Level sets of continuous functions defined on some domain $F\subseteq \mathbb{R}^p$ convey information about the dimensionality of $F$. Roughly speaking, a generic continuous function should have large level sets
if $F$ itself is thick in an intuitive sense, and small otherwise. This heuristic prompted the investigation of the size of such level sets in terms of Hausdorff dimension
(see \cite{BBElevel, BBEtoph, BK}), with the quantity providing the answer (the topological Hausdorff dimension) attracting interest in its own right \cite{MaZha}, even in physics
\cite{Balankintoph, Balankinfracspace, Balankintransport, Balankintoph2, Balankinfluid}.

The authors with Gáspár Vértesy initiated the study of the Hausdorff and box dimension of level sets of generic 1-Hölder-$\alpha$ functions \cite{ BuczolichMagaVertesy2025, sierc, sier, BUCZOLICH2024531}.
Under self-similarity conditions on the domain $F$, they proved that for every $\alpha$ and for either of the Hausdorff/lower box/upper box dimension, there is a constant which is the dimension
of Lebesgue almost every level set of the generic 1-Hölder-$\alpha$ function. For any underlying dimension, these values form an interesting spectrum describing the geometry of $F$.
However, dealing with this spectrum is far from trivial in any of the cases: the existing literature focuses on particular choices of $F$ and provides mostly estimates at the price of
a significant amount of technicalities, highly depending on the structure of $F$. In this direction, in \cite{BuczolichMagaVertesy2025},
we managed to provide lower and upper bounds on the Hausdorff spectrum for the Sierpiński triangle which asymptotically coincide as $\alpha\to 0+$.
However, no explicit formulae or general machinery have been found before for describing any of the spectra for a larger class of fractals. In this respect this paper is the first one.

\subsection{The main result}

We give an explicit formula for the upper box dimension $\overline{\dim}_B$ of Lebesgue almost every level set of a generic 1-Hölder-$\alpha$ function
valid for connected self-similar sets subject to the geometric condition of having {\it finitely many directions}, i.e.,
whose defining similarities have orthogonal parts generating a finite subgroup of the orthogonal group.

\begin{abstract}
 In the previous decades, the size of level sets of functions have been extensively studied in various setups involving different regularity properties and
 size notions. In the case of Hölder functions, the authors have provided various bounds, but to date no explicit formulae have been found for any studied dimension
 and the results were valid only about very specific fractals.
 In this paper, for the first time, we have a result valid for a large class of self-similar sets, namely we prove that for these
 fractals Lebesgue almost every level set of the generic 1-Hölder-$\alpha$ function defined on $F\subseteq \mathbb{R}^p$ has
 upper box dimension $\dim_H F - \alpha$.
\end{abstract}

We give an explicit formula for the upper box dimension $\overline{\dim}_B$ of Lebesgue almost every level set of a generic 1-Hölder-$\alpha$ function
valid for connected self-similar sets subject to the geometric condition of having {\it finitely many directions}, i.e.,
whose defining similarities have orthogonal parts generating a finite subgroup of the orthogonal group.

\subsection{Organization of the paper}

Given a function $f:F\to \mathbb{R}$, we put $D_{\overline{B}}^f(r,F)=\overline{\dim}_B f^{-1}(r)$. For $f:F\to \mathbb{R}$, we denote by $D_{\overline{B}*}^f(F)$ the essential infimum
of the upper box dimension of non-empty level sets, i.e.,
\begin{displaymath}
 D_{\overline{B}*}^f(F)= \begin{cases}
 \inf\{d: \lambda\{r : r\in f(F) \text{ and } D_{\overline{B}}^f(r,F)\leq{d}\}>0\}, & \text{if } \lambda(f(F))>0, \\
 0, & \text{if } \lambda(f(F))=0.
 \end{cases}
\end{displaymath}
We denote by $\mg_{1,\aaa}(F)$, or by simply $\mg_{1,\aaa}$ the set of dense $G_{\ddd}$
sets in $C_{1}^{\aaa}(F)$. Whether $D_{\overline{B}*}^f(F)$ has a generic value is a priori unclear, nevertheless the following quantity is always well-defined:
\begin{equation}\label{def:generic_upper}
 D_{\overline{B}*}(\aaa,F)=\inf_{\cag\in \mg_{1,\aaa}}\sup\{ D_{\overline{B}*}^{f}(F):f\in \cag \}.
 \end{equation}
As established by \cite[Theorem~3.8]{BUCZOLICH2024531}, subject to certain conditions on the connectivity properties of $F$, $D_{\overline{B}*}(\aaa,F)$ is indeed simply the generic value of $D_{\overline{B}^*}^f(F)$.
In Section \ref{sec:box_dimension_locally_connected} we cite these results and discuss how they extend to locally connected $F$.
We note that even if $D_{\overline{B}*}(\aaa,F)$ is the generic value of $D_{\overline{B}^*}^f(F)$, it does not necessarily coincide with $D_{\overline{B}}^f(r,F)$ for almost every $r\in f(F)$.
In particular it is not clear a priori why in the setup of Theorem \ref{thm:gen_upper_box_precise} almost every level set of the generic 1-Hölder-$\alpha$ should have the same upper box dimension.

\begin{lemma}[{\cite[Lemma~3.9]{BUCZOLICH2024531}}] \label{lemma:from_covering_to_box}
 Assume that $0<\alpha\leq 1$, and the measurable set $F\subseteq \mathbb{R}^p$ has coverings $(\mathcal{S}_n)_{n=1}^{\infty}$,
 such that with some constants $C,l,\rho>1$:
 \begin{itemize}
 \item the cardinality of $\mathcal{S}_n$ is at most $Cl^n$ for some $C,l>1$,
 \item if $S\in \mathcal{S}_n$, then $\diam(S)\leq C\rho^{-n}$.
 \end{itemize}
 Then for any $f\in C_{\alpha}^{1}(F)$ and almost every $r\in \mathbb{R}$, we have
 $\overline{\dim}_B(f^{-1}(r))\leq \frac{\log l}{\log \rho} - \alpha$. In particular,
 $D_{\overline{B}*}(\alpha, F)\leq \frac{\log l}{\log \rho} - \alpha$.
\end{lemma}

\begin{corollary} \label{corollary:lower_box_dim_upper_bound}
 Assume that $0<\alpha\leq 1$, and let $F\subseteq \mathbb{R}^p$ be measurable. Then for every $f\in C_{\alpha}^{1}(F)$ and almost every $r\in \mathbb{R}$, we have
 $\overline{\dim}_B(f^{-1}(r))\leq \underline{\dim}_B(F) - \alpha$. In particular,
 $D_{\overline{B}*}(\alpha, F)\leq \underline{\dim}_B(F) - \alpha$.
\end{corollary}

\begin{theorem}[{\cite[Corollary~1]{wenxi}}] \label{prop:wenxi}
 Assume that $A\subseteq \mathbb{R}^p$ is a self-similar set with finitely many directions and $\dim_H A>1$.
 Then for a full measure set $\mathcal{W}_A$ of the $(p-1)$-dimensional linear subspaces, if $W\in \mathcal{W}_A$, then
 $$\mathcal{H}^1(\mathrm{Pr}_{W^{\perp}}(A)\setminus \{a\in W^{\perp}: \dim (A\cap (W+a))= \dim_H A -1\}) = 0,$$
 that is standard dimension drop occurs on a set of full measure. (Here $W^{\perp}$ denotes the 1-dimensional orthocomplement of $W$,
 and $\mathrm{Pr}_{W^{\perp}}$ is the projection onto it.)
\end{theorem}

\begin{lemma} \label{lemma:values_admitted_in_simplices}
 Let $F\subseteq \mathbb{R}^p$ be compact and $\mathcal{S}$ be a finite family of simplices covering $F$ such that $\inte (S) \cap F\neq \emptyset$ for all $S\in \mathcal{S}$.
 Assume moreover that $f:\bigcup_{S\in\mathcal{S}} S \to \mathbb{R}$ is a piecewise affine function, affine on members of $\mathcal{S}$.
 Then for almost every $u\in\mathbb{R}^p$, we have
 \begin{equation} \label{eq:values_admitted_in_interior}
 \bigcup_{S\in\mathcal{S}} f(S\cap (F+u)) = \bigcup_{S\in\mathcal{S}} f\left(\inte S\cap (F+u)\right) \text{ a. e.}
 \end{equation}
 i.e., for almost every $u$ almost every value admitted on $F+u$ by $f$ is admitted in the interior of a simplex.
\end{lemma}

\begin{theorem} \label{thm:gen_upper_box_precise}
    Assume that $F\subseteq \mathbb{R}^p$ is a connected self-similar set with finitely many directions
    and $\dim_H F = s > 1$.
    Then for the generic 1-Hölder-$\alpha$ function $f:F\to \mathbb{R}$ and Lebesgue almost every $r\in f(F)$,
    \begin{displaymath}
    \overline{\dim}_B f^{-1}(r)=s-\alpha.
    \end{displaymath}
\end{theorem}

\subsection{Organization of the paper}

In Section \ref{sec:prelim} we introduce some notation and recall some relevant tools.

In Section \ref{sec:box_dimension_locally_connected} we provide a lemma on how a dense set of functions over which the upper box dimension of level sets is controlled can be used
to get bounds over a dense $G_\delta$ set of functions. Such a lemma was previously provided by \cite{BUCZOLICH2024531} assuming a more elaborate connectivity property,
this time it will be sufficient to assume local connectivity.

In Section \ref{sec:lemmas} we prove two key lemmas, which do the heavy lifting for Section \ref{sec:main}, in which we prove Theorem \ref{thm:gen_upper_box_precise}.
We note that in \cite[Theorem~3.6]{BUCZOLICH2024531}, we proved a lower bound for the upper box dimension of level sets under certain conditions using Theorem \ref{prop:wenxi} (identical to
\cite[Proposition~2.6]{BUCZOLICH2024531}). A subtle point in applying this statement was not addressed explicitly there.
Lemma \ref{lemma:values_admitted_in_simplices} fills this important geometric gap, essentially stating
that if we have a piecewise affine function, defined on simplices containing our fractal $F$ then with an arbitrarily small translation
we can make sure that the function takes Lebesgue almost all of its values in the interior of the simplices. This fact is quite easy if the Hausdorff dimension of $F$ is less than $2$,
for higher dimensions it needs some extra care.

We conclude the paper with some open problems.

In Section \ref{sec:lemmas} the authors “prove two key lemmas, which do the heavy lifting for Section \ref{sec:main}, in which we prove Theorem \ref{thm:gen_upper_box_precise}.” They also note that a “subtle point in applying” a previously used statement (Theorem \ref{prop:wenxi}) “was not addressed explicitly” earlier, and that Lemma \ref{lemma:values_admitted_in_simplices} “fills this important geometric gap,” namely: for a piecewise affine function on simplices containing the fractal $F$, “with an arbitrarily small translation we can make sure that the function takes Lebesgue almost all of its values in the interior of the simplices.” This geometric fact is said to be “quite easy if the Hausdorff dimension of $F$ is less than $2$,” but “for higher dimensions it needs some extra care.” Additionally, Section \ref{sec:box_dimension_locally_connected} provides a lemma showing how “a dense set of functions over which the upper box dimension of level sets is controlled can be used to get bounds over a dense $G_\delta$ set of functions,” now assuming local connectivity.

Next the authors prove two key lemmas, which do the heavy lifting for the later section where they establish the main theorem. They also note that a subtle point in applying the following previously used statement was not addressed explicitly earlier.

\begin{theorem}[{\cite[Corollary~1]{wenxi}}] \label{prop:wenxi}
    Assume that $A\subseteq \mathbb{R}^p$ is a self-similar set with finitely many directions and $\dim_H A>1$.
    Then for a full measure set $\mathcal{W}_A$ of the $(p-1)$-dimensional linear subspaces, if $W\in \mathcal{W}_A$, then
    $$\mathcal{H}^1(\mathrm{Pr}_{W^{\perp}}(A)\setminus \{a\in W^{\perp}: \dim (A\cap (W+a))= \dim_H A  -1\}) = 0,$$
    that is standard dimension drop occurs on a set of full measure. (Here $W^{\perp}$ denotes the 1-dimensional orthocomplement of $W$,
    and $\mathrm{Pr}_{W^{\perp}}$ is the projection onto it.)
\end{theorem}

They explain that the following lemma fills this important geometric gap, namely: for a piecewise affine function on simplices containing the fractal $F$, with an arbitrarily small translation we can make sure that the function takes Lebesgue almost all of its values in the interior of the simplices.

\begin{lemma} \label{lemma:values_admitted_in_simplices}
    Let $F\subseteq \mathbb{R}^p$ be compact and $\mathcal{S}$ be a finite family of simplices covering $F$ such that $\inte (S) \cap F\neq \emptyset$ for all $S\in \mathcal{S}$.
    Assume moreover that $f:\bigcup_{S\in\mathcal{S}} S \to \mathbb{R}$ is a piecewise affine function, affine on members of $\mathcal{S}$.
    Then for almost every $u\in\mathbb{R}^p$, we have
    \begin{equation} \label{eq:values_admitted_in_interior}
    \bigcup_{S\in\mathcal{S}} f(S\cap (F+u)) = \bigcup_{S\in\mathcal{S}} f\left(\inte S\cap (F+u)\right) \text{ a. e.}
    \end{equation}
    i.e., for almost every $u$ almost every value admitted on $F+u$ by $f$ is admitted in the interior of a simplex.
\end{lemma}

This geometric fact is said to be quite easy if the Hausdorff dimension of $F$ is less than $2$, but for higher dimensions it needs some extra care. Additionally, later the authors provide a lemma showing how a dense set of functions over which the upper box dimension of level sets is controlled can be used to get bounds over a dense $G_\delta$ set of functions, now assuming local connectivity.

Question: Fix $0<\alpha\le 1$. Let $F\subseteq \mathbb{R}^p$ be a connected self-similar set with finitely many directions, meaning that in some defining iterated function system for $F$, the orthogonal parts of the similarities generate a finite subgroup of the orthogonal group, and assume that $\dim_H F=s>1$. Let $C_\alpha^1(F)$ denote the space of 1-H\"older-$\alpha$ functions $f:F\to\mathbb{R}$, i.e. functions satisfying $|f(x)-f(y)|\le |x-y|^\alpha$ for all $x,y\in F$. Which statement holds for generic $f\in C_\alpha^1(F)$ (that is, for all $f$ in a dense $G_\delta$ subset of $C_\alpha^1(F)$) concerning the level sets $f^{-1}(r)$ for Lebesgue almost every $r\in f(F)$?

