[Dynamical Non-Commutative Bifurcation]\label{thm:main_bifurcation}
Let $P_0(x)$ be a central polynomial over $\mathbb{A} \in \{\mathbb{H}, \mathbb{O}\}$ such that its root variety $V_0 = Z(P_0)$ contains a non-real manifold of dimension $d_{\rm M} > 0$. Let $P_\epsilon(x)$ be a generically non-central analytic deformation. Let $V_\epsilon = Z(P_\epsilon)$. Let $d_{\rm A} = \dim(\mathbb{A})$.
\begin{enumerate}
    \item \textbf{Topological Collapse:} The Hausdorff dimension of the root variety changes discontinuously at $\epsilon=0$.
    \item \textbf{Algebraic Singularity:} The central fiber $V_0$ is characterized by a Jacobian rank deficiency: $\rank(J_{P_0}(x)) = d_{\rm A} - d_{\rm M}$.
    \item \textbf{Dynamical Retraction:} Let $\mathcal{V}_\epsilon(x) = \|P_\epsilon(x)\|^2$. The gradient flow $\dot{x} = -\nabla\mathcal{V}_\epsilon(x)$ realizes the collapse as a deformation retract. The timescale exhibits critical slowing down: $T_{\rm collapse} \propto \epsilon^{-2}$.
    \item \textbf{Thermodynamic Phase Transition:} In a statistical ensemble at temperature $T$, the collapse manifests as a phase transition. An alignment order parameter $m(\epsilon,T)$ exhibits a discontinuity at $(\epsilon, T) = (0, 0)$.
\end{enumerate}

The study of polynomial roots over non-commutative division algebras has a rich history, dating back to the foundational work on quaternions ($\mathbb{H}$) by Niven \cite{niven1941equations} and Eilenberg and Niven \cite{eilenberg1944fundamental}. A key observation, formalized by Gordon and Motzkin \cite{gordon1965zeros}, is that the algebraic structure of the underlying space profoundly affects the geometry of the solution set (see also \cite{lam2001first}). The failure of commutativity (and associativity in the octonions $\mathbb{O}$) induces a dimensional inflation of the root set. For central polynomials (coefficients in $\mathbb{R}$), 0-dimensional sets in $\mathbb{C}$ inflate into continuous manifolds ($S^2$ in $\mathbb{H}$, $S^6$ in $\mathbb{O}$). The associated symmetry group transitions from a discrete group to a compact Lie group ($SO(3)$ or $G_2$), arising from the action of the automorphism group.

This paper introduces a dynamical framework, Dynamical Non-Commutative Algebraic Geometry (DNCAG), where we treat the polynomial coefficients as parameters. We move beyond the static characterization of root sets to analyze their evolution, stability, and bifurcations. While the underlying algebraic structures are established, our approach utilizes tools from dynamical systems (cf. \cite{guckenheimer1983nonlinear}), algebraic deformation theory (cf. \cite{gerstenhaber1964deformation}), and Morse-Bott theory (cf. \cite{bott1954nondegenerate}) to analyze the stability of these structures under perturbations.

The motivation for developing DNCAG extends beyond pure mathematics into theoretical physics. The geometric structures analyzed here—spherical manifolds arising from algebraic constraints—are ubiquitous. In quaternionic quantum mechanics \cite{adler1995quaternionic}, these manifolds may represent parameter spaces or emergent symmetries. The dynamical collapse of topology offers a novel algebraic model for phenomena involving dimensional reduction and symmetry breaking, such as those hypothesized near black hole singularities or characterizing transitions between topological phases of matter. Furthermore, the extension to octonions is motivated by their potential role in unified theories and their unique algebraic properties \cite{baez2002octonions}.

