0,1)$, which form a simple class of multiscale substitution tilings \cite{SmilanskySolomon-2021}. These tilings are defined by a substitution rule on a closed interval $I\subset\R$ of unit length that splits $I$ into two subintervals, a left one of length $\alpha$ and a right one of length $1-\alpha$. Starting with a copy of $I$, which we think of as a \emph{tile}, we apply a uniform inflation and substitute any tile of length greater than $1$. This results in a growing family of patches that exhausts the real line, and limits taken in an appropriate way define a space of tilings of $\R$, all made of interval-tiles of lengths between $\min\{\alpha,1-\alpha\}$ and $1$. Given such a tiling, an \emph{associated} Delone set is obtained by selecting one point from each tile, with the requirements that distinct tiles yield distinct points and that the resulting set is uniformly discrete. A natural choice is to take the left endpoint of each tile. However, since all Delone sets associated with a fixed tiling are BD equivalent, the particular choice of points is immaterial.

Our main result is the following classification.

Our main result is the following classification.

\subsection{Commensurable tilings as primitive substitution tilings}\label{subsec:construction of rho_alpha} Recall our assumption that $\alpha=\min\{\alpha,1-\alpha\}\in(0,1/2]$, and consider a commensurable $\alpha$-Kakutani rule. Let $n\geq m>0$ be coprime integers so that
\begin{equation}\label{eq:rational ratio}
 \frac{n}{m}=r_\alpha=\frac{\log\alpha}{\log(1-\alpha)}.
\end{equation}
If $n=m$ then $\alpha=1/2$, and any $\alpha$-Kakutani tiling is a translation of a lattice tiling in $\R$, with all associated Delone sets uniformly spread. Assume from now on that $n> m$, and consider the associated graph $G_\alpha$. By \eqref{eq:rational ratio} we have
$$
g_\alpha :=\frac{1}{n}\log\frac{1}{\alpha}=\frac{1}{m}\log\frac{1}{1-\alpha}.
$$
Then by adding vertices to $G_\alpha$, the loop of length $\log(1/\alpha)$ and the loop of length $\log(1/(1-\alpha))$ can be partitioned into $n$ and $m$ edges of equal length $g_\alpha$, respectively. This defines a directed graph $G'_\alpha$ with $n+m-1$ vertices and $n+m$ edges of equal length. Label the original vertex with $1$, and assign labels $2,\ldots,n$ and $n+1,\ldots,n+m-1$ to the $(n-1)+(m-1)$ new vertices that are added along the two loops.

\begin{prop}\label{prop:generating patches}
 Consider a commensurable $\alpha$-Kakutani substitution rule with $r_\alpha=n/m$ for $n,m\in\N$ coprime. Let $\ell\in\Z_{\geq 0}$ and put
 \begin{equation}\label{eq:t_l}
 t_\ell:= \ell\cdot g_\alpha=\ell\cdot \frac{1}{n}\log\frac{1}{\alpha}.
 \end{equation}
 Then the (Kakutani) generating patch $F_{t_\ell}(I)$ and the (primitive substitution) generating patch $(\xi\rho_\alpha)^\ell(I)$ are geometrically the same, where $\xi=\alpha^{-1/n}$.
\end{prop}
\begin{proof}
 We prove by induction on $\ell\geq0$. For $\ell=0$, both patches contain only the single tile $I$ and so are geometrically the same. Assume the claim holds for some $\ell\geq0$. Tiles in $F_{t_\ell}(I)$ and $(\xi\rho_\alpha)^\ell(I)$ are either intervals of unit length, that is, copies of $I$, or intervals of length at most $|T_n|=|T_{n+m-1}|=\alpha^{1/n}$. By construction, the patch $F_{t_{\ell+1}}(I)$ is defined by inflating the patch $F_{t_\ell}(I)$ by $\alpha^{-1/n}$ and then substituting every tile of length strictly greater than $1$ according to the $\alpha$-Kakutani rule. The tiles that are substituted are exactly the unit intervals from $F_{t_\ell}(I)$. On the other hand, the tiles in $(\xi\rho_\alpha)^{\ell+1}(I)$ are the result of one application of $\xi\rho_\alpha$ on the tiles in $(\xi\rho_\alpha)^\ell(I)$. For copies of $I$ this amounts to an application of the $\alpha$-Kakutani rule once and inflation by $\xi=\alpha^{-1/n}$, while all other tiles in the patch are simply inflated by the same constant, and so $F_{t_{\ell+1}}(I)$ and $(\xi\rho_\alpha)^{\ell+1}(I)$ are the same.
\end{proof}