For Lebesgue almost every $r\in f(F)$, the upper box dimension of the level set satisfies
$$\overline{\dim}_B f^{-1}(r)=s-\alpha.$$

Choices:
B.
For Lebesgue almost every $r\in f(F)$, the upper box dimension of the level set satisfies
$$\overline{\dim}_B f^{-1}(r)=s-1.$$
C.
For Lebesgue almost every $r\in f(F)$, the upper box dimension of the level set satisfies
$$\overline{\dim}_B f^{-1}(r)\le s-\alpha.$$
D.
For every generic $f\in C_\alpha^1(F)$, there is a full measure set $R_f\subseteq f(F)$ such that for every $r\in R_f$,
$$\dim_H f^{-1}(r)=s-\alpha.$$
E.
There exists a dense $G_\delta$ set $\mathcal G\subseteq C_\alpha^1(F)$ such that for every $f\in \mathcal G$ there exists a full measure set $R_f\subseteq f(F)$ with
$$\overline{\dim}_B f^{-1}(r)\ge s-\alpha$$
for all $r\in R_f$.

dimension-drop amount replaced by fixed codimension one

dropped the lower-bound/equality conclusion, keeping only the upper bound

upper box dimension replaced by Hausdorff dimension

retained only the lower-bound side coming from interior-value construction and omitted the matching upper-bound mechanism

The stem does not reveal the exact conclusion. It specifies the hypotheses and asks which quantitative level-set statement is valid, but it does not explicitly give the equality, quantifiers, or generic/almost-every distinctions needed to identify the correct choice.

The item is close to a theorem-recall question: the stem lists the hypotheses and the options are variants of the theorem’s conclusion. However, it is not purely tautological, since the choices differ in dimension formula, equality vs. inequality, and quantifier strength.

Some reasoning is required to choose between nearby claims, especially the weaker true upper bound, the strengthened quantifiers, and the incorrect replacement of alpha by 1. Still, the question mainly tests recognition of the precise theorem statement rather than deeper derivation.

The distractors are strong and mathematically meaningful. They reflect common failure modes: confusing s-alpha with s-1, accepting a weaker bound instead of the sharp equality, and overlooking the importance of 'generic' and 'almost every r' quantifiers.

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


Question: Fix $0<\alpha\le 1$. Let $F\subseteq \mathbb{R}^p$ be a connected self-similar set with finitely many directions, meaning that in some defining iterated function system for $F$, the orthogonal parts of the similarities generate a finite subgroup of the orthogonal group, and assume that $\dim_H F=s>1$. Let $C_\alpha^1(F)$ denote the space of 1-H\"older-$\alpha$ functions $f:F\to\mathbb{R}$, i.e. functions satisfying $|f(x)-f(y)|\le |x-y|^\alpha$ for all $x,y\in F$. Which statement holds for generic $f\in C_\alpha^1(F)$ (that is, for all $f$ in a dense $G_\delta$ subset of $C_\alpha^1(F)$) concerning the level sets $f^{-1}(r)$ for Lebesgue almost every $r\in f(F)$?

Choices:
B.
For Lebesgue almost every $r\in f(F)$, the upper box dimension of the level set satisfies
$$\overline{\dim}_B f^{-1}(r)=s-1.$$
C.
For Lebesgue almost every $r\in f(F)$, the upper box dimension of the level set satisfies
$$\overline{\dim}_B f^{-1}(r)\le s-\alpha.$$
D.
For every generic $f\in C_\alpha^1(F)$, there is a full measure set $R_f\subseteq f(F)$ such that for every $r\in R_f$,
$$\dim_H f^{-1}(r)=s-\alpha.$$
E.
There exists a dense $G_\delta$ set $\mathcal G\subseteq C_\alpha^1(F)$ such that for every $f\in \mathcal G$ there exists a full measure set $R_f\subseteq f(F)$ with
$$\overline{\dim}_B f^{-1}(r)\ge s-\alpha$$
for all $r\in R_f$.

The stem does not reveal the conclusion explicitly or by an obvious cue. It states the hypotheses and asks which of several nearby level-set statements is true.

This is essentially a direct theorem-recall item: the stem lists the theorem hypotheses and asks for its conclusion. The correct option is the theorem statement almost verbatim.

There is some pressure to distinguish equality from weaker bounds and to separate upper box dimension from Hausdorff dimension, but the task is still mainly recognition/recall rather than substantial derivation.

The distractors are mathematically plausible and target common confusions: codimension-1 heuristics, weakening equality to an upper bound, swapping upper box for Hausdorff dimension, and keeping only one side of the estimate.

A technically well-constructed theorem-identification MCQ with strong distractors and no leakage, but it is largely a direct restatement of a known result rather than a genuinely generative reasoning question.


\label{thm:main}
Let $\tau^{\phi}$ be a Teichm\"{u}ller disk in $ \Tei_{2,0}$.
    The Teichm\"uller metric and Carath\'{e}odory metric agree on $\tau^{\phi}$  if and only if  $\phi$ is a quadratic differential all of whose zeros are of even order.

Let $\Tei_{g,n}$ be the Teichm\"{u}ller space of Riemann surfaces of genus $g$ with $n$ marked points.
Throughout this paper, we assume that $3g-3+n\geq 2$.
The Teichm\"uller space $\Tei_{g,n}$ is the (orbifold) universal cover of the moduli space of Riemann surfaces
 $\mathcal{M}_{g,n}$ and  is naturally a complex manifold of dimension $3g-3+n$. It is known that
 $\Tei_{g,n}$ can be realized as a bounded domain in $\mathbb{C}^{3g-3+n}$, by the Bers
 embedding \cite{Bers}.

Let $\mathbb{H}$ be the upper half-plane, equipped with the Poincar\'e metric $$d_{\mathbb{H}}= \frac{|dz|}{2 \operatorname{Im} z}.$$
The \emph{Kobayashi metric} $d_{\mathcal{K}}$ on $\Tei_{g,n}$ is the largest metric so that every holomorphic map $$f: \mathbb{H}\rightarrow\Tei_{g,n}$$ is nonexpanding.
Royden \cite{Royden1969} proved that the Kobayashi metric on $\Tei_{g,n}$ coincides with the Teichm\"{u}ller metric $d_{\mathcal{T}}$.
Furthermore,  the Teichm\"uller metric is not homogeneous at any point and its isometry group is essentially the mapping class group.

Another natural metric on $\Tei_{g,n}$ satisfying the Schwarz-Pick inequality is the \emph{Carath\'{e}odory metric}, which can be defined as the smallest metric so that every holomorphic map $$F:  \Tei_ {g,n} \rightarrow \mathbb{H}$$ is nonexpanding.
Let $d_\mathcal{C}$ denote the \emph{Carath\'{e}odory metric} on $\Tei_{g,n}$.
 By the Schwarz Lemma, the inequality
$$d_\mathcal{C} \leq d_\mathcal{T}$$ holds  on $\Tei_{g,n}$.

A longstanding open  problem is whether the two metrics $d_\mathcal{T}$ and $d_\mathcal{C}$ agree on $\Tei_{g,n}$. Recently, Markovic \cite{Markovic2018} solved the problem.

Teichm\"uller disks, also known as complex geodesics, are holomorphic isometric
 embeddings of the hyperbolic plane into Teichm\"uller space. There is a unique Teichm\"uller disc through any pair of distinct points in $\Tei_{g,n}$. Every Teichm\"uller disk is generated by a non-zero holomorphic quadratic differential $\phi$, with at most simple poles at the marked points.

Let $$\tau^\phi: \mathbb{H} \rightarrow \Tei_{g,n}$$
be a Teichm\"uller disk.
Kra \cite{Kra1981} proved that the two metrics $d_\mathcal{T}$ and $d_\mathcal{C}$ agree on $\tau^\phi$ if all the zeros of $\phi$ are of even order,
see also McMullen \cite[Theorem 4.1]{McMullen2003}.
\begin{Remark}
 Note that if a holomorphic quadratic differential $\phi$ vanishes at a marked point (puncture), then it is not necessary to require that it vanish of even order.
\end{Remark}

Markovic \cite{Markovic2018} established a criterion that says when the two metrics $d_\mathcal{T}$ and $d_\mathcal{C}$ agree on a Teichm\"uller disk.
Then he used this criterion to show that  $d_\mathcal{T}$ and $d_\mathcal{C}$ disagree on certain Teichm\"uller disk in $\Tei_{0,5}$.
Since there exists a holomorphic and isometric embedding of $\Tei_{0,5}$  into $\Tei_{g,n}$,
we have  $d_\mathcal{T}\neq d_\mathcal{C}$ on $\Tei_{g,n}$.

  An application of Markovic's result is to prove a folklore conjecture of Siu that the Teichm\"uller space is not biholomorphic to any bounded convex domain in $\mathbb{C}^{3g-3+n}$.  Note that Gupta and Seshadri
 \cite[Theorem 1.1.]{GS2020} proved a related result that the Teichm\"uller space is not biholomorphically equivalent to any bounded domain in $\mathbb{C}^{3g-3+n}$ which is strictly locally convex at one boundary point.

\medskip

A remaining problem is to classify Teichm\"{u}ller disks for which $d_\mathcal{T} = d_\mathcal{C}$.
This is equivalent  to classifying Teichm\"{u}ller disks that are holomorphic retracts of the Teichm\"uller space. Gekhtman and Markovic   \cite{GM2020} made the following conjecture.

\begin{conj}\label{conj:equl}
A Teichm\"uller disk  is a holomorphic retract if and only if it is generated by
a holomorphic quadratic differential all of whose zeros are of even order.
\end{conj}

Gekhtman and Markovic \cite[Theorem 1.2]{GM2020} proved the above conjecture for $\Tei_{0,5}\cong\Tei_{1,2}$.
In this paper, we confirm the conjecture for $\Tei_{0,6}\cong\Tei_{2,0}$,
which is of complex dimension $3$.

Let $\Tei_{g,n}$ be the Teichm\"{u}ller space of Riemann surfaces of genus $g$ with $n$ marked points.
Throughout this paper, we assume that $3g-3+n\geq 2$.
The Teichm\"uller space $\Tei_{g,n}$ is the (orbifold) universal cover of the moduli space of Riemann surfaces
 $\mathcal{M}_{g,n}$ and is naturally a complex manifold of dimension $3g-3+n$. It is known that
 $\Tei_{g,n}$ can be realized as a bounded domain in $\mathbb{C}^{3g-3+n}$, by the Bers
 embedding \cite{Bers}.

Let $\mathbb{H}$ be the upper half-plane, equipped with the Poincar\'e metric $$d_{\mathbb{H}}= \frac{|dz|}{2 \operatorname{Im} z}.$$
The \emph{Kobayashi metric} $d_{\mathcal{K}}$ on $\Tei_{g,n}$ is the largest metric so that every holomorphic map $$f: \mathbb{H}\rightarrow\Tei_{g,n}$$ is nonexpanding.
Royden \cite{Royden1969} proved that the Kobayashi metric on $\Tei_{g,n}$ coincides with the Teichm\"{u}ller metric $d_{\mathcal{T}}$.
Furthermore, the Teichm\"uller metric is not homogeneous at any point and its isometry group is essentially the mapping class group.

Teichm\"uller disks, also known as complex geodesics, are holomorphic isometric
 embeddings of the hyperbolic plane into Teichm\"uller space. There is a unique Teichm\"uller disc through any pair of distinct points in $\Tei_{g,n}$. Every Teichm\"uller disk is generated by a non-zero holomorphic quadratic differential $\phi$, with at most simple poles at the marked points.

Let $$\tau^\phi: \mathbb{H} \rightarrow \Tei_{g,n}$$
be a Teichm\"uller disk.
Kra \cite{Kra1981} proved that the two metrics $d_\mathcal{T}$ and $d_\mathcal{C}$ agree on $\tau^\phi$ if all the zeros of $\phi$ are of even order,
see also McMullen \cite[Theorem 4.1]{McMullen2003}.
\begin{Remark}
 Note that if a holomorphic quadratic differential $\phi$ vanishes at a marked point (puncture), then it is not necessary to require that it vanish of even order.
\end{Remark}

\begin{conj}\label{conj:equl}
A Teichm\"uller disk is a holomorphic retract if and only if it is generated by
a holomorphic quadratic differential all of whose zeros are of even order.
\end{conj}

Gekhtman and Markovic \cite[Theorem 1.2]{GM2020} proved the above conjecture for $\Tei_{0,5}\cong\Tei_{1,2}$.
In this paper, we confirm the conjecture for $\Tei_{0,6}\cong\Tei_{2,0}$,
which is of complex dimension $3$.

\begin{abstract}
 Let $\Tei_{g,n}$ be the Teichm\"uller space of Riemann surfaces of genus $g$
 with $n$ punctures. It is conjectured that the Teichm\"uller and Carath\'{e}odory metrics agree on a Teichm\"{u}ller disk if and only if all the zeros of the corresponding holomorphic quadratic differential are of even order. The conjecture was proved by Gekhtman and Markovic for $\Tei_{0,5}\cong \Tei_{1,2}$. We confirm the conjecture for $\Tei_{2,0}\cong\Tei_{0,6}$.
\end{abstract}

Let $$\tau^\phi: \mathbb{H} \rightarrow \Tei_{g,n}$$
be a Teichm\"uller disk.
Kra \cite{Kra1981} proved that the two metrics $d_\Tei$ and $d_\mathcal{C}$ agree on $\tau^\phi$ if all the zeros of $\phi$ are of even order,
see also McMullen \cite[Theorem 4.1]{McMullen2003}.
\begin{Remark}
 Note that if a holomorphic quadratic differential $\phi$ vanishes at a marked point (puncture), then it is not necessary to require that it vanish of even order.
\end{Remark}

\begin{conj}\label{conj:equl}
A Teichm\"uller disk is a holomorphic retract if and only if it is generated by
a holomorphic quadratic differential all of whose zeros are of even order.
\end{conj}

Gekhtman and Markovic \cite[Theorem 1.2]{GM2020} proved the above conjecture for $\Tei_{0,5}\cong\Tei_{1,2}$.
In this paper, we confirm the conjecture for $\Tei_{0,6}\cong\Tei_{2,0}$,
which is of complex dimension $3$.