Our main original contributions are:
\begin{enumerate}
    \item The analysis of central dynamics (breathing modes), including spectral characterization of non-linear coupling and the classification of dynamics near central bifurcations (transversal vs. tangential crossings).
    \item The formalization of the topological collapse dynamics using gradient flow, including the proof of critical slowing down and the analysis of basin decomposition.
    \item The introduction of a thermodynamic formalism (Gibbs measure, Entropy Scaling Law, Order Parameter) to characterize the algebraic collapse as a statistical phase transition.
\end{enumerate}

\subsection{Main Results}

The central contribution of this paper is the characterization of the topological instability induced by non-central perturbations across division algebras.

\begin{definition}[Analytic Deformation]\label{def:analytic_deformation}
Let $\mathbb{A}$ be a real division algebra. Let $P_0(x) \in \mathbb{R}[x]$ be a central polynomial. Let $G_P$ be the automorphism stabilizer of $P$ (Definition~\ref{def:aut_stabilizer}). A generically non-central analytic deformation is a family of polynomials $P_\epsilon(x) \in \mathbb{A}[x]$ parameterized by $\epsilon \in \mathbb{R}$, such that the coefficients depend analytically on $\epsilon$, $P_\epsilon(x) = P_0(x)$ when $\epsilon=0$, and $G_{P_\epsilon}$ is trivial for $\epsilon \neq 0$ in a punctured neighborhood of $\epsilon=0$.
\end{definition}

The study of polynomial roots over non-commutative division algebras has a rich history, dating back to the foundational work on quaternions ($\mathbb{H}$) by Niven \cite{niven1941equations} and Eilenberg and Niven \cite{eilenberg1944fundamental}. A key observation, formalized by Gordon and Motzkin \cite{gordon1965zeros}, is that the algebraic structure of the underlying space profoundly affects the geometry of the solution set (see also \cite{lam2001first}). The failure of commutativity (and associativity in the octonions $\mathbb{O}$) induces a dimensional inflation of the root set. For central polynomials (coefficients in $\mathbb{R}$), 0-dimensional sets in $\mathbb{C}$ inflate into continuous manifolds ($S^2$ in $\mathbb{H}$, $S^6$ in $\mathbb{O}$). The associated symmetry group transitions from a discrete group to a compact Lie group ($SO(3)$ or $G_2$), arising from the action of the automorphism group.

This paper introduces a dynamical framework, Dynamical Non-Commutative Algebraic Geometry (DNCAG), where we treat the polynomial coefficients as parameters. We move beyond the static characterization of root sets to analyze their evolution, stability, and bifurcations. While the underlying algebraic structures are established, our approach utilizes tools from dynamical systems (cf. \cite{guckenheimer1983nonlinear}), algebraic deformation theory (cf. \cite{gerstenhaber1964deformation}), and Morse-Bott theory (cf. \cite{bott1954nondegenerate}) to analyze the stability of these structures under perturbations.

The motivation for developing DNCAG extends beyond pure mathematics into theoretical physics. The geometric structures analyzed here—spherical manifolds arising from algebraic constraints—are ubiquitous. In quaternionic quantum mechanics \cite{adler1995quaternionic}, these manifolds may represent parameter spaces or emergent symmetries. The dynamical collapse of topology offers a novel algebraic model for phenomena involving dimensional reduction and symmetry breaking, such as those hypothesized near black hole singularities or characterizing transitions between topological phases of matter. Furthermore, the extension to octonions is motivated by their potential role in unified theories and their unique algebraic properties \cite{baez2002octonions}.

Our main original contributions are:
\begin{enumerate}
 \item The analysis of central dynamics (breathing modes), including spectral characterization of non-linear coupling and the classification of dynamics near central bifurcations (transversal vs. tangential crossings).
 \item The formalization of the topological collapse dynamics using gradient flow, including the proof of critical slowing down and the analysis of basin decomposition.
 \item The introduction of a thermodynamic formalism (Gibbs measure, Entropy Scaling Law, Order Parameter) to characterize the algebraic collapse as a statistical phase transition.
\end{enumerate}

The central contribution of this paper is the characterization of the topological instability induced by non-central perturbations across division algebras.