We focus here on Delone sets in $\R$.
For a detailed discussion about equivalence relations on Delone sets in $\R^d$ we refer to the book chapter \cite{SmilanskySolomon-2025}.
Let $|U|$ denote the Lebesgue measure of a bounded measurable set $U\subset \R$. A Delone set $\Lambda\subset\R$ has \emph{asymptotic density} $d_\Lambda$ if
$$
\lim_{n\rightarrow\infty}\frac{\#(\Lambda\cap U_n)}{|U_n|}
$$
exists and is equal to $d_\Lambda$ for any sequence \emph{van Hove sequence} $(U_n)$, that is, a sequence of bounded measurable sets so that $|(\partial U_n)^{+\varepsilon}|/|U_n|\rightarrow0$ for all $\varepsilon>0$, where $A^{+\varepsilon}$ is the $\varepsilon$ neighbourhood of $A\subset\R$. The \emph{discrepancy} of $\Lambda$ with respect to the parameter $\beta>0$ and a measurable set $B\subset\R$ is defined as
$$
\disc{\Lambda}{\beta}{U}:=\left|\#(\Lambda \cap U)-\beta\cdot|U|\right|.
$$
A criterion established by Laczkovich \cite{Laczkovich-1992} provides a necessary and sufficient condition for a Delone set in $\R^d$ to be uniformly spread. In the case of a Delone set $\Lambda\subset \R$ with asymptotic density $d_\Lambda$, Laczkovich's criterion can be stated in the following way.
\begin{thm}\label{thm:Laczkovich}
 A Delone set $\Lambda\subset \R$ with asymptotic density $d_\Lambda$ is uniformly spread if and only if
 $$
 \disc{\Lambda}{d_\Lambda}{U}\leq C\cdot|\partial U|^{+1}
 $$
 for all bounded measurable sets $U\subset\R$.
\end{thm}

\begin{thm}\label{thm:BD_substitution}
 Let $\rho$ be a primitive substitution rule in $\R^d$ on a set of prototiles that are biLipschitz homeomorphic to closed balls, and let $\Lambda$ be a Delone set associated with a tiling generated by $\rho$. Let $\ell\ge 2$ be the minimal index such that the corresponding eigenspace satisfies $W_{\lambda_\ell} \nsubseteq \mathbf{1}^\perp$, where $\mathbf{1}=(1,\ldots,1)\in \R^k$. Then
\begin{enumerate}
\item
If $\absolute{\lambda_\ell}<\lambda_1^{\frac{d-1}{d}}$ then $\Lambda$ is uniformly spread.
\item
If $\absolute{\lambda_\ell}>\lambda_1^{\frac{d-1}{d}}$ then $\Lambda$ is not uniformly spread.
\end{enumerate}
\end{thm}
We note that if $\absolute{\lambda_\ell}=\lambda_1^{\frac{d-1}{d}}$ then $\Lambda$ may exhibit either property, for an example see \cite{FrettlohSmilanskySolomon-2021}.
In the case $d=1$, checking the conditions in Theorem \ref{thm:BD_substitution} is reduced to determining if $\lambda_\ell$ is in the interior of the closed unit disk or in its complement.