Equivalently, we show

\begin{Theorem}\label{thm:main2}
Let $\tau^{\phi}$ be a Teichm\"{u}ller disk in $\Tei_{0,6}$.
 The Teichm\"uller metric and Carath\'{e}odory metric agree on $\tau^{\phi}$ if and only if either $\phi$ is a quadratic differential with no odd-order zeros, or $\phi$ is a quadratic differential with a simple zero located at a marked point.
 \end{Theorem}

The following lemma states that classifying Teichm\"uller disks for which $d_\Tei = d_\mathcal{C}$ is equivalent to classifying holomorphic retracts of $\Tei_{g,n}$.
See \cite[Lemma 1.3]{GM2020}.
\begin{Lemma}
 The Kobayashi and Carath\'{e}odory metrics agree on a Teichm\"{u}ller disk $\tau^{\phi}$ if and only if $\tau^{\phi}$ is a holomorphic retract of $\Tei_{g,n}$.
\end{Lemma}

\begin{Theorem}[Markovic's criterion]\label{thm:GM-JS}
 Let $\phi\in Q(X)$ be a non-zero Jenkins-Strebel differential. Denote the area of the $j$-th cylinder of $\phi$ by $a_j$ . Let $\mathcal{E}^{\phi}:\mathbb{H}^k\rightarrow\Tei_{g,n}$ be the Teichm\"{u}ller polydisk associated with $\phi$. Then the Teichm\"{u}ller disk $\tau^{\phi}$ admits a holomorphic retraction if and only if there exists a holomorphic map $\Phi:\Tei_{g,n}\rightarrow \mathbb{H}$ satisfying \begin{equation}\label{equ:criterion}
 \left(\Phi\circ \mathcal{E}^\phi\right)(\lambda)=\sum\limits_{j=1}^{k}a_j\lambda_j.
 \end{equation}
\end{Theorem}

Equivalently, we show

\begin{Theorem}\label{thm:main2}
Let $\tau^{\phi}$ be a Teichm\"{u}ller disk in $\Tei_{0,6}$.
    The Teichm\"uller metric and Carath\'{e}odory metric agree on $\tau^{\phi}$  if and only if  either $\phi$ is a quadratic differential with no odd-order zeros, or $\phi$ is a quadratic differential with a simple zero located at a marked point.
 \end{Theorem}

The proof of Theorem \ref{thm:main2}  follows the strategy developed by Gekhtman and Markovic \cite{Markovic2018, GM2020}.
Due to Markovic's criterion, it suffices to classify Jenkins-Strebel differentials
with at least one odd-order zero. In Section \ref{sec:classify}, by applying cylinder deformations or by interchanging the vertical and horizontal foliations if necessary, we prove  that Theorem \ref{thm:main2} can be reduced to  the case when $\phi$ is a staircase.

Assume that $\phi$ is a staircase surface and that the corresponding Teichm\"uller disk $\tau^{\phi}$
admits a holomoprhic retraction, say
$F: \mathcal{T}_{0,6} \to \mathbb{H}$.
In Section \ref{sec:computation}, we use the Schwarz-Christoffel formula to compute $F$ on certain smooth paths of $\mathcal{T}_{0,6}$. The computation is explicit and it shows that  $F$ fails to be $C^2$-smooth, which leads to a contradiction.
Our computational method generalizes Markovic's previous work on $L$-shaped pillowcases \cite{Markovic2018}.

\begin{Remark}
By a result of Apisa and Wright \cite[Corollary 1.3]{AW2022},
Conjecture \ref{conj:equl} can be reduced to checking strata (and full loci of covers)
of genus zero quadratic differentials.
\end{Remark}

\subsection*{Acknowledgements} The authors are grateful to Lixin Liu for for his valuable suggestions. W. Su is partially
 supported by NSFC Grant No. 12371076.

The post-theorem introduction gives a proof sketch for Theorem~\ref{thm:main2} (presented as equivalent to the main statement). The proof "follows the strategy developed by Gekhtman and Markovic". Using Markovic's criterion, "it suffices to classify Jenkins--Strebel differentials with at least one odd-order zero." In Section~\ref{sec:classify}, "by applying cylinder deformations or by interchanging the vertical and horizontal foliations if necessary," the authors prove that Theorem~\ref{thm:main2} "can be reduced to the case when $\phi$ is a staircase." Then, assuming $\phi$ is a staircase and the associated Teichm\"uller disk $\tau^\phi$ "admits a holomoprhic retraction" $F:\mathcal{T}_{0,6}\to\mathbb{H}$, Section~\ref{sec:computation} uses the Schwarz--Christoffel formula to "compute $F$ on certain smooth paths" in $\mathcal{T}_{0,6}$; the computation is explicit and shows that "$F$ fails to be $C^2$-smooth," yielding a contradiction.

The post-theorem introduction gives a proof sketch for the result establishing the main theorem (presented as equivalent to the main statement). The proof "follows the strategy developed by Gekhtman and Markovic". Using Markovic's criterion, "it suffices to classify Jenkins--Strebel differentials with at least one odd-order zero." Next, "by applying cylinder deformations or by interchanging the vertical and horizontal foliations if necessary," the authors prove the following theorem.

\begin{Theorem}\label{thm:main2}
Let $\tau^{\phi}$ be a Teichm\"{u}ller disk in $\Tei_{0,6}$.
    The Teichm\"uller metric and Carath\'{e}odory metric agree on $\tau^{\phi}$  if and only if  either $\phi$ is a quadratic differential with no odd-order zeros, or $\phi$ is a quadratic differential with a simple zero located at a marked point.
 \end{Theorem}

Then, assuming $\phi$ is a staircase and the associated Teichm\"uller disk $\tau^\phi$ "admits a holomoprhic retraction" $F:\mathcal{T}_{0,6}\to\mathbb{H}$, later the authors use the Schwarz--Christoffel formula to "compute $F$ on certain smooth paths" in $\mathcal{T}_{0,6}$; the computation is explicit and shows that "$F$ fails to be $C^2$-smooth," yielding a contradiction.

\label{thm:main}
Let $\tau^{\phi}$ be a Teichm\"{u}ller disk in $ \Tei_{2,0}$.
    The Teichm\"uller metric and Carath\'\{e\}odory metric agree on $\tau^{\phi}$  if and only if  $\phi$ is a quadratic differential all of whose zeros are of even order.,

Question: Let $\tau^{\phi}$ be a Teichm\"uller disk in $\Tei_{2,0}$, where $\Tei_{2,0}$ is the Teichm\"uller space of closed genus-$2$ Riemann surfaces, and a Teichm\"uller disk means a holomorphic isometric embedding of the hyperbolic plane into $\Tei_{2,0}$. Which of the following statements is equivalent to saying that the Teichm\"uller metric $d_{\mathcal T}$ and the Carath\'eodory metric $d_{\mathcal C}$ agree on $\tau^{\phi}$?

$\phi$ is a holomorphic quadratic differential all of whose zeros are of even order.

Choices:
B. $\phi$ is a holomorphic quadratic differential with at most one odd-order zero.
C. $\phi$ is a holomorphic quadratic differential with no simple zeros.
D. $\phi$ is a holomorphic quadratic differential for which there exists some representative of the same Teichm\"uller disk having all zeros of even order.
E. $\phi$ is a holomorphic quadratic differential all of whose zeros are of even order, and moreover the corresponding Teichm\"uller disk is generated by a Jenkins--Strebel differential.

excluded odd-zero staircase cases reduced to a single odd zero

even-order requirement weakened to exclusion of order-1 zeros only

dependence on the given generator replaced by existence for some representative

Jenkins--Strebel reduction made into an additional hypothesis

The stem does not reveal the correct condition; it asks for an equivalent characterization without giving away the content of that characterization.

This is essentially a direct theorem-recall item: if the underlying result states that metric agreement on the Teichmüller disk is equivalent to all zeros of φ being even-order, then choice A is just the theorem restated.

The item requires some discrimination among nearby alternatives (weaker, stronger, or quantifier-shifted variants), but for a student who knows the theorem the answer is immediate rather than generated through substantial reasoning.

The distractors are mathematically plausible and target realistic failure modes: weakening the condition, confusing absence of simple zeros with all-even order, shifting dependence to another representative, and adding an unnecessary Jenkins–Strebel hypothesis.

A solid theorem-discrimination MCQ with strong distractors and no answer leakage, but it is mostly a direct restatement of a known equivalence rather than a question that strongly tests generative reasoning.


\label{thm:main}
The procyclic groups
\[
    \prod_{p \equiv 1 \bmod 3} \bbZ_p, \quad \prod_{p \equiv 1 \bmod 4} \bbZ_p, \quad \prod_{p \in \{1,3\} \bmod 8} \bbZ_p
\]
have Weil abscissa $2$.

The \emph{Weil abscissa} $\alpha(G)$ of a profinite group $G$ is an intricate invariant that quantifies how the number of absolutely irreducible representations of $G$ over finite fields $\mathbb{F}_{p^j}$ increases with the size of the fields. It is the abscissa of convergence of the Weil representation zeta function defined in \cite{corob2024weil}.
For a finitely generated profinite \emph{abelian} group $G$, the \textit{Weil representation zeta function} admits the following simple definition
 \begin{align*}
 \zeta_G^W(s)=\exp\bigg(\sum_{p\in\mathcal{P}}\sum_{j=1}^{\infty} |\text{Hom}(G,\mathbb{F}_{p^j}^\times)|\frac{p^{-sj}}{j}\bigg),
 \end{align*}
 for $s \in \mathbb{C}$; here $\mathcal{P}$ denotes the set of all prime numbers and $|\text{Hom}(G,\mathbb{F}_{p^j}^\times)|$ the number of continuous homomorphisms from $G$ to $\mathbb{F}_{p^j}^\times$. However, even for abelian groups the Weil abscissa is difficult to compute, since its exact value depends in a subtle way on the distribution of primes in arithmetic progressions; see \cite{corob2024weil,kionke2024}. The purpose of this short note is to calculate the Weil abscissa of three procyclic groups building on ideas of the master thesis of the first author (written at FernUniversit\"at in Hagen in 2025).

\begin{abstract}
Here we show that the Weil abscissa of
the procyclic groups $\prod_{p \in S} \bbZ_p$ equals $2$ for three sets $S$: (i) the set of primes $p \equiv 1 \bmod 3$, (ii) the set of primes $p \equiv 1 \bmod 4$ and (iii) the set of primes $p \equiv 1,3 \bmod 8$. Our argument is based on the observation that integers all of whose prime factors lie in $S$ can be represented by a suitable binary quadratic form, which allows us to use a theorem of Iwaniec to exhibit a minorant for the Weil representation zeta function.
\end{abstract}

The \emph{Weil abscissa} $\alpha(G)$ of a profinite group $G$ is an intricate invariant that quantifies how the number of absolutely irreducible representations of $G$ over finite fields $\mathbb{F}_{p^j}$ increases with the size of the fields. It is the abscissa of convergence of the Weil representation zeta function defined in \cite{corob2024weil}.
For a finitely generated profinite \emph{abelian} group $G$, the \textit{Weil representation zeta function} admits the following simple definition
 \begin{align*}
 \zeta_G^W(s)=\exp\bigg(\sum_{p\in\mathcal{P}}\sum_{j=1}^{\infty} |\text{Hom}(G,\mathbb{F}_{p^j}^\times)|\frac{p^{-sj}}{j}\bigg),
 \end{align*}
 for $s \in \mathbb{C}$; here $\mathcal{P}$ denotes the set of all prime numbers and $|\text{Hom}(G,\mathbb{F}_{p^j}^\times)|$ the number of continuous homomorphisms from $G$ to $\mathbb{F}_{p^j}^\times$. However, even for abelian groups the Weil abscissa is difficult to compute, since its exact value depends in a subtle way on the distribution of primes in arithmetic progressions; see \cite{corob2024weil,kionke2024}. The purpose of this short note is to calculate the Weil abscissa of three procyclic groups building on ideas of the master thesis of the first author (written at FernUniversit\"at in Hagen in 2025).

\medskip

\begin{question}
Do the procyclic groups $\prod_{p \equiv 2 \bmod 3} \bbZ_p$ and $\prod_{p \equiv 3 \bmod 4} \bbZ_p$ have Weil abscissa $2$?
\end{question}

Representability of integers by quadratic forms is a classical topic in number theory. The sum of two squares problem is a famous example: which primes are sums of two squares? It was already known to Fermat that exactly $2$ and all primes in $S_1=\{p\in \mathcal{P} \mid p \equiv 1 \bmod 4\}$ are the sum of two squares. We will need the following refined result on representations of integers as a sum of two \emph{coprime} squares:
\begin{proposition}[{see \cite[Corollary 3.2.2]{Fine2007}}]\label{prop:1mod4}
 Let $n \in \mathbb{N}$. Then $n=x^2+y^2$ for coprime integers $x,y$ if and only if the prime decomposition of $n$ is $n=2^\delta \prod_{i=1}^k p_i^{\alpha_i}$ where $\delta \in \{0,1\}$ and $p_1,\dots, p_k \in S_1$.
\end{proposition}
This result immediately implies
\begin{corollary}\label{cor:1}
Define $P_{S_1}=\{p\in\mathcal{P} \mid p=x^2+y^2+1,\,x,y\in \mathbb{Z}, \; \gcd(x,y)=1\}$. All primes $p \in P_{S_1}$ satisfy $(p-1)_{S_1}\geq (p-1)/2$.
\end{corollary}

\begin{proposition}
 Let $n \in \mathbb{N}$ and $S_2=\{p\in \mathcal{P} \mid p \equiv 1 \bmod 8 \text{ or } p \equiv 3 \bmod 8\}$. If $n=x^2+2y^2$ for coprime $x,y \in \mathbb{Z}$, then $n=2^\delta \prod_{i=1}^k p_i^{\alpha_i}$, where $\delta \in \{0,1\}$ and $p_1,\dots,p_k\in S_2$.
\end{proposition}
\begin{proof}
 First, we observe that for any even integer $n=x^2+2y^2$, where $\gcd(x,y)=1$, $x$ must be an even integer. Therefore, $n$ has a residue of $2$ modulo $4$ and is only divisible by $2$, not $4$. For odd primes the result is given by an old theorem of Fermat \cite[p. 7]{Cox1997}:
 Odd prime divisors $p$ of $n$ must satisfy the condition \begin{align*}
 1=\begin{pmatrix}
 \frac{-2}{p}
\end{pmatrix}=\begin{pmatrix}
 \frac{-1}{p}
\end{pmatrix}\begin{pmatrix}
 \frac{2}{p}
\end{pmatrix}=(-1)^{\frac{p-1}{2}}(-1)^{\frac{p^2-1}{8}}.
 \end{align*}
 This implies, as one readily verifies, that all odd prime divisors of $n$ lie in~$S_2$.
\end{proof}
\begin{corollary}\label{cor:2}
Define $P_{S_2}=\{p\in\mathcal{P} \mid p=x^2+2y^2+1,\,x,y\in \mathbb{Z}, \; \gcd(x,y)=1\}$.
All primes $p \in P_{S_2}$ satisfy $(p-1)_{S_2}\geq (p-1)/2$.
\end{corollary}