\begin{definition}[Analytic Deformation]\label{def:analytic_deformation}
Let $\mathbb{A}$ be a real division algebra. Let $P_0(x) \in \mathbb{R}[x]$ be a central polynomial. Let $G_P$ be the automorphism stabilizer of $P$ (Definition~\ref{def:aut_stabilizer}). A generically non-central analytic deformation is a family of polynomials $P_\epsilon(x) \in \mathbb{A}[x]$ parameterized by $\epsilon \in \mathbb{R}$, such that the coefficients depend analytically on $\epsilon$, $P_\epsilon(x) = P_0(x)$ when $\epsilon=0$, and $G_{P_\epsilon}$ is trivial for $\epsilon \neq 0$ in a punctured neighborhood of $\epsilon=0$.
\end{definition}

\begin{definition}[Automorphism Stabilizer of the Polynomial]\label{def:aut_stabilizer}
Let $P(x) \in \A[x]$. The automorphism stabilizer of $P$ is the subgroup of $\Aut(\A)$ that fixes all coefficients:
\begin{equation}
G_P = \{g \in \Aut(\A) \mid g(a_k) = a_k \text{ for all } k\}.
\end{equation}
\end{definition}

\abstract{We develop a framework for dynamical non-commutative algebraic geometry (DNCAG) by analyzing the evolution and stability of polynomial root manifolds in real normed division algebras ($\HH$ and $\OO$). We establish a Generalized Inflation Theorem, demonstrating that for central polynomials, the root set forms a homogeneous space $G/H$, where $G$ is the automorphism group of the algebra ($SO(3)$ for $\HH$, $G_2$ for $\OO$). This mechanism generates continuous geometry from non-commutativity. We analyze the dynamics under central modulation (breathing modes), classifying topological bifurcations ($\Delta=0$). We then analyze the topological collapse induced by non-central perturbations, governed by symmetry reduction. We utilize the Localization Theorem (Gordon-Motzkin) to explain the alignment of roots with coefficient subalgebras. We formalize the dynamics of collapse using gradient flow on the potential landscape $\mathcal{V}(x) = \|P(x)\|^2$, characterizing it as a deformation retract and proving that the collapse timescale exhibits critical slowing down with quadratic scaling ($T_{\rm collapse} \propto \epsilon^{-2}$). Finally, we introduce a thermodynamic formalism, proving an Entropy Scaling Law that rigorously characterizes the collapse as a symmetry-breaking phase transition.}

Our main original contributions are:
\begin{enumerate}
 \item The analysis of central dynamics (breathing modes), including spectral characterization of non-linear coupling and the classification of dynamics near central bifurcations (transversal vs. tangential crossings).
 \item The formalization of the topological collapse dynamics using gradient flow, including the proof of critical slowing down and the analysis of basin decomposition.
 \item The introduction of a thermodynamic formalism (Gibbs measure, Entropy Scaling Law, Order Parameter) to characterize the algebraic collapse as a statistical phase transition.
\end{enumerate}

\begin{definition}[Analytic Deformation]\label{def:analytic_deformation}
Let $\A$ be a real division algebra. Let $P_0(x) \in \R[x]$ be a central polynomial. Let $G_P$ be the automorphism stabilizer of $P$ (Definition~\ref{def:aut_stabilizer}). A generically non-central analytic deformation is a family of polynomials $P_\epsilon(x) \in \A[x]$ parameterized by $\epsilon \in \R$, such that the coefficients depend analytically on $\epsilon$, $P_\epsilon(x) = P_0(x)$ when $\epsilon=0$, and $G_{P_\epsilon}$ is trivial for $\epsilon \neq 0$ in a punctured neighborhood of $\epsilon=0$.
\end{definition}

\begin{proof}
We synthesize the results established in the subsequent sections.