The discrepancy can be large however, and by \cite[\S 6]{SmilanskySolomon-2025} there exist arbitrarily large intervals $U$ for which
\begin{equation}\label{eq:incomm_discrepancy}
 \disc{\Lambda_\alpha}{d_{\Lambda_\alpha}}{U}=\Omega\left(\frac{|U|}{\log(|U|)}\right).
\end{equation}
By Laczkovich's criterion as stated in Theorem \ref{thm:Laczkovich}, if $\Lambda_\alpha$ is uniformly spread then there exists a constant $C>0$ so that for any interval $U$ the discrepancy is bounded by $\disc{\Lambda_\alpha}{d_{\Lambda_\alpha}}{U}\leq 2C$. By \eqref{eq:incomm_discrepancy} this is clearly not the case, and so $\Lambda_\alpha$ is not uniformly spread. In fact, for any fixed incommensurable $\alpha$-Kakutani rule there are continuously many associated Delone sets that are pairwise BD non-equivalent, see \cite{SmilanskySolomon1-2022}.

Assume now that $r_\alpha=n/m$ with $n>m>0$ coprime integers. By Proposition \ref{prop:primitive cover}, any $\alpha$-Kakutani tiling is also generated by a primitive substitution rule $\rho_\alpha$ with substitution matrix $M_\alpha$ as in \eqref{eq:substitution matrix}. All tiles are closed interval and are therefore closed balls in $\R$, and so we can apply Solomon's criterion for uniform spreadness, stated above as Theorem \ref{thm:BD_substitution}, with $d=1$. By
Proposition \ref{prop:eigenspacesnotperp} and the preceding discussion, the eigenvalue $\lambda_2$ of $M_\alpha$ is non-zero and satisfies $W_{\lambda_2}\nsubseteq{\bf 1}^\perp$, and so $\ell=2$. Since $\gcd(n-m,n)=\gcd(m,n)=1$, Proposition \ref{prop:nounitroots} applies to $f_\alpha(x)$ as defined in \eqref{eq:charpoly}, and so $|\lambda_2|\neq1$. Combining the above, we deduce that an associated Delone set $\Lambda_\alpha$ is uniformly spread if and only if $|\lambda_2|<1$. By Proposition \ref{prop:PVpolynomials}, for this to hold $f_\alpha(x)$ must be one of the four polynomials listed in \eqref{eq:PVpoly}. We thus conclude that the values of $\alpha\in(0,1/2)$ for which $\Lambda_\alpha$ is uniformly spread are given by
\begin{equation}\label{eq:PVvalues}
 r_\alpha=\frac{n}{m}\in\left\{\frac32,\frac21,\frac31,\frac41\right\},
\end{equation}
and the proof is complete.
\end{proof}

For simplicity of presentation, we will assume throughout that $\alpha\in(0,1/2]$, and so $\alpha=\min\{\alpha,1-\alpha\}$. We note that if $r_\alpha=1$ then $\alpha=1/2$, and the left endpoints of tiles in any corresponding tiling are simply translated lattices in $\R$. When $r_\alpha=2$ then $\alpha=1/\phi^2$, where $\phi$ is the \emph{golden ratio}. The other values of $\alpha$ are harder to present explicitly, though the cases $r_\alpha=3/2$ and $r_\alpha=3$ are related to the \emph{supergolden ratio} and the \emph{plastic ratio}, respectively. The latter is the smallest \emph{Pisot–Vijayaraghavan (PV) number}, and the case $r_\alpha=4$ is related to the second smallest PV number, see \S \ref{sec:remarks}. For all other values of $\alpha$ the corresponding Delone sets are never uniformly spread.

\subsection{Key steps of the proof}

A complete proof requires some further preparation, but  we can aready outline the main steps of the argument. First, in the \emph{incommensurable} case $r_\alpha\notin\Q$, the discrepancy in $\alpha$-Kakutani tilings, that is, the fluctuation of the number of tiles that appear in large patches relative to their expected number (namely, the density), can be shown to be large. This, when combined with Laczkovich's criterion for uniform spreadness \cite{Laczkovich-1992}, is enough to establish that any associated Delone set is not BD equivalent to a lattice. This is a special case of \cite[Theorem 8.2]{SmilanskySolomon-2021}, which states that any Delone set associated with an incommensurable multiscale substitution tiling in $\R^d$ is not uniformly spread.