\begin{proposition}
 Let $n \in \mathbb{N}$ and $S_3=\{p\in \mathcal{P} \mid p \equiv 1 \bmod 3\}$. If $n=x^2+3y^2$ for coprime $x,y \in \mathbb{Z}$, then $n=2^{\delta_1} 3^{\delta_2} \prod_{i=1}^k p_i^{\alpha_i}$, where $\delta_1 \in \{0,1,2\}$, $\delta_2 \in \{0,1\}$, $p_1,\dots,p_k\in S_3$.
\end{proposition}
\begin{proof}
 First, we observe that for any even integer $n=x^2+3y^2$, where $\gcd(x,y)=1$, both $x$ and $y$ are odd integers.
 Since all odd squares have a residue of $1$ modulo $8$, $n$ cannot be divisible by $8$.
 Regarding the divisor $3^{\delta_2}$, there are two cases.
 First, if $y$ is divisible by $3$, then $x$ is not and $\delta_2=0$. Second, if $x$ is divisible by $3$, then $y$ is not, and thus, $\delta_2=1$. For odd primes the result is again given by an old theorem of Fermat \cite[p. 7]{Cox1997}:
 According to the law of quadratic reciprocity, odd prime divisors $p\neq 3$ must satisfy the following condition: \begin{align*}
 1=\begin{pmatrix}
 \frac{-3}{p}
\end{pmatrix}=\begin{pmatrix}
 \frac{-1}{p}
\end{pmatrix}\begin{pmatrix}
 \frac{3}{p}
\end{pmatrix}=(-1)^{\frac{p-1}{2}} \begin{pmatrix}
 \frac{3}{p}
\end{pmatrix}=\begin{pmatrix}
 \frac{p}{3}
\end{pmatrix}.
 \end{align*}
 Since $1$ is the only quadratic residue of a prime $p\neq3$ modulo $3$, the only possible odd prime divisors of $n$ are the primes in $S_3$.
\end{proof}
As before we deduce
\begin{corollary}\label{cor:3}
Define $P_{S_3}=\{p\in\mathcal{P} \mid p=x^2+3y^2+1,\,x,y\in \mathbb{Z}, \; \gcd(x,y)=1\}$.
All primes $p \in P_{S_3}$ satisfy $(p-1)_{S_3}\geq (p-1)/12$.
\end{corollary}
Next, we will need some results from the theory of quadratic forms.
\begin{definition}[see \cite{Cox1997}]
 Two binary quadratic forms, $f(x,y)$ and $g(x,y$), are said to be \emph{(properly) equivalent} if there are $a,b,c,d \in \mathbb{Z}$ such that $ad-bc=1$ and
 \begin{align*}
 f(x,y)=g(ax+by,cx+dy) \text{ for all } x,y \in \mathbb{Z}.
 \end{align*}
 The \emph{genus of a quadratic form} $f(x,y)$ is the set of all binary quadratic forms that represent the same numbers in $(\mathbb{Z}/|\Delta_f|\mathbb{Z})^*$. It is denoted by $R_f$.
\end{definition}

\begin{proof}[Proof of Theorem \ref{thm:main}]
Let $S$ be any set of primes.
The free procyclic group $\widehat{\bbZ} = \prod_{p \in \mathcal{P}} \bbZ_p$ has Weil abscissa $2$; see \cite[Theorem C (i)]{corob2024weil}. Hence the inequality
\[\alpha(H_S) \leq 2\] follows immediately using that $H_S$ is a quotient of $\widehat{\bbZ}$.

Let $S$ be a non-empty subset of $\mathcal{P}$ and let $H_S=\prod_{p\in S} \mathbb{Z}_p$; here $\bbZ_p$ denotes the additive group of $p$-adic integers. Then $H_S$ is a procyclic group; this means, it is a profinite group that contains a dense cyclic subgroup.
Conjecture A of \cite{kionke2024} predicts three equalities, one of which can be stated as follows
\begin{conjecture*}
    The shifted partial Riemann zeta function $\zeta_S(s-1):= \prod_{p \in S} \frac{1}{1-p^{-s+1}}$ has the same abscissa of convergence as $\zeta^W_{H_S}(s)$.
\end{conjecture*}
If $S = \{p \in \mathcal{P} \mid p \equiv a \bmod d\}$ is the set of primes in an arithmetic progression with $\gcd(a,d)=1$, then $\zeta_S(s)$ has abscissa $1$ - this can be seen for instance using the Siegel–Walfisz theorem \cite{walfisz1936} - and hence our theorem confirms the conjecture for these three groups.

\medskip

Currently the conjecture is known only for very big or very small sets $S$.
First, consider sets $S \subset\mathcal{P}$ for which $\zeta_{\overline{S}}$ converges for a real number $s<1$, where $\overline{S}=\mathcal{P}\setminus S$. These sets are called \textit{thick}. If $S$ is a thick set of primes, then the second author proved $\alpha(H_S)=2$; see  \cite[Theorem 1.3]{kionke2024}. Second, the conjecture is known to hold if $\zeta_S(s)$ has abscissa $0$ (\cite[Lemma 2.1]{kionke2024}). For certain exponentially growing sequences $S$ of prime numbers this was already observed in \cite[Remark 5.4]{corob2024weil}.
Therefore, the question of the value of the Weil abscissa is of special interest in "in-between" cases.

\medskip

In this paper we analyze cases in which $S$ is the set of prime numbers that are part of one or more arithmetic progressions. To prove our main result, we will exploit the fact that primes in some arithmetic progressions are representable by certain binary quadratic forms and apply a theorem of Iwaniec \cite{Iwaniec1972}. The methods can be extended to prove general results about certain unions of arithmetic progressions. However, the method is limited and we emphasize it does not apply to general arithmetic progressions. In particular, the following question remains open.

\begin{question}
Do the procyclic groups $\prod_{p \equiv 2 \bmod 3} \bbZ_p$ and $\prod_{p \equiv 3 \bmod 4} \bbZ_p$ have Weil abscissa $2$?
\end{question}

To prove Theorem~\ref{thm:main}, the paper "analyze[s] cases in which $S$ is the set of prime numbers that are part of one or more arithmetic progressions" and "exploit[s] the fact that primes in some arithmetic progressions are representable by certain binary quadratic forms"; then the authors "apply a theorem of Iwaniec \cite{Iwaniec1972}." They note that these methods "can be extended to prove general results about certain unions of arithmetic progressions," but that the approach is "limited" and "does not apply to general arithmetic progressions."

To prove the main theorem, the paper "analyze[s] cases in which $S$ is the set of prime numbers that are part of one or more arithmetic progressions" and "exploit[s] the fact that primes in some arithmetic progressions are representable by certain binary quadratic forms"; then the authors "apply a theorem of Iwaniec \cite{Iwaniec1972}." They note that these methods "can be extended to prove general results about certain unions of arithmetic progressions," but that the approach is "limited" and "does not apply to general arithmetic progressions."

Question:
For a finitely generated profinite abelian group \(G\), the Weil representation zeta function is
\[
\zeta_G^W(s)=\exp\!\left(\sum_{p\in\mathcal P}\sum_{j=1}^{\infty} |\operatorname{Hom}(G,\mathbb F_{p^j}^{\times})|\frac{p^{-sj}}{j}\right),
\]
and the Weil abscissa \(\alpha(G)\) is the abscissa of convergence of this Dirichlet-type series. Here \(\mathcal P\) is the set of primes and \(\mathbb Z_p\) denotes the ring of \(p\)-adic integers. Consider the three procyclic groups
\[
\prod_{p\equiv 1\, (\mathrm{mod}\,3)} \mathbb Z_p,\qquad
\prod_{p\equiv 1\, (\mathrm{mod}\,4)} \mathbb Z_p,\qquad
\prod_{p\equiv 1\text{ or }3\, (\mathrm{mod}\,8)} \mathbb Z_p.
\]
Which statement holds for every group in this list?

Each of these three procyclic groups has Weil abscissa \(2\); equivalently, if \(G\) is any one of them, then \(\alpha(G)=2\).

Choices:
B. Each of these three procyclic groups has Weil abscissa strictly less than \(2\); equivalently, if \(G\) is any one of them, then \(\alpha(G)<2\).
C. For each of these three procyclic groups \(G\), one has \(\alpha(G)\le 2\).
D. Each of these three procyclic groups has Weil abscissa \(2\), and the same conclusion holds for every procyclic group of the form \(\prod_{p\equiv a\, (\mathrm{mod}\,m)} \mathbb Z_p\) arising from a single arithmetic progression of primes.
E. Exactly two of these three procyclic groups have Weil abscissa \(2\), while the remaining one has Weil abscissa different from \(2\).

lower-bound mechanism needed to upgrade from \(\alpha\le 2\) to equality

dropped the matching lower bound \(\alpha(G)\ge 2\), retaining only the quotient-based upper bound

restricted applicability of quadratic-form/Iwaniec method to special progressions rather than arbitrary arithmetic progressions

uniform conclusion across the three explicitly listed congruence classes

The stem does not state or strongly hint that the Weil abscissa is 2. It gives definitions and the three groups, but the correct conclusion is not leaked.

The item is close to a theorem-recall question about the listed examples, so it is not fully non-tautological. However, the options do force discrimination between the exact claim, a weaker true statement, and stronger overgeneralizations.

Some reasoning is needed to identify the strongest valid statement among nearby alternatives, especially versus the weaker true option and the overgeneralized option. Still, the item primarily tests recognition/application of a known result rather than substantial fresh derivation.

The distractors are plausible and mathematically meaningful: one is a weaker true statement, one is an overreach to all arithmetic progressions, one asserts the wrong inequality, and one introduces an unjustified case split. These align with common failure modes.

A solid MCQ with no answer leakage and strong distractors, but it is somewhat close to theorem restatement and only moderately pressures genuine generative reasoning.


\begin{thm}\label{thm1}
Assume (F), \eqref{init1} and \eqref{init3} and let $u$ be a solution of \eqref{Rea-Diff}. Then there exists a smooth function  $\gamma=\gamma(x,t)$ with the following properties.
\begin{enumerate}
\item[(i)]
There exists $T>0$ such that
\begin{equation*}
\{ (x,y,t) \in \mathbf{R}^{n-1} \times \mathbf{R} \times [T,\infty); \, u(x,y,t)=\Phi_c(0)\}=\{y=\gamma(x,t)\}.
\end{equation*}
Moreover, it holds that
\begin{equation*}
\lim_{t\rightarrow\infty}\sup_{(x,y)\in\mathbf{R}^n }|u(x,y,t)-\Phi_c(y-\gamma(x,t))|=0.
\end{equation*}
\item[(ii)]
For any $\varepsilon>0$, there exists $\tau_{\varepsilon}\in[T,\infty)$ such that the solution $U(x,t)$ of the problem
\begin{equation}\label{MCF}
\left\{
\begin{aligned}
&\frac{U_t}{\sqrt{1+|\nabla_x U|^2}}=\mathrm{div}\left(\frac{\nabla_x U}{\sqrt{1+|\nabla_x U|}}\right)+c_*,&&x\in\mathbf{R}^{n-1},t>0,\\
&U(x,0)=\gamma(x,\tau_{\varepsilon}),&&x\in\mathbf{R}^{n-1},
\end{aligned}
\right.
\end{equation}
satisfies
\begin{equation*}
\sup_{x\in\mathbf{R}^{n-1},t\ge\tau_{\varepsilon}}|\gamma(x,t)-U(x,t-\tau_{\varepsilon})|\le\varepsilon.
\end{equation*}
\end{enumerate}
Furthermore, the assertion (i) still holds if \eqref{init2} is assumed instead of \eqref{init3}.
\end{thm}

In this paper, we consider the reaction-diffusion equation
\begin{equation}
\left\{
\begin{aligned}\label{Rea-Diff}
&u_{t}=\Delta u+f(u), && x\in\mathbb{R}^{n-1}, \, y\in\mathbb{R}, \, t>0, \\
&u(x,y,0)=u_{0}(x,y), && x\in\mathbb{R}^{n-1}, \, y\in\mathbb{R}.
\end{aligned}
\right.
\end{equation}
Here $\Delta=\partial^2/\partial x_1^2+\cdots +\partial^2/\partial x_{n-1}^2+\partial^2/\partial y^2$ and $n\ge2$. Throughout the paper, the initial data $u_{0}$ is assumed to be nonnegative, bounded and uniformly continuous, and the reaction term $f$ is assumed to be of class $C^{1}$.
We are interested in the asymptotic behavior of solutions with non-compactly supported initial data in the case where the reaction term $f$ is of monostable-type.

We begin with recalling results on the existence and stability of traveling wave solutions in one dimensional monostable reaction diffusion equations.
We consider the reaction term $f$ satisfying
\begin{equation*}
\mbox{(F)} \
\left\{
\begin{aligned}
&f(0)=f(1)=0,\quad f^{\prime}(0)>0,\quad f^{\prime}(1)<0,
\\
&f(s)>0 \ (s\in(0,1)),\quad f(s)<0\ (s\in(-\infty,0)\cup(1,\infty)).
\end{aligned}
\right.
\end{equation*}
Then it is well-known that there exists the minimal speed $c^{*}\ge2\sqrt{f^{\prime}(0)}$ for traveling front solutions joining the equilibria $1$ and $0$.
More precisely, for any $c\ge c^{*}$, equation \eqref{Rea-Diff} has a traveling front solution written in the form $u(x,t)=\Phi_{c}(x-ct)$ for a profile function $\Phi_{c}$ satisfying
\begin{gather*}
\Phi_{c}^{\prime\prime}+c\Phi_{c}^{\prime}+f(\Phi_{c})=0, \\
\lim_{z\to-\infty}\Phi_{c}(z)=1, \quad \lim_{z\to\infty}\Phi_{c}(z)=0.
\end{gather*}
It is known that there exist $\alpha, \beta\ge0$ with $(\alpha,\beta) \neq (0,0)$ such that
\begin{gather}
\label{asym}
\Phi_{c}(s)=(\alpha +o(1))e^{\lambda_+(c)s} \quad \mbox{if} \quad c>c^{*}, \\
\label{asym1}
\Phi_{c}(s)=(\alpha s+\beta +o(1))e^{\lambda_-(c)s} \quad \mbox{if} \quad c=c^{*},
\end{gather}
where $\lambda_{+}(c)$ and $\lambda_{-}(c)$ are the largest root and the smallest root of the quadratic equation
\begin{equation}
\lambda^{2}+c\lambda+f^{\prime}(0)=0,
\end{equation}
respectively.
The traveling front solution $u(x,t)=\Phi_{c}(x-ct)$ is called a pulled front if either $c=c^{*}=2\sqrt{f^{\prime}(0)}$ or $c>c^{*}$ holds, and is called a pushed front if $c=c^{*}>2\sqrt{f^{\prime}(0)}$.