\begin{proposition}[Topological Collapse]\label{prop:collapse}
For a generically non-central perturbation $P_\epsilon(x)$, the root manifold undergoes a topological phase transition. The Hausdorff dimension of the solution set $Z(P_\epsilon)$ changes discontinuously at $\epsilon=0$. (Part 1 of Main Theorem~\ref{thm:main_bifurcation}).
\end{proposition}

\begin{figure}[htbp]
 \centering
 \includegraphics[width=\textwidth]{topological_collapse.pdf}
 \caption{Visualization of Topological Collapse via Gradient Flow (in $\HH$). The sequence illustrates the transition from $S^2$ to $S^0$ under a non-central perturbation aligned with the $i$-axis. (1) Initial state $V_0$. (2-4) Evolution under the gradient flow $-\nabla\mathcal{V}_\epsilon$. (5) Final state $V_\epsilon$: The manifold has collapsed onto the isolated roots (Alignment Principle). An animation of the collapse is shown in \url{http://youtu.be/yaJQOuftZjE}}
 \label{fig:topological_collapse}
\end{figure}

\begin{theorem}[Symmetry Breaking and Phase Transition]\label{thm:phase_transition}
Consider a perturbation $P_\epsilon(x)$ breaking the symmetry along the $\nu$-axis. We analyze the behavior in the zero temperature limit ($T \to 0$). (Part 4 of Main Theorem~\ref{thm:main_bifurcation}).
\begin{enumerate}
 \item \textbf{Symmetric Phase (Disordered):} For $\epsilon=0$, the measure concentrates uniformly on $S^{d_{\rm A}-2}$. $m(0, 0) = 1/(d_{\rm A}-1)$.
 \item \textbf{Ordered Phase (Aligned):} For $\epsilon \neq 0$, the measure concentrates on the isolated roots aligned with the $\nu$-axis. $m(\epsilon, 0) = 1$.
\end{enumerate}
\end{theorem}

The analysis reveals a unified picture across the real normed division algebras:
\begin{enumerate}
 \item The geometry of central roots is determined by the automorphism group of the algebra (Generalized Inflation Theorem).
 \item Central dynamics are governed by the auxiliary polynomial and its discriminant, independent of associativity (due to power-associativity).
 \item Topological collapse is driven by symmetry reduction (Generalized Symmetry Reduction Theorem) and the Alignment Principle (Localization Theorem), which holds due to alternativity.
 \item The dynamics of collapse follow a gradient flow characterized by Critical Slowing Down ($T \propto \epsilon^{-2}$).
 \item The collapse can be rigorously characterized as a thermodynamic phase transition (Entropy Scaling Law and Order Parameter analysis).
\end{enumerate}

\begin{definition}[Automorphism Stabilizer of the Polynomial]\label{def:aut_stabilizer}
Let $P(x) \in \A[x]$. The automorphism stabilizer of $P$ is the subgroup of $\Aut(\A)$ that fixes all coefficients:
\begin{equation}
G_P = \{g \in \Aut(\A) \mid g(a_k) = a_k \text{ for all } k\}.
\end{equation}
\end{definition}

\begin{proof}
We synthesize the results established in the subsequent sections.

\textbf{Part 1 (Topological Collapse):} For $\epsilon=0$, by the Dimensional Inflation Law (Corollary~\ref{cor:inflation_law}), $d_{\rm M} = d_{\rm A} - 2$. For $\epsilon \neq 0$, $G_{P_\epsilon}$ is trivial. By the Generalized Symmetry Reduction Theorem (Theorem~\ref{thm:generalized_symmetry_reduction}), the roots are isolated ($d_{\rm M}=0$). The discontinuity follows (Proposition~\ref{prop:collapse}).

\textbf{Part 2 (Algebraic Singularity):} The tangent space $T_x V_0$ is related to $\Ker(J_{P_0}(x))$. The rank deficiency follows from the positive dimension of the manifold (Theorem~\ref{thm:jacobian_singularity}).

\textbf{Part 3 (Dynamical Retraction):} The potential $\mathcal{V}_\epsilon(x)$ is real-analytic and coercive. The Lojasiewicz gradient inequality \cite{lojasiewicz1963propriete} guarantees convergence of the gradient flow (Theorem~\ref{thm:deformation_retract}). The timescale scaling $T_{\rm collapse} \propto \epsilon^{-2}$ is derived from the quadratic scaling of the restricted potential (Theorem~\ref{thm:collapse_timescale}) and the generic Lojasiewicz exponent (Proposition~\ref{prop:lojasiewicz_exponent}).