The case $r_\alpha\in\Q$, namely the \emph{commensurable} case, is rather different, and is the new result contained in this contribution.
The first step is to observe that, by introducing labelled prototiles and modifying the substitution rule in a suitable way, one can construct a standard primitive substitution that ``covers'' the commensurable multiscale substitution construction, in the sense that it generates substitution tilings that are geometrically identical to the corresponding $\alpha$-Kakutani tilings described above.
 The problem of uniform spreadness of Delone set associated with primitive substitution tilings can be approached using Solomon's criterion \cite{Solomon-2014}. This criterion, which is a careful application of the aforementioned Laczkovich criterion, is given in terms of a comparison between the leading Perron-Frobenius eigenvalue of the associated substitution matrix, and the subsequent eigenvalues. In the special case considered here, this becomes a question of classification of certain \emph{Pisot–Vijayaraghavan (PV) polynomials} \cite{DubickasJankauskas-2014}, from which the special values that appear in \eqref{eq:special values} can be extracted.

We complete our discussion by showing that such scarcity of uniformly spread constructions does not necessarily carry over when moving away from $\alpha$-Kakutani tilings to more complicated multiscale substitution tilings. For example, when considering $(\alpha,\beta)$-Kakutani tilings, generated by a substitution rule on $I$, now split into three tiles of lengths $\alpha,\beta,1-\alpha-\beta$, we present a countable set of parameters $(\alpha,\beta)$ for which the associated Delone sets are uniformly spread.

A complete proof requires some further preparation, but the post-theorem introduction outlines these main steps.

1. **Assume** $\alpha\in(0,1/2]$, so $\alpha=\min\{\alpha,1-\alpha\}$.

2. **Incommensurable case** ($r_\alpha\notin\Q$): show that the *discrepancy* in $\alpha$-Kakutani tilings—“the fluctuation of the number of tiles that appear in large patches relative to their expected number (namely, the density)”—“can be shown to be large.” Combine this with “Laczkovich's criterion for uniform spreadness” to conclude that any associated Delone set “is not BD equivalent to a lattice,” hence is not uniformly spread. This is noted as a special case of \cite[Theorem 8.2]{SmilanskySolomon-2021}.

3. **Commensurable case** ($r_\alpha\in\Q$): this is described as “rather different” and the “new result.”
   - **Cover by a standard primitive substitution**: “by introducing labelled prototiles and modifying the substitution rule in a suitable way,” construct “a standard primitive substitution that ‘covers’ the commensurable multiscale substitution construction,” meaning it generates substitution tilings “geometrically identical” to the $\alpha$-Kakutani tilings.
   - **Apply Solomon's criterion**: study uniform spreadness for the resulting primitive substitution tilings via “Solomon's criterion,” formulated as “a comparison between the leading Perron-Frobenius eigenvalue of the associated substitution matrix, and the subsequent eigenvalues.”
   - **Reduce to PV polynomial classification**: “In the special case considered here, this becomes a question of classification of certain Pisot–Vijayaraghavan (PV) polynomials,” and “from [this] the special values that appear in \eqref{eq:special values} can be extracted,” yielding exactly the listed set $\left\{1,\frac{3}{2},2,3,4\right\}$.

4. **Further remark beyond the theorem**: they add that this “scarcity” need not persist for more general multiscale substitution tilings (e.g. $(\alpha,\beta)$-Kakutani tilings), where they “present a countable set of parameters $(\alpha,\beta)$ for which the associated Delone sets are uniformly spread.”

A complete proof requires some further preparation, but the post-theorem introduction outlines these main steps.

1. **Assume** $\alpha\in(0,1/2]$, so $\alpha=\min\{\alpha,1-\alpha\}$.

2. **Incommensurable case** ($r_\alpha\notin\Q$): show that the *discrepancy* in $\alpha$-Kakutani tilings—“the fluctuation of the number of tiles that appear in large patches relative to their expected number (namely, the density)”—“can be shown to be large.” Combine this with “Laczkovich's criterion for uniform spreadness” to conclude that any associated Delone set “is not BD equivalent to a lattice,” hence is not uniformly spread. This is noted as a special case of Smilansky–Solomon, *[title and year not provided]*, Theorem 8.2.