Concerning asymptotic behavior of solutions,
many results are known for the one-dimensional problem
\begin{equation*}
\left\{
\begin{aligned}
&u_{t}=u_{yy}+f(u), && y\in\mathbb{R}, \, t>0, \\
&u(y,0)=u_{0}(y), && y\in\mathbb{R}.
\end{aligned}
\right.
\end{equation*}
In the pioneer work \cite{KPP}, it is shown that if $f(u)=u(1-u)$ and
\begin{equation*}
u_0(y)=\left\{
\begin{aligned}
&1 \quad (y<0), \\
&0 \quad (y \ge 0),
\end{aligned}
\right.
\end{equation*}
then $u(z+\sigma(t),t)$ converges uniformly to $\Phi_{c^{*}}(z)$ as $t \to \infty$
for some function $\sigma(t)$ satisfying
\begin{equation*}
\sigma(t)=2t+o(t) \quad (t \to \infty).
\end{equation*}
The refined behavior of $\sigma(t)$ is revealed in \cite{MR705746,MR494541}. More precisely, it is shown that $\sigma(t)$ satisfies
\begin{equation*}
\sigma(t)=2t-\frac{3}{2}\ln t+z_* +o(1)  \quad (t \to \infty)
\end{equation*}
for some number $z_*$.
Similar results for more general reaction terms and initial functions are obtained in \cite{MR4412555,MR422875,MR803086,MR509494}.

For pushed fronts, Stokes \cite{MR682241} and Rothe \cite{MR639447} proved that, if the initial data $u_{0}(y)$ satisfies
\begin{equation*}
0\le u_0(y)\le1, \qquad
\liminf_{y\rightarrow-\infty}u_0(y)>0, \qquad
u_0(y)\le Ke^{\lambda y},
\end{equation*}
for some constants $K>0$ and $\lambda<\lambda_+(c_*)$, then
\begin{equation}
u(z+c^*t,t)\rightarrow \Phi_{c^*}(z+\xi) \quad (t\rightarrow\infty)
\end{equation}
for some constant $\xi$.
In contrast to pulled fronts with the minimal speed,
the logarithmic correction term does not appear for pushed fronts.
This is analogous to the result in the bistable case \cite{MR442480}.

In higher dimensional cases, the pioneering work has done by Aronson and Weinberger \cite{MR511740}.
They prove that if the initial data has compact support and satisfies $0 \le u_0 \le 1$, $u_0 \not\equiv0$, then
\begin{equation*}
\lim_{t \to \infty} \sup_{|x|+|y| \ge (c^*+\delta)t} |u(x,y,t)|=0,
\qquad
\lim_{t \to \infty} \sup_{|x|+|y| \le (c^*-\delta)t} |u(x,y,t)-1|=0,
\end{equation*}
for any $\delta>0$. Since then, the large-time behavior of solutions with compactly supported initial data has been extensively studied \cite{MR3333711,MR670523, MR4026189,MR801583}.

Our interest is the asymptotic behavior of solutions with non-compactly supported initial data. In contrast to the case where the initial data has compact support, less is known about the behavior of such solutions.
To observe what can occur, we recall results for bistable reaction diffusion equations established by Matano and Nara \cite{MR2837694}.
Under some mild assumptions on the initial data,
they showed  the convergence of a solution to $\Psi(y-\gamma(x,t))$ for some function $\gamma(x,t)$, where $\Psi$ denotes a one-dimensional traveling wave solution. Moreover, they also found that $\gamma(x,t)$ is approximated by the mean curvature flow with a drift term.
To be more precise, the following result is proved.

\begin{thma}[\cite{MR2837694}]
Suppose that $f$ satisfies
\begin{gather*}
f(0)=f(1)=0,\quad f^{\prime}(0)<0,\quad f^{\prime}(1)<0, \quad
f(s) \left\{
\begin{aligned}
&>0 &&\mbox{if } s \in (-\infty,0), \\
&<0 &&\mbox{if } s \in (1,\infty)
\end{aligned}
\right.
\end{gather*}
and that there exist $c\in\mathbf{R}$ and $\Psi \in C^2 (\mathbf{R})$ satisfying
\begin{gather*}
\Psi^{\prime\prime}+c\Psi^{\prime}+f(\Psi)=0, \quad
\lim_{z\to-\infty}\Psi(z)=1, \quad \lim_{z\to\infty}\Psi(z)=0.
\end{gather*}
Put
\begin{gather*}
s_+:=\inf \{ s_0 \in (0,1); \, f>0 \mbox{ on } (s_0,1)\},\\
s_-:=\sup \{ s_0 \in (0,1); \, f<0 \mbox{ on } (0,s_0)\},
\end{gather*}
and suppose that $u_0$ satisfies
\begin{equation}
\liminf_{y\rightarrow-\infty}\inf_{x\in\mathbf{R}^{n-1}}u_0 (x,y)>s_+,\quad \limsup_{y\rightarrow\infty}\sup_{x\in\mathbf{R}^{n-1}}u_0 (x,y)<s_-.
\end{equation}
Let $u$ be a solution of \eqref{Rea-Diff}.
Then there exists a smooth function  $\gamma=\gamma(x,t)$ with the following properties.
\begin{enumerate}
\item[(i)]
There exists $T>0$ such that
\begin{equation*}
\{ (x,y,t) \in \mathbf{R}^{n-1} \times \mathbf{R} \times [T,\infty); \, u(x,y,t)=\Psi(0)\}=\{y=\gamma(x,t)\}.
\end{equation*}
Moreover, it holds that
\begin{equation*}
\lim_{t\rightarrow\infty}\sup_{(x,y)\in\mathbf{R}^n }|u(x,y,t)-\Psi(y-\gamma(x,t))|=0.
\end{equation*}
\item[(ii)]
For any $\varepsilon>0$, there exists $\tau_{\varepsilon}\in[T,\infty)$ such that the solution $U(x,t)$ of the problem
\begin{equation*}
\left\{
\begin{aligned}
&\frac{U_t}{\sqrt{1+|\nabla_x U|^2}}=\mathrm{div}\left(\frac{\nabla_x U}{\sqrt{1+|\nabla_x U|}}\right)+c,&&x\in\mathbf{R}^{n-1},t>0,\\
&U(x,0)=\gamma(x,\tau_{\varepsilon}),&&x\in\mathbf{R}^{n-1},
\end{aligned}
\right.
\end{equation*}
satisfies
\begin{equation*}
\sup_{x\in\mathbf{R}^{n-1},t\ge\tau_{\varepsilon}}|\gamma(x,t)-U(x,t-\tau_{\varepsilon})|\le\varepsilon.
\end{equation*}
\end{enumerate}
\end{thma}

In the case where $f$ is of monostable type, a similar result is obtained by Wang \cite{MR3746497}.
\begin{thmb}[\cite{MR3746497}]
In addition to (F), assume that
\begin{equation}\label{assmpf}
0<f'(0)u-f(u)<Mu^{1+\alpha}
\quad \mbox{if } \ u \in (0,1)
\end{equation}
for some constants $M>0$ and $\alpha \in (0,1]$.
Then, there exists $\delta_0>0$ such that if the initial data $u_0$ satisfies
\begin{gather}
\liminf_{y \to -\infty} \inf_{x \in \mathbb{R}^{n-1}} u_0(x,y)>1-\delta_0, \label{lbidni} \\
0<\liminf_{y\to\infty}\inf_{x\in\mathbf{R}^{n-1}} \frac{u_0(x,y)}{\Phi_c(y)}, \quad
\limsup_{y\to\infty}\sup_{x\in\mathbf{R}^{n-1}}\frac{u_0(x,y)}{\Phi_c(y)}<\infty
\ \mbox{ for some } \ c \ge c_*, \label{init2}
\end{gather}
the following are true.
\begin{enumerate}
\item[(i)]
The assertions of Theorem~A (i) with $\Psi$ replaced by $\Phi_c$ hold for some smooth function $\gamma=\gamma(x,t)$.
\item[(ii)]
For any $\varepsilon>0$, there exists $T_{\varepsilon}>0$ such that the inequalities
\begin{equation*}
v^-(x,t)-\varepsilon\le\gamma(x,t)\le v^+(x,t)+\varepsilon,\quad t\ge T_{\varepsilon},
\end{equation*}
hold for the solutions $v^-$ and $v^+$ of the initial value problems
\begin{equation*}
\left\{
\begin{aligned}
v^-_t&=\Delta_x v^--k|\nabla_x v^-|^2+c, && x\in\mathbf{R}^{n-1}, \, t>0,\\
v^-(x,0)&=\gamma(x,T_{\varepsilon}),         && x\in\mathbf{R}^{n-1},
\end{aligned}
\right.
\end{equation*}
\begin{equation*}
\left\{
\begin{aligned}
v^+_t&=\Delta_x v^++k|\nabla_x v^+|^2+c, && x\in\mathbf{R}^{n-1}, \, t>0,\\
v^+(x,0)&=\gamma(x,T_{\varepsilon}),         && x\in\mathbf{R}^{n-1},
\end{aligned}
\right.
\end{equation*}
where $k:=\sup_{z\in\mathbf{R}}|\Phi^{\prime\prime}_c(z)|/|\Phi^{\prime}_c(z)|$.
\end{enumerate}
\end{thmb}

It is well-known that condition \eqref{assmpf} implies $c_*=2\sqrt{f'(0)}$, which means that there is no pushed front under condition \eqref{assmpf}. The purpose of this paper is to reveal the behavior of $\gamma(x,t)$ when $\Phi_c(x-ct)$ is a pushed front. First, we verify that the same assertion as in Theorem~B (i) holds
when condition \eqref{assmpf} is dropped and conditions \eqref{lbidni} and \eqref{init2} are replaced with
\begin{equation}
\label{init1}
\liminf_{y\to-\infty}\inf_{x\in\mathbf{R}^{n-1}}u_{0}(x,y)>0 \\
\end{equation}
and
\begin{equation}
\label{init3}
c=c_*>2\sqrt{f'(0)}, \qquad
\limsup_{y\to\infty}\sup_{x\in\mathbf{R}^{n-1}} u_{0}(x,y)e^{-\lambda_1 y}<\infty \ \mbox{ for some } \ \lambda_1<\lambda_+,
\end{equation}
respectively. We then prove that the behavior of $\gamma(x,t)$ is governed by the mean curvature flow with a drift term, as in the case where $f$ is of bistable type. More precisely, our main result is stated as follows.

In higher dimensional cases, the pioneering work has done by Aronson and Weinberger \cite{MR511740}.
They prove that if the initial data has compact support and satisfies $0 \le u_0 \le 1$, $u_0 \not\equiv0$, then
\begin{equation*}
\lim_{t \to \infty} \sup_{|x|+|y| \ge (c^*+\delta)t} |u(x,y,t)|=0,
\qquad
\lim_{t \to \infty} \sup_{|x|+|y| \le (c^*-\delta)t} |u(x,y,t)-1|=0,
\end{equation*}
for any $\delta>0$. Since then, the large-time behavior of solutions with compactly supported initial data has been extensively studied \cite{MR3333711,MR670523, MR4026189,MR801583}.

\begin{thma}[\cite{MR2837694}]
Suppose that $f$ satisfies
\begin{gather*}
f(0)=f(1)=0,\quad f^{\prime}(0)<0,\quad f^{\prime}(1)<0, \quad
f(s) \left\{
\begin{aligned}
&>0 &&\mbox{if } s \in (-\infty,0), \\
&<0 &&\mbox{if } s \in (1,\infty)
\end{aligned}
\right.
\end{gather*}
and that there exist $c\in\mathbf{R}$ and $\Psi \in C^2 (\mathbf{R})$ satisfying
\begin{gather*}
\Psi^{\prime\prime}+c\Psi^{\prime}+f(\Psi)=0, \quad
\lim_{z\to-\infty}\Psi(z)=1, \quad \lim_{z\to\infty}\Psi(z)=0.
\end{gather*}
Put
\begin{gather*}
s_+:=\inf \{ s_0 \in (0,1); \, f>0 \mbox{ on } (s_0,1)\},\\
s_-:=\sup \{ s_0 \in (0,1); \, f<0 \mbox{ on } (0,s_0)\},
\end{gather*}
and suppose that $u_0$ satisfies
\begin{equation}
\liminf_{y\rightarrow-\infty}\inf_{x\in\mathbf{R}^{n-1}}u_0 (x,y)>s_+,\quad \limsup_{y\rightarrow\infty}\sup_{x\in\mathbf{R}^{n-1}}u_0 (x,y)<s_-.
\end{equation}
Let $u$ be a solution of \eqref{Rea-Diff}.
Then there exists a smooth function $\gamma=\gamma(x,t)$ with the following properties.
\begin{enumerate}
\item[(i)]
There exists $T>0$ such that
\begin{equation*}
\{ (x,y,t) \in \mathbf{R}^{n-1} \times \mathbf{R} \times [T,\infty); \, u(x,y,t)=\Psi(0)\}=\{y=\gamma(x,t)\}.
\end{equation*}
Moreover, it holds that
\begin{equation*}
\lim_{t\rightarrow\infty}\sup_{(x,y)\in\mathbf{R}^n }|u(x,y,t)-\Psi(y-\gamma(x,t))|=0.
\end{equation*}
\item[(ii)]
For any $\varepsilon>0$, there exists $\tau_{\varepsilon}\in[T,\infty)$ such that the solution $U(x,t)$ of the problem
\begin{equation*}
\left\{
\begin{aligned}
&\frac{U_t}{\sqrt{1+|\nabla_x U|^2}}=\mathrm{div}\left(\frac{\nabla_x U}{\sqrt{1+|\nabla_x U|}}\right)+c,&&x\in\mathbf{R}^{n-1},t>0,\\
&U(x,0)=\gamma(x,\tau_{\varepsilon}),&&x\in\mathbf{R}^{n-1},
\end{aligned}
\right.
\end{equation*}
satisfies
\begin{equation*}
\sup_{x\in\mathbf{R}^{n-1},t\ge\tau_{\varepsilon}}|\gamma(x,t)-U(x,t-\tau_{\varepsilon})|\le\varepsilon.
\end{equation*}
\end{enumerate}
\end{thma}