\textbf{Part 4 (Thermodynamic Phase Transition):} We define the Gibbs measure (Definition~\ref{def:gibbs_measure}) and the alignment order parameter $m(\epsilon, T)$ (Definition~\ref{def:order_parameter}). In the limit $T \to 0$, the measure concentrates on $V_\epsilon$. For $\epsilon=0$, symmetry implies $m(0, 0) = 1/(d_{\rm A}-1)$. For $\epsilon \neq 0$, the Localization Theorem (Theorem~\ref{thm:algebraic_alignment}) implies alignment, $m(\epsilon, 0) = 1$. The discontinuity characterizes the phase transition (Theorem~\ref{thm:phase_transition}).
\end{proof}

To prove Theorem~\ref{thm:main_bifurcation} the proof \"synthesizes the results established in the subsequent sections\" and proceeds by parts:

\textbf{Part 1 (Topological Collapse):} For $\epsilon=0$, \"by the Dimensional Inflation Law (Corollary~\ref{cor:inflation_law}), $d_{\rm M} = d_{\rm A} - 2$.\" For $\epsilon\neq 0$, \"$G_{P_\epsilon}$ is trivial\" and \"by the Generalized Symmetry Reduction Theorem (Theorem~\ref{thm:generalized_symmetry_reduction}), the roots are isolated ($d_{\rm M}=0$).\" Hence \"the discontinuity follows (Proposition~\ref{prop:collapse}).\"

\textbf{Part 2 (Algebraic Singularity):} \"The tangent space $T_x V_0$ is related to $\Ker(J_{P_0}(x))$.\" Therefore \"the rank deficiency follows from the positive dimension of the manifold (Theorem~\ref{thm:jacobian_singularity}).\"

\textbf{Part 3 (Dynamical Retraction):} \"The potential $\mathcal{V}_\epsilon(x)$ is real-analytic and coercive.\" Then \"the Lojasiewicz gradient inequality ... guarantees convergence of the gradient flow (Theorem~\ref{thm:deformation_retract}).\" The scaling \"$T_{\rm collapse} \propto \epsilon^{-2}$ is derived from the quadratic scaling of the restricted potential (Theorem~\ref{thm:collapse_timescale}) and the generic Lojasiewicz exponent (Proposition~\ref{prop:lojasiewicz_exponent}).\"

\textbf{Part 4 (Thermodynamic Phase Transition):} \"We define the Gibbs measure (Definition~\ref{def:gibbs_measure}) and the alignment order parameter $m(\epsilon, T)$ (Definition~\ref{def:order_parameter}).\" As \"$T\to 0$, the measure concentrates on $V_\epsilon$.\" For $\epsilon=0$, \"symmetry implies $m(0, 0) = 1/(d_{\rm A}-1)$.\" For $\epsilon\neq 0$, \"the Localization Theorem (Theorem~\ref{thm:algebraic_alignment}) implies alignment, $m(\epsilon, 0) = 1$.\" \"The discontinuity characterizes the phase transition (Theorem~\ref{thm:phase_transition}).\"

To prove the main theorem, the proof "synthesizes the results established in the subsequent sections" and proceeds by parts:

\textbf{Part 1 (Topological Collapse):} For $\epsilon=0$, by the following result.

\begin{corollary}[Dimensional Inflation Law]\label{cor:inflation_law}
Let $\A$ be a real normed division algebra ($\C, \HH, \OO$) of dimension $d_{\rm A} \ge 2$. The dimension of the root manifold $M_{x_0}$ is $d_{\rm M} = d_{\rm A} - 2$.
\end{corollary}

For $\epsilon\neq 0$, "$G_{P_\epsilon}$ is trivial" and by the following theorem, the roots are isolated ($d_{\rm M}=0$).

\begin{theorem}[Generalized Symmetry Reduction Theorem]\label{thm:generalized_symmetry_reduction}
The root set $Z(P)$ is invariant under the action of $G_P$. The geometry of the root manifolds is determined by the orbits of $G_P$.
\end{theorem}