3. **Commensurable case** ($r_\alpha\in\Q$): this is described as “rather different” and the “new result.”
   - **Cover by a standard primitive substitution**: “by introducing labelled prototiles and modifying the substitution rule in a suitable way,” construct “a standard primitive substitution that ‘covers’ the commensurable multiscale substitution construction,” meaning it generates substitution tilings “geometrically identical” to the $\alpha$-Kakutani tilings.
   - **Apply Solomon's criterion**: study uniform spreadness for the resulting primitive substitution tilings via “Solomon's criterion,” formulated as “a comparison between the leading Perron-Frobenius eigenvalue of the associated substitution matrix, and the subsequent eigenvalues.”
   - **Reduce to PV polynomial classification**: “In the special case considered here, this becomes a question of classification of certain Pisot–Vijayaraghavan (PV) polynomials,” and “from [this] the special values that appear in
\begin{equation}\label{eq:special values}
            r_\alpha:=\frac{\log(\min\{\alpha,1-\alpha\})}{\log(1-\min\{\alpha,1-\alpha\})}\in\left\{1,\frac{3}{2},2,3,4  \right\}.
    \end{equation}
can be extracted,” yielding exactly the listed set $\left\{1,\frac{3}{2},2,3,4\right\}$.

4. **Further remark beyond the theorem**: they add that this “scarcity” need not persist for more general multiscale substitution tilings (e.g. $(\alpha,\beta)$-Kakutani tilings), where they “present a countable set of parameters $(\alpha,\beta)$ for which the associated Delone sets are uniformly spread.”

Question:
Let \(\alpha\in(0,1)\), and let \(\Lambda_\alpha\) be a Delone set associated with an \(\alpha\)-Kakutani tiling of \(\mathbb{R}\) (for example, by choosing one point from each tile so that the resulting set is uniformly discrete). Say that \(\Lambda_\alpha\) is uniformly spread if it is bounded-displacement equivalent to a lattice in \(\mathbb{R}\). Define
\[
r_\alpha:=\frac{\log(\min\{\alpha,1-\alpha\})}{\log(1-\min\{\alpha,1-\alpha\})}.
\]
Which condition on \(r_\alpha\) is exactly equivalent to \(\Lambda_\alpha\) being uniformly spread?

\(\Lambda_\alpha\) is uniformly spread if and only if \(r_\alpha\in\left\{1,\tfrac{3}{2},2,3,4\right\}.\)

Choices:
B. \(\Lambda_\alpha\) is uniformly spread if and only if \(r_\alpha\in\mathbb{Q}.\)
C. If \(\Lambda_\alpha\) is uniformly spread, then \(r_\alpha\in\mathbb{Q}.\)
D. \(\Lambda_\alpha\) is uniformly spread if and only if \(r_\alpha\in\left\{\tfrac{3}{2},2,3,4\right\}.\)
E. \(\Lambda_\alpha\) is uniformly spread if and only if \(r_\alpha\in\left\{1,\tfrac{3}{2},2,3\right\}.\)

PV-classification refinement from rational to finite exceptional set

replace exact finite classification by necessary commensurability only

top endpoint value 4 in the extracted special set

The stem does not reveal the finite exceptional set or otherwise point directly to the correct option; it only defines the quantity and asks for the exact criterion.

This is essentially a direct recall of a classification theorem: the correct choice states the exact equivalence being asked for, with little reformulation beyond theorem-to-MCQ conversion.

Some discrimination is needed among nearby alternatives (all rationals, one-way implication, omitted exceptional values), but success depends mainly on recall of the theorem rather than substantial reasoning or synthesis.

The distractors are plausible and mathematically targeted: they reflect common errors such as overgeneralizing to all rationals, weakening an iff to a necessary condition, or missing boundary exceptional values.

A solid recall-based MCQ with no answer leakage and strong distractors, but it is largely a direct theorem restatement and does not strongly test generative reasoning.