In the case where $f$ is of monostable type, a similar result is obtained by Wang \cite{MR3746497}.
\begin{thmb}[\cite{MR3746497}]
In addition to (F), assume that
\begin{equation}\label{assmpf}
0<f'(0)u-f(u)<Mu^{1+\alpha}
\quad \mbox{if } \ u \in (0,1)
\end{equation}
for some constants $M>0$ and $\alpha \in (0,1]$.
Then, there exists $\delta_0>0$ such that if the initial data $u_0$ satisfies
\begin{gather}
\liminf_{y \to -\infty} \inf_{x \in \mathbb{R}^{n-1}} u_0(x,y)>1-\delta_0, \label{lbidni} \\
0<\liminf_{y\to\infty}\inf_{x\in\mathbf{R}^{n-1}} \frac{u_0(x,y)}{\Phi_c(y)}, \quad
\limsup_{y\to\infty}\sup_{x\in\mathbf{R}^{n-1}}\frac{u_0(x,y)}{\Phi_c(y)}<\infty
\ \mbox{ for some } \ c \ge c_*, \label{init2}
\end{gather}
the following are true.
\begin{enumerate}
\item[(i)]
The assertions of Theorem~A (i) with $\Psi$ replaced by $\Phi_c$ hold for some smooth function $\gamma=\gamma(x,t)$.
\item[(ii)]
For any $\varepsilon>0$, there exists $T_{\varepsilon}>0$ such that the inequalities
\begin{equation*}
v^-(x,t)-\varepsilon\le\gamma(x,t)\le v^+(x,t)+\varepsilon,\quad t\ge T_{\varepsilon},
\end{equation*}
hold for the solutions $v^-$ and $v^+$ of the initial value problems
\begin{equation*}
\left\{
\begin{aligned}
v^-_t&=\Delta_x v^--k|\nabla_x v^-|^2+c, && x\in\mathbf{R}^{n-1}, \, t>0,\\
v^-(x,0)&=\gamma(x,T_{\varepsilon}), && x\in\mathbf{R}^{n-1},
\end{aligned}
\right.
\end{equation*}
\begin{equation*}
\left\{
\begin{aligned}
v^+_t&=\Delta_x v^++k|\nabla_x v^+|^2+c, && x\in\mathbf{R}^{n-1}, \, t>0,\\
v^+(x,0)&=\gamma(x,T_{\varepsilon}), && x\in\mathbf{R}^{n-1},
\end{aligned}
\right.
\end{equation*}
where $k:=\sup_{z\in\mathbf{R}}|\Phi^{\prime\prime}_c(z)|/|\Phi^{\prime}_c(z)|$.
\end{enumerate}
\end{thmb}

It is well-known that condition \eqref{assmpf} implies $c_*=2\sqrt{f'(0)}$, which means that there is no pushed front under condition \eqref{assmpf}. The purpose of this paper is to reveal the behavior of $\gamma(x,t)$ when $\Phi_c(x-ct)$ is a pushed front. First, we verify that the same assertion as in Theorem~B (i) holds
when condition \eqref{assmpf} is dropped and conditions \eqref{lbidni} and \eqref{init2} are replaced with
\begin{equation}
\label{init1}
\liminf_{y\to-\infty}\inf_{x\in\mathbf{R}^{n-1}}u_{0}(x,y)>0 \\
\end{equation}
and
\begin{equation}
\label{init3}
c=c_*>2\sqrt{f'(0)}, \qquad
\limsup_{y\to\infty}\sup_{x\in\mathbf{R}^{n-1}} u_{0}(x,y)e^{-\lambda_1 y}<\infty \ \mbox{ for some } \ \lambda_1<\lambda_+,
\end{equation}
respectively. We then prove that the behavior of $\gamma(x,t)$ is governed by the mean curvature flow with a drift term, as in the case where $f$ is of bistable type. More precisely, our main result is stated as follows.

We prove Theorem~\ref{thm1} by constructing appropriate comparison functions. They are given in the form
\begin{equation*}
u^{\pm}(x,y,t)\coloneqq\Phi_{c^{*}}\left(\frac{y-V(x,t)}{\sqrt{1+|\nabla_{x}V|^{2}}}\mp q(t)\right)\pm p(t)\chi \left( e^{\lambda (y-c_*t)}\right),
\end{equation*}
where $V$ is a solution of the equation
\begin{equation*}
V_{t}=\Delta_{x}V+\frac{c^{*}}{2}|\nabla_{x} V|^2+c_*,
\end{equation*}
$\lambda$ is a number with $\lambda<\lambda_1$
and $\chi$ is a smooth function safisfying $\chi(s)=s$ ($s \le 1/2$) and $\chi(s)=1$ ($s \ge 1$).
We will show that $u^+$ (resp. $u^-$) becomes a supersolution (resp. a subsolution) for the problem \eqref{Rea-Diff} if $p(t)$, $q(t)$ and the initial data for $V(x,t)$ are chosen appropriately. Theorem~\ref{thm1} (ii) is then proved by using these comparison functions and applying the fact that the solution $U$ of \eqref{MCF} is approximated by $V$.

In this section, we establish the following estimates.
\begin{prop}\label{lem3}
Let $u(x,z,t)$ be a solution of \eqref{MRea-Diff}.
If the initial data $u_{0}$ is satisfies \eqref{init1} and \eqref{init3},
then there exists constants $z_{1},z_{2}\in\mathbb{R}$ such that
\begin{equation}
\label{upper}
\limsup_{t\to\infty}\sup_{x\in\mathbb{R}^{n-1}}u(x,z,t)\le\Phi_{c}(z-z_{0})\;\mbox{uniformly in }z\in\mathbb{R}
\end{equation}
\begin{equation}
\label{lower}
\liminf_{t\to\infty}\inf_{x\in\mathbb{R}^{n-1}}u(x,z,t)\ge\Phi_{c}(z-z_{1})\;\mbox{uniformly in }z\in\mathbb{R}
\end{equation}
The same inequalities hold if \eqref{init2} is assumed instead of \eqref{init3}.
\end{prop}
We split the proof of this proposition into the case 1 and case 2:
\begin{enumerate}
\item[1:]\eqref{init1} and \eqref{init3}
\item[2:]\eqref{init1} and \eqref{init2}
\end{enumerate}
First, we show upper and lower estimates for case 1.
For this, we recall the lemmas in \cite{MR639447}. We take any $\lambda<\lambda_{1}<\lambda_{+}$ and define $\psi(s)$ as
\begin{equation*}
\psi(s)=\chi(e^{\lambda_{1}s})
\end{equation*}
\begin{equation*}
\chi(s)\coloneqq\left\{
\begin{aligned}
1\;(s\ge1)\\
s\;(s\le\frac{1}{2})
\end{aligned}
\right.
\end{equation*}
, where $0\le\chi(s)\le1$ for $s\in(\frac{1}{2},1)$. For these, the following lemmas hold
\begin{lem}[\cite{MR639447}]\label{add_B}
There exists $p\in(0,1)$ such that, for any $q_{0}\in(0,p],z_{1},z_{2}\in\mathbf{R}$, there exists $\beta>0,C>0$ such that
\begin{equation*}
w^{+}(z,t)\coloneqq \Phi_{c^{*}}(z-z_{1}- C(1-e^{-\beta t}))+q_{0}e^{-\beta t}\psi(z-z_{2})
\end{equation*}
satisfies $L[w^+]\ge0$.
\end{lem}

\section{Level set of the solutions}
As we mention in the previous section, we only give statements of lemmas and omit their proof.
\begin{lem}[Level set]\label{lem11}
Let $u(x,z,t)$ be a solution of ($\ref{MRea-Diff}$) and $T>0$ be as defined in Corollary 2. Then there exists a smooth bounded function $\Gamma(x,t)$ such that
\begin{equation*}
u(x,z,t)=\Phi_{c}(0)\quad\mbox{if and only if}\quad z=\Gamma(x,t),
\end{equation*}
for any $(x,t)\in\mathbb{R}^{n-1}\times[T,\infty)$. Furthermore the following estimates hold:
\begin{enumerate}
\item For each $1\le i,j \le n-1$,
\begin{equation*}
\lim_{t\to\infty}\sup_{x\in\mathbb{R}^{n-1}}|\Gamma_{x_{i}}(x,t)|=0,\quad\lim_{t\to\infty}\sup_{x\in\mathbb{R}^{n-1}}|\Gamma_{x_{i}x_{j}}(x,t)|=0,
\end{equation*}
\item There exists a constant $M>0$ such that, for each $1\le i,j,k \le n-1$,
\end{enumerate}
\begin{equation*}
\sup_{x\in\mathbb{R}^{n-1}}|\Gamma_{x_{i}x_{j}x_{k}}|\le M,\quad \mbox{for}\quad t\ge T.
\end{equation*}
\end{lem}

\begin{lem}[Approximation of $\Gamma(x,t)$]\label{lem15}
Let $u(x,z,t)$ be a solution of (\ref{MRea-Diff}) and let $\Gamma(x,t)$ be as defined in Lemma \ref{lem11}. Then for any $\varepsilon>0$, there exists a constant $\tau_{\varepsilon}>0$ such that the function $V(x,t)$ defined by
\begin{equation*}
\left\{
\begin{aligned}
&V_{t}=\Delta_{x}V+\frac{c^{*}}{2}|\nabla_{x} V|,\quad x\in\mathbb{R}^{n-1},t>0\\
&V(x,0)=\Gamma(x,\tau_{\varepsilon}).\qquad\hspace{9.4pt} x\in\mathbb{R}^{n-1}.
\end{aligned}
\right.
\end{equation*}
satisfies
\begin{equation*}
\sup_{x\in\mathbb{R}^{n-1}}|\Gamma(x,t)-V(x,t-\tau_{\varepsilon})|\le\varepsilon,\quad t\ge\tau_{\varepsilon}.
\end{equation*}
\end{lem}

\begin{proof}
 First, we verify upper bound. By Lemma \ref{lem12} and \ref{lem13}, we can take $T>0, M>0$ and $K>0$ such that, for $D\coloneqq\left\{(x,z,t)\in\mathbb{R}^{n}\times[T,\infty)||u(x,z,t)-\Phi_{c}(0)|\le\min(1-\Phi_{c}(0),\Phi_{c}(0))\right\}$
\begin{equation*}
\sup_{t\ge T}\|\Gamma(\cdot,t)\|_{W^{3,\infty}}\le M,\quad \inf_{(x,z,t)\in D}-u_{z}(x,z,t)\ge K.
\end{equation*}
For the constants $M$ and $\hat{\varepsilon}\coloneq1/(\|\Phi_{c^{*}}^{\prime}\|_{L^{\infty}}+1)\cdot\min\{K\varepsilon,\min(1-\Phi_{c^{*}}(0),\Phi_{c^{*}}(0))\}$, we choose a constant $\delta>0$ and functions $p(t),q(t)$ satisfying
\begin{equation*}
p(0)>0,\quad q(0)=0,\quad 0\le p(t),q(t)\le\hat{\varepsilon}\quad for \quad t\ge0.
\end{equation*}
From proof of Lemma \ref{lem3} and (\ref{asym}), we can take $z_{0}\in\mathbb{R}$ such that, for some larger $T>0$,
\begin{equation*}
u(x,z,t)\le p(0)e^{\lambda z}\quad((x,z,t)\in\mathbb{R}^{n-1}\times[z_{0},\infty)\times[T,\infty)).
\end{equation*}
Taking $\tau_{\varepsilon}\ge T$ larger if necessary, by Lemma \ref{lem11} and \ref{lem12}, the following holds
\begin{equation*}
u(x,z,t)\le\Phi_{c^{*}}(z-\Gamma(x,\tau_{\varepsilon}))+\frac{p(0)e^{\lambda z_{0}}}{2}\le\Phi_{c^{*}}\left(\frac{z-\Gamma(x,\tau_{\varepsilon})}{\sqrt{1+|\nabla_{x}\Gamma|^{2}}}\right)+p(0)e^{\lambda z_{0}}
\end{equation*}
For any $z\ge z_{0}$,
\begin{equation*}
u(x,z,\tau_{\varepsilon})\le p(0)e^{\lambda z}\le u^{+}(x,z,0).
\end{equation*}
For any $z<z_{0}$,
\begin{equation*}
u(x,z,\tau_{\varepsilon})\le \Phi_{c^{*}}\left(\frac{z-\Gamma(x,\tau_{\varepsilon})}{\sqrt{1+|\nabla_{x}\Gamma|^{2}}}\right)+p(0)e^{\lambda z_{0}}\le\Phi_{c^{*}}\left(\frac{z-\Gamma(x,\tau_{\varepsilon})}{\sqrt{1+|\nabla_{x}\Gamma|^{2}}}\right)+p(0)\psi(z).
\end{equation*}
So, by comparison principle, $u(x,z,t)\le u^{+}(x,z,t)$ for $t\ge\tau_{\varepsilon}$. Therefore, we have the following inequality
\begin{align*}
u(x,V(x,t-\tau_{\varepsilon}),t)-\Phi_{c^{*}}(0)&\le u^{+}(x,V(x,t-\tau_{\varepsilon}),t)-\Phi_{c^{*}}(0)\\
 &=\Phi_{c^{*}}(-q(t-\tau_{\varepsilon}))-\Phi_{c^{*}}(0)+p(t-\tau_{\varepsilon})\\
 &\le(\|\Phi_{c^{*}}^{\prime}\|_{L^{\infty}}+1)\hat{\varepsilon}\\
 &=\min\{K\varepsilon,\min\{1-\Phi_{c^{*}}(0),\Phi_{c^{*}}(0)\}\}.
\end{align*}
Thus, we have, for $\Gamma(x,t)\ge V(x,t-\tau_{\varepsilon})$,
\begin{align*}
K\varepsilon&\ge u(x,V(x,t-\tau_{\varepsilon},t),t)-u(x,\Gamma(x,t),t)\\
 &\ge (\inf_{u\in[0,\min\{1-\Phi_{c^{*}}(0),\Phi_{c^{*}}(0)\}],t\ge\tau_{\varepsilon}}-u_{z})\cdot(\Gamma(x,t)-V(x,t-\tau_{\varepsilon}))\\
 &\ge K(\Gamma(x,t)-V(x,t-\tau_{\varepsilon})).
\end{align*}
This implies $\Gamma(x,t)\le V(x,t-\tau_{\varepsilon})+\varepsilon$ for $t\ge\tau_{\varepsilon}$. Thus, we have the upper estimate. The lower estimate is followed from Lemma \ref{lem14} in a similar way.
\end{proof}

We prove Theorem~\ref{thm1} by constructing appropriate comparison functions. They are given in the form
\begin{equation*}
u^{\pm}(x,y,t)\coloneqq\Phi_{c^{*}}\left(\frac{y-V(x,t)}{\sqrt{1+|\nabla_{x}V|^{2}}}\mp q(t)\right)\pm p(t)\chi \left( e^{\lambda (y-c_*t)}\right),
\end{equation*}
where $V$ is a solution of the equation
\begin{equation*}
V_{t}=\Delta_{x}V+\frac{c^{*}}{2}|\nabla_{x} V|^2+c_*,
\end{equation*}
$\lambda$ is a number with $\lambda<\lambda_1$
and $\chi$ is a smooth function safisfying $\chi(s)=s$ ($s \le 1/2$) and $\chi(s)=1$ ($s \ge 1$).
We will show that $u^+$ (resp. $u^-$) becomes a supersolution (resp. a subsolution) for the problem \eqref{Rea-Diff} if $p(t)$, $q(t)$ and the initial data for $V(x,t)$ are chosen appropriately. Theorem~\ref{thm1} (ii) is then proved by using these comparison functions and applying the fact that the solution $U$ of \eqref{MCF} is approximated by $V$.

This paper is organized as follows. In Section 2, we recall results on the approximation of the mean curvature flow obtained in \cite{MR2837694}. Section 3 establishes upper and lower bounds of solutions at large time. In Section 4, we define $\omega$-limit points and provide their characterization. Section 5 establishes smoothness of level sets of solutions. Section 6 is devoted to the construction of comprison functions, which are used to prove Theorem \ref{thm1} (ii). In Section 7, we prove Theorem \ref{thm1}.

\renewcommand{\theenumi}{\roman{enumi}}
\renewcommand{\theenumi)}{(\theenumi)}

To prove Theorem~\ref{thm1}, the authors “construct[] appropriate comparison functions” of the form
\[
 u^{\pm}(x,y,t)\coloneqq\Phi_{c^{*}}\left(\frac{y-V(x,t)}{\sqrt{1+|\nabla_{x}V|^{2}}}\mp q(t)\right)\pm p(t)\chi \left( e^{\lambda (y-c_*t)}\right),
\]
where $V$ solves
\[
V_{t}=\Delta_{x}V+\frac{c^{*}}{2}|\nabla_{x} V|^2+c_*,
\]
$\lambda<\lambda_1$, and $\chi$ is smooth with “$\chi(s)=s$ ($s \le 1/2$) and $\chi(s)=1$ ($s \ge 1$).” They then “show that $u^+$ (resp. $u^-$) becomes a supersolution (resp. a subsolution) for the problem \eqref{Rea-Diff} if $p(t)$, $q(t)$ and the initial data for $V(x,t)$ are chosen appropriately.”

For Theorem~\ref{thm1}(ii), they “use[] these comparison functions and apply[] the fact that the solution $U$ of \eqref{MCF} is approximated by $V$.”

To prove the main theorem, the authors “construct[] appropriate comparison functions” of the form
\[
 u^{\pm}(x,y,t)\coloneqq\Phi_{c^{*}}\left(\frac{y-V(x,t)}{\sqrt{1+|\nabla_{x}V|^{2}}}\mp q(t)\right)\pm p(t)\chi \left( e^{\lambda (y-c_*t)}\right),
\]
where $V$ solves
\[
V_{t}=\Delta_{x}V+\frac{c^{*}}{2}|\nabla_{x} V|^2+c_*,
\]
$\lambda<\lambda_1$, and $\chi$ is smooth with “$\chi(s)=s$ ($s \le 1/2$) and $\chi(s)=1$ ($s \ge 1$).” They then “show that $u^+$ (resp. $u^-$) becomes a supersolution (resp. a subsolution) for the problem
\begin{aligned}\label{Rea-Diff}
&u_{t}=\Delta u+f(u), && x\in\mathbb{R}^{n-1}, \, y\in\mathbb{R}, \, t>0, \\
&u(x,y,0)=u_{0}(x,y), && x\in\mathbb{R}^{n-1}, \, y\in\mathbb{R}.
\end{aligned}
if $p(t)$, $q(t)$ and the initial data for $V(x,t)$ are chosen appropriately.”

In establishing part (ii) of the main theorem, they “use[] these comparison functions and apply[] the fact that the solution $U$ of
\begin{equation}\label{MCF}
\left\{
\begin{aligned}
&\frac{U_t}{\sqrt{1+|\nabla_x U|^2}}=\mathrm{div}\left(\frac{\nabla_x U}{\sqrt{1+|\nabla_x U|}}\right)+c_*,&&x\in\mathbf{R}^{n-1},t>0,\\
&U(x,0)=\gamma(x,\tau_{\varepsilon}),&&x\in\mathbf{R}^{n-1},
\end{aligned}
\right.
\end{equation}
is approximated by $V$.”

\label{thm1}
Assume (F),
\begin{equation}
\label{init1}
\liminf_{y\to-\infty}\inf_{x\in\mathbf{R}^{n-1}}u_{0}(x,y)>0 \\
\end{equation}
and
\begin{equation}
\label{init3}
c=c_*>2\sqrt{f'(0)}, \qquad
\limsup_{y\to\infty}\sup_{x\in\mathbf{R}^{n-1}} u_{0}(x,y)e^{-\lambda_1 y}<\infty \ \mbox{ for some } \ \lambda_1<\lambda_+,
\end{equation}
and let $u$ be a solution of
\begin{aligned}\label{Rea-Diff}
&u_{t}=\Delta u+f(u), && x\in\mathbb{R}^{n-1}, \, y\in\mathbb{R}, \, t>0, \\
&u(x,y,0)=u_{0}(x,y), && x\in\mathbb{R}^{n-1}, \, y\in\mathbb{R}.
\end{aligned}.
Then there exists a smooth function  $\gamma=\gamma(x,t)$ with the following properties.
\begin{enumerate}
\item[(i)]
There exists $T>0$ such that
\begin{equation*}
\{ (x,y,t) \in \mathbf{R}^{n-1} \times \mathbf{R} \times [T,\infty); \, u(x,y,t)=\Phi_c(0)\}=\{y=\gamma(x,t)\}.
\end{equation*}
Moreover, it holds that
\begin{equation*}
\lim_{t\rightarrow\infty}\sup_{(x,y)\in\mathbf{R}^n }|u(x,y,t)-\Phi_c(y-\gamma(x,t))|=0.
\end{equation*}
\item[(ii)]
For any $\varepsilon>0$, there exists $\tau_{\varepsilon}\in[T,\infty)$ such that the solution $U(x,t)$ of the problem
\begin{equation}\label{MCF}
\left\{
\begin{aligned}
&\frac{U_t}{\sqrt{1+|\nabla_x U|^2}}=\mathrm{div}\left(\frac{\nabla_x U}{\sqrt{1+|\nabla_x U|}}\right)+c_*,&&x\in\mathbf{R}^{n-1},t>0,\\
&U(x,0)=\gamma(x,\tau_{\varepsilon}),&&x\in\mathbf{R}^{n-1},
\end{aligned}
\right.
\end{equation}
satisfies
\begin{equation*}
\sup_{x\in\mathbf{R}^{n-1},t\ge\tau_{\varepsilon}}|\gamma(x,t)-U(x,t-\tau_{\varepsilon})|\le\varepsilon.
\end{equation*}
\end{enumerate}
Furthermore, the assertion (i) still holds if
\begin{gather}
\liminf_{y \to -\infty} \inf_{x \in \mathbb{R}^{n-1}} u_0(x,y)>1-\delta_0, \label{lbidni} \\
0<\liminf_{y\to\infty}\inf_{x\in\mathbf{R}^{n-1}} \frac{u_0(x,y)}{\Phi_c(y)}, \quad
\limsup_{y\to\infty}\sup_{x\in\mathbf{R}^{n-1}}\frac{u_0(x,y)}{\Phi_c(y)}<\infty
\ \mbox{ for some } \ c \ge c_*, \\
\end{gather}
is assumed instead of the preceding condition.

Question:
Let $u=u(x,y,t)$ solve the Cauchy problem
\[
\begin{aligned}
&u_t=\Delta u+f(u), && x\in\mathbb{R}^{n-1},\ y\in\mathbb{R},\ t>0,\\
&u(x,y,0)=u_0(x,y), && x\in\mathbb{R}^{n-1},\ y\in\mathbb{R}.
\end{aligned}
\]
Assume $f$ satisfies the standing hypothesis $(F)$ from the traveling-front setting, and let $\Phi_c$ denote the associated one-dimensional front profile, i.e. a $C^2(\mathbb R)$ solution of
\[
\Phi_c''+c\Phi_c'+f(\Phi_c)=0,\qquad \Phi_c(-\infty)=1,\qquad \Phi_c(+\infty)=0,
\]
with minimal wave speed $c_*$. Let $\lambda_+$ be the larger root of $\lambda^2-c_*\lambda+f'(0)=0$. Assume moreover that
\[
\liminf_{y\to-\infty}\inf_{x\in\mathbb{R}^{n-1}}u_0(x,y)>0,
\]
and
\[
c=c_*>2\sqrt{f'(0)},\qquad
\limsup_{y\to\infty}\sup_{x\in\mathbb{R}^{n-1}} u_0(x,y)e^{-\lambda_1 y}<\infty
\quad\text{for some }\lambda_1<\lambda_+.
\]
Also consider the alternative initial-data regime, for a constant $\delta_0>0$,
\[
\liminf_{y\to -\infty} \inf_{x \in \mathbb{R}^{n-1}} u_0(x,y)>1-\delta_0,
\]
\[
0<\liminf_{y\to\infty}\inf_{x\in\mathbb{R}^{n-1}} \frac{u_0(x,y)}{\Phi_c(y)}, \qquad
\limsup_{y\to\infty}\sup_{x\in\mathbb{R}^{n-1}}\frac{u_0(x,y)}{\Phi_c(y)}<\infty
\quad\text{for some } c\ge c_*.
\]
Under these assumptions, which existence statement holds?

There exists a smooth function $\gamma=\gamma(x,t)$ such that: (i) for some $T>0$,
\[
\{(x,y,t)\in\mathbb{R}^{n-1}\times\mathbb{R}\times[T,\infty):\ u(x,y,t)=\Phi_c(0)\}=\{y=\gamma(x,t)\},
\]
and
\[
\lim_{t\to\infty}\sup_{(x,y)\in\mathbb{R}^n}\big|u(x,y,t)-\Phi_c(y-\gamma(x,t))\big|=0.
\]
(ii) For every $\varepsilon>0$, there exists $\tau_\varepsilon\in[T,\infty)$ such that if $U(x,t)$ solves
\[
\left\{
\begin{aligned}
&\frac{U_t}{\sqrt{1+|\nabla_x U|^2}}=\mathrm{div}\left(\frac{\nabla_x U}{\sqrt{1+|\nabla_x U|}}\right)+c_*,&&x\in\mathbb{R}^{n-1},\ t>0,\\
&U(x,0)=\gamma(x,\tau_\varepsilon),&&x\in\mathbb{R}^{n-1},
\end{aligned}
\right.
\]
then
\[
\sup_{x\in\mathbb{R}^{n-1},\,t\ge\tau_\varepsilon}\big|\gamma(x,t)-U(x,t-\tau_\varepsilon)\big|\le\varepsilon.
\]
Furthermore, if one replaces the initial-data assumptions above by the alternative regime
\[
\liminf_{y\to -\infty} \inf_{x \in \mathbb{R}^{n-1}} u_0(x,y)>1-\delta_0,
\]
\[
0<\liminf_{y\to\infty}\inf_{x\in\mathbb{R}^{n-1}} \frac{u_0(x,y)}{\Phi_c(y)},\qquad
\limsup_{y\to\infty}\sup_{x\in\mathbb{R}^{n-1}}\frac{u_0(x,y)}{\Phi_c(y)}<\infty
\quad\text{for some } c\ge c_*,
\]
then conclusion (i) still holds.

Choices:
B.
There exists a smooth function $\gamma=\gamma(x,t)$ such that: (i) for some $T>0$,
\[
\{(x,y,t)\in\mathbb{R}^{n-1}\times\mathbb{R}\times[T,\infty):\ u(x,y,t)=\Phi_c(0)\}=\{y=\gamma(x,t)\},
\]
and
\[
\lim_{t\to\infty}\sup_{(x,y)\in\mathbb{R}^n}\big|u(x,y,t)-\Phi_c(y-\gamma(x,t))\big|=0.
\]
(ii) There exists $\tau\in[T,\infty)$ such that if $U(x,t)$ solves
\[
\left\{
\begin{aligned}
&\frac{U_t}{\sqrt{1+|\nabla_x U|^2}}=\mathrm{div}\left(\frac{\nabla_x U}{\sqrt{1+|\nabla_x U|}}\right)+c_*,&&x\in\mathbb{R}^{n-1},\ t>0,\\
&U(x,0)=\gamma(x,\tau),&&x\in\mathbb{R}^{n-1},
\end{aligned}
\right.
\]
then
\[
\lim_{t\to\infty}\sup_{x\in\mathbb{R}^{n-1}}\big|\gamma(x,t)-U(x,t-\tau)\big|=0.
\]
Furthermore, if one replaces the initial-data assumptions above by the alternative regime
\[
\liminf_{y\to -\infty} \inf_{x \in \mathbb{R}^{n-1}} u_0(x,y)>1-\delta_0,
\]
\[
0<\liminf_{y\to\infty}\inf_{x\in\mathbb{R}^{n-1}} \frac{u_0(x,y)}{\Phi_c(y)},\qquad
\limsup_{y\to\infty}\sup_{x\in\mathbb{R}^{n-1}}\frac{u_0(x,y)}{\Phi_c(y)}<\infty
\quad\text{for some } c\ge c_*,
\]
then conclusions (i) and (ii) still hold.
C.
There exists a smooth function $\gamma=\gamma(x,t)$ such that for some $T>0$,
\[
\{(x,y,t)\in\mathbb{R}^{n-1}\times\mathbb{R}\times[T,\infty):\ u(x,y,t)=\Phi_c(0)\}=\{y=\gamma(x,t)\},
\]
and
\[
\lim_{t\to\infty}\sup_{(x,y)\in\mathbb{R}^n}\big|u(x,y,t)-\Phi_c(y-\gamma(x,t))\big|=0.
\]
Moreover, if one replaces the initial-data assumptions above by the alternative regime
\[
\liminf_{y\to -\infty} \inf_{x \in \mathbb{R}^{n-1}} u_0(x,y)>1-\delta_0,
\]
\[
0<\liminf_{y\to\infty}\inf_{x\in\mathbb{R}^{n-1}} \frac{u_0(x,y)}{\Phi_c(y)},\qquad
\limsup_{y\to\infty}\sup_{x\in\mathbb{R}^{n-1}}\frac{u_0(x,y)}{\Phi_c(y)}<\infty
\quad\text{for some } c\ge c_*,
\]
then the same conclusion still holds.
D.
There exists a smooth function $\gamma=\gamma(x,t)$ such that: (i) for some $T>0$,
\[
\{(x,y,t)\in\mathbb{R}^{n-1}\times\mathbb{R}\times[T,\infty):\ u(x,y,t)=\Phi_c(0)\}=\{y=\gamma(x,t)\},
\]
and
\[
\lim_{t\to\infty}\sup_{(x,y)\in\mathbb{R}^n}\big|u(x,y,t)-\Phi_c(y-\gamma(x,t))\big|=0.
\]
(ii) For every $\varepsilon>0$, there exists $\tau_\varepsilon\in[T,\infty)$ such that if $U(x,t)$ solves
\[
\left\{
\begin{aligned}
&U_t=\Delta_x U+\frac{c_*}{2}|\nabla_x U|^2+c_*,&&x\in\mathbb{R}^{n-1},\ t>0,\\
&U(x,0)=\gamma(x,\tau_\varepsilon),&&x\in\mathbb{R}^{n-1},
\end{aligned}
\right.
\]
then
\[
\sup_{x\in\mathbb{R}^{n-1},\,t\ge\tau_\varepsilon}\big|\gamma(x,t)-U(x,t-\tau_\varepsilon)\big|\le\varepsilon.
\]
Furthermore, if one replaces the initial-data assumptions above by the alternative regime
\[
\liminf_{y\to -\infty} \inf_{x \in \mathbb{R}^{n-1}} u_0(x,y)>1-\delta_0,
\]
\[
0<\liminf_{y\to\infty}\inf_{x\in\mathbb{R}^{n-1}} \frac{u_0(x,y)}{\Phi_c(y)},\qquad
\limsup_{y\to\infty}\sup_{x\in\mathbb{R}^{n-1}}\frac{u_0(x,y)}{\Phi_c(y)}<\infty
\quad\text{for some } c\ge c_*,
\]
then conclusion (i) still holds.
E.
There exists a smooth function $\gamma=\gamma(x,t)$ such that: (i) for some $T>0$,
\[
\{(x,y,t)\in\mathbb{R}^{n-1}\times\mathbb{R}\times[T,\infty):\ u(x,y,t)=\Phi_c(0)\}=\{y=\gamma(x,t)\},
\]
and
\[
\lim_{t\to\infty}\sup_{(x,y)\in\mathbb{R}^n}\big|u(x,y,t)-\Phi_c(y-\gamma(x,t))\big|=0.
\]
(ii) For every $\varepsilon>0$, there exists $\tau_\varepsilon\in[T,\infty)$ such that if $U(x,t)$ solves
\[
\left\{
\begin{aligned}
&\frac{U_t}{\sqrt{1+|\nabla_x U|^2}}=\mathrm{div}\left(\frac{\nabla_x U}{\sqrt{1+|\nabla_x U|}}\right)+c,&&x\in\mathbb{R}^{n-1},\ t>0,\\
&U(x,0)=\gamma(x,\tau_\varepsilon),&&x\in\mathbb{R}^{n-1},
\end{aligned}
\right.
\]
then
\[
\sup_{x\in\mathbb{R}^{n-1},\,t\ge\tau_\varepsilon}\big|\gamma(x,t)-U(x,t-\tau_\varepsilon)\big|\le\varepsilon.
\]
Furthermore, if one replaces the initial-data assumptions above by the alternative regime
\[
\liminf_{y\to -\infty} \inf_{x \in \mathbb{R}^{n-1}} u_0(x,y)>1-\delta_0,
\]
\[
0<\liminf_{y\to\infty}\inf_{x\in\mathbb{R}^{n-1}} \frac{u_0(x,y)}{\Phi_c(y)},\qquad
\limsup_{y\to\infty}\sup_{x\in\mathbb{R}^{n-1}}\frac{u_0(x,y)}{\Phi_c(y)}<\infty
\quad\text{for some } c\ge c_*,
\]
then conclusion (i) still holds.

quantifier on epsilon and extension of alternative-regime conclusion to (ii)

dropped part (ii) approximation by the geometric-flow solution

replaced the mean-curvature-type evolution approximated by V with the viscous Hamilton-Jacobi equation solved by V itself

forcing term in geometric law uses c instead of the theorem’s c_* despite the special pushed-front assumption c=c_* in part (ii)

The stem does not state the conclusion explicitly and does not single out the correct option. It provides hypotheses only, while the answer choices differ in subtle but nontrivial ways (speed, quantifiers, and scope of the alternative assumptions).

This is essentially a theorem-recall item: the stem gives the full hypotheses and asks for the corresponding conclusion. The correct choice is basically the theorem statement itself rather than a consequence derived from fresh problem-solving.

Selecting the correct answer requires some careful reasoning about quantifiers and which parts persist under modified hypotheses, but it mainly tests recognition/matching of a known result rather than substantial generative mathematical reasoning.

The distractors are strong and plausible: they alter the front speed, overextend the alternative-data conclusion, weaken or change the approximation statement, or tamper with epsilon-dependent quantifiers. These reflect realistic mathematical failure modes.

Well-constructed in terms of leakage control and distractor quality, but it is largely a theorem-statement recognition question rather than a genuinely generative reasoning task.


Question:
Let $u=u(x,y,t)$ solve the Cauchy problem
\[
\begin{aligned}
&u_t=\Delta u+f(u), && x\in\mathbb{R}^{n-1},\ y\in\mathbb{R},\ t>0,\\
&u(x,y,0)=u_0(x,y), && x\in\mathbb{R}^{n-1},\ y\in\mathbb{R}.
\end{aligned}
\]
Assume $f$ satisfies the standing hypothesis $(F)$ from the traveling-front setting, and let $\Phi_c$ denote the associated one-dimensional front profile, i.e. a $C^2(\mathbb R)$ solution of
\[
\Phi_c''+c\Phi_c'+f(\Phi_c)=0,\qquad \Phi_c(-\infty)=1,\qquad \Phi_c(+\infty)=0,
\]
with minimal wave speed $c_*$. Let $\lambda_+$ be the larger root of $\lambda^2-c_*\lambda+f'(0)=0$. Assume moreover that
\[
\liminf_{y\to-\infty}\inf_{x\in\mathbb{R}^{n-1}}u_0(x,y)>0,
\]
and
\[
c=c_*>2\sqrt{f'(0)},\qquad
\limsup_{y\to\infty}\sup_{x\in\mathbb{R}^{n-1}} u_0(x,y)e^{-\lambda_1 y}<\infty
\quad\text{for some }\lambda_1<\lambda_+.
\]
Also consider the alternative initial-data regime, for a constant $\delta_0>0$,
\[
\liminf_{y\to -\infty} \inf_{x \in \mathbb{R}^{n-1}} u_0(x,y)>1-\delta_0,
\]
\[
0<\liminf_{y\to\infty}\inf_{x\in\mathbb{R}^{n-1}} \frac{u_0(x,y)}{\Phi_c(y)}, \qquad
\limsup_{y\to\infty}\sup_{x\in\mathbb{R}^{n-1}}\frac{u_0(x,y)}{\Phi_c(y)}<\infty
\quad\text{for some } c\ge c_*.
\]
Under these assumptions, which existence statement holds?

Choices:
B.
There exists a smooth function $\gamma=\gamma(x,t)$ such that: (i) for some $T>0$,
\[
\{(x,y,t)\in\mathbb{R}^{n-1}\times\mathbb{R}\times[T,\infty):\ u(x,y,t)=\Phi_c(0)\}=\{y=\gamma(x,t)\},
\]
and
\[
\lim_{t\to\infty}\sup_{(x,y)\in\mathbb{R}^n}\big|u(x,y,t)-\Phi_c(y-\gamma(x,t))\big|=0.
\]
(ii) There exists $\tau\in[T,\infty)$ such that if $U(x,t)$ solves
\[
\left\{
\begin{aligned}
&\frac{U_t}{\sqrt{1+|\nabla_x U|^2}}=\mathrm{div}\left(\frac{\nabla_x U}{\sqrt{1+|\nabla_x U|}}\right)+c_*,&&x\in\mathbb{R}^{n-1},\ t>0,\\
&U(x,0)=\gamma(x,\tau),&&x\in\mathbb{R}^{n-1},
\end{aligned}
\right.
\]
then
\[
\lim_{t\to\infty}\sup_{x\in\mathbb{R}^{n-1}}\big|\gamma(x,t)-U(x,t-\tau)\big|=0.
\]
Furthermore, if one replaces the initial-data assumptions above by the alternative regime
\[
\liminf_{y\to -\infty} \inf_{x \in \mathbb{R}^{n-1}} u_0(x,y)>1-\delta_0,
\]
\[
0<\liminf_{y\to\infty}\inf_{x\in\mathbb{R}^{n-1}} \frac{u_0(x,y)}{\Phi_c(y)},\qquad
\limsup_{y\to\infty}\sup_{x\in\mathbb{R}^{n-1}}\frac{u_0(x,y)}{\Phi_c(y)}<\infty
\quad\text{for some } c\ge c_*,
\]
then conclusions (i) and (ii) still hold.
C.
There exists a smooth function $\gamma=\gamma(x,t)$ such that for some $T>0$,
\[
\{(x,y,t)\in\mathbb{R}^{n-1}\times\mathbb{R}\times[T,\infty):\ u(x,y,t)=\Phi_c(0)\}=\{y=\gamma(x,t)\},
\]
and
\[
\lim_{t\to\infty}\sup_{(x,y)\in\mathbb{R}^n}\big|u(x,y,t)-\Phi_c(y-\gamma(x,t))\big|=0.
\]
Moreover, if one replaces the initial-data assumptions above by the alternative regime
\[
\liminf_{y\to -\infty} \inf_{x \in \mathbb{R}^{n-1}} u_0(x,y)>1-\delta_0,
\]
\[
0<\liminf_{y\to\infty}\inf_{x\in\mathbb{R}^{n-1}} \frac{u_0(x,y)}{\Phi_c(y)},\qquad
\limsup_{y\to\infty}\sup_{x\in\mathbb{R}^{n-1}}\frac{u_0(x,y)}{\Phi_c(y)}<\infty
\quad\text{for some } c\ge c_*,
\]
then the same conclusion still holds.
D.
There exists a smooth function $\gamma=\gamma(x,t)$ such that: (i) for some $T>0$,
\[
\{(x,y,t)\in\mathbb{R}^{n-1}\times\mathbb{R}\times[T,\infty):\ u(x,y,t)=\Phi_c(0)\}=\{y=\gamma(x,t)\},
\]
and
\[
\lim_{t\to\infty}\sup_{(x,y)\in\mathbb{R}^n}\big|u(x,y,t)-\Phi_c(y-\gamma(x,t))\big|=0.
\]
(ii) For every $\varepsilon>0$, there exists $\tau_\varepsilon\in[T,\infty)$ such that if $U(x,t)$ solves
\[
\left\{
\begin{aligned}
&U_t=\Delta_x U+\frac{c_*}{2}|\nabla_x U|^2+c_*,&&x\in\mathbb{R}^{n-1},\ t>0,\\
&U(x,0)=\gamma(x,\tau_\varepsilon),&&x\in\mathbb{R}^{n-1},
\end{aligned}
\right.
\]
then
\[
\sup_{x\in\mathbb{R}^{n-1},\,t\ge\tau_\varepsilon}\big|\gamma(x,t)-U(x,t-\tau_\varepsilon)\big|\le\varepsilon.
\]
Furthermore, if one replaces the initial-data assumptions above by the alternative regime
\[
\liminf_{y\to -\infty} \inf_{x \in \mathbb{R}^{n-1}} u_0(x,y)>1-\delta_0,
\]
\[
0<\liminf_{y\to\infty}\inf_{x\in\mathbb{R}^{n-1}} \frac{u_0(x,y)}{\Phi_c(y)},\qquad
\limsup_{y\to\infty}\sup_{x\in\mathbb{R}^{n-1}}\frac{u_0(x,y)}{\Phi_c(y)}<\infty
\quad\text{for some } c\ge c_*,
\]
then conclusion (i) still holds.
E.
There exists a smooth function $\gamma=\gamma(x,t)$ such that: (i) for some $T>0$,
\[
\{(x,y,t)\in\mathbb{R}^{n-1}\times\mathbb{R}\times[T,\infty):\ u(x,y,t)=\Phi_c(0)\}=\{y=\gamma(x,t)\},
\]
and
\[
\lim_{t\to\infty}\sup_{(x,y)\in\mathbb{R}^n}\big|u(x,y,t)-\Phi_c(y-\gamma(x,t))\big|=0.
\]
(ii) For every $\varepsilon>0$, there exists $\tau_\varepsilon\in[T,\infty)$ such that if $U(x,t)$ solves
\[
\left\{
\begin{aligned}
&\frac{U_t}{\sqrt{1+|\nabla_x U|^2}}=\mathrm{div}\left(\frac{\nabla_x U}{\sqrt{1+|\nabla_x U|}}\right)+c,&&x\in\mathbb{R}^{n-1},\ t>0,\\
&U(x,0)=\gamma(x,\tau_\varepsilon),&&x\in\mathbb{R}^{n-1},
\end{aligned}
\right.
\]
then
\[
\sup_{x\in\mathbb{R}^{n-1},\,t\ge\tau_\varepsilon}\big|\gamma(x,t)-U(x,t-\tau_\varepsilon)\big|\le\varepsilon.
\]
Furthermore, if one replaces the initial-data assumptions above by the alternative regime
\[
\liminf_{y\to -\infty} \inf_{x \in \mathbb{R}^{n-1}} u_0(x,y)>1-\delta_0,
\]
\[
0<\liminf_{y\to\infty}\inf_{x\in\mathbb{R}^{n-1}} \frac{u_0(x,y)}{\Phi_c(y)},\qquad
\limsup_{y\to\infty}\sup_{x\in\mathbb{R}^{n-1}}\frac{u_0(x,y)}{\Phi_c(y)}<\infty
\quad\text{for some } c\ge c_*,
\]
then conclusion (i) still holds.

The stem does not explicitly reveal the correct option, and no single phrase directly points to choice A. Although the setup is highly specialized and theorem-specific, the answer is not leaked in a trivial way.

This is essentially a theorem-recall item: the stem lists the full hypotheses and asks which existence statement holds. The correct choice is basically the theorem conclusion, so the task is close to matching premises to a restated result rather than reasoning from competing conclusions.

There is some reasoning pressure because the choices differ by subtle quantifiers, PDE laws, and scope of the alternative-regime conclusion. However, the item mainly tests precise recognition/recall of the theorem statement, not genuine generative mathematical reasoning.

The distractors are mathematically plausible and targeted: one is weaker-but-true, others alter quantifiers or replace the geometric evolution law in realistic ways. These reflect common failure modes in reading advanced theorem statements.

Technically well-constructed with strong distractors, but it is largely a theorem-restatement/recognition question rather than a non-tautological test of generative reasoning.