Hence the discontinuity follows from the following proposition.

\begin{proposition}[Topological Collapse]\label{prop:collapse}
For a generically non-central perturbation $P_\epsilon(x)$, the root manifold undergoes a topological phase transition. The Hausdorff dimension of the solution set $Z(P_\epsilon)$ changes discontinuously at $\epsilon=0$. (Part 1 of Main Theorem~\ref{thm:main_bifurcation}).
\end{proposition}

\textbf{Part 2 (Algebraic Singularity):} "The tangent space $T_x V_0$ is related to $\Ker(J_{P_0}(x))$." Therefore "the rank deficiency follows from the positive dimension of the manifold" by the following theorem.

\begin{theorem}[Jacobian Singularity of the Central Fiber]\label{thm:jacobian_singularity}
Let $V_0$ be the root manifold of a central polynomial $P_0$. The Jacobian $J_{P_0}(x)$ (viewed as a map $\R^{d_{\rm A}} \to \R^{d_{\rm A}}$) is singular for all $x \in V_0$. (Part 2 of Main Theorem~\ref{thm:main_bifurcation}).
\end{theorem}

\textbf{Part 3 (Dynamical Retraction):} "The potential $\mathcal{V}_\epsilon(x)$ is real-analytic and coercive." Then "the Lojasiewicz gradient inequality ... guarantees convergence of the gradient flow" via the following theorem.

\begin{theorem}[Deformation Retract]\label{thm:deformation_retract}
The gradient flow defines a deformation retract from the initial manifold $V_0$ onto the final set of attractors $V_\epsilon$. (Part 3 of Main Theorem~\ref{thm:main_bifurcation}).
\end{theorem}

The scaling "$T_{\rm collapse} \propto \epsilon^{-2}$ is derived from the quadratic scaling of the restricted potential" using the following theorem and proposition.

\begin{theorem}[Critical Slowing Down (Quadratic Scaling)]\label{thm:collapse_timescale}
The characteristic relaxation time $T_{\rm collapse}$ for the topological collapse under a generic non-central perturbation of magnitude $\epsilon$ scales as $T_{\rm collapse} \propto 1/\epsilon^2$.
\end{theorem}

\begin{proposition}[Lojasiewicz Exponent]\label{prop:lojasiewicz_exponent}
For a generic analytic perturbation (Morse function $f_\epsilon$), the Lojasiewicz exponent characterizing the flatness of the potential near the critical manifold $V_0$ is $\theta = 1/2$. This yields the quadratic timescale $T_{\rm collapse} \propto \epsilon^{-2}$. At non-generic perturbations (where the restricted potential is flatter than quadratic), $\theta$ may be smaller ($0 < \theta < 1/2$), leading to slower collapse timescales $T_{\rm collapse} \propto \epsilon^{-(1+\theta)/\theta}$.
\end{proposition}

\textbf{Part 4 (Thermodynamic Phase Transition):} "We define the Gibbs measure" as follows.

\begin{definition}[Gibbs Measure and Partition Function]\label{def:gibbs_measure}
The Gibbs probability measure $\mu_T$ on $\A$ is defined by
\begin{equation}
d\mu_T(x) = \frac{1}{Z(T)} \exp\left(-\frac{\mathcal{V}(x)}{T}\right) dx,
\end{equation}
where $Z(T) = \int_{\A} \exp(-\mathcal{V}(x)/T) dx$ is the partition function.
\end{definition}

and "the alignment order parameter $m(\epsilon, T)$" as follows.

\begin{definition}[Alignment Order Parameter]\label{def:order_parameter}
Let $\nu$ be a unit imaginary vector defining the perturbation axis. The alignment order parameter $m_\nu$ is the normalized mean square alignment of the coordinate along $\nu$ under the Gibbs measure. For $\HH$ ($d_{\rm A}=4$) aligned along $i$ (coordinate $b$):
\begin{equation}
m(\epsilon, T) = \frac{\langle b^2 \rangle_{\mu_T(\epsilon)}}{\langle b^2+c^2+d^2 \rangle_{\mu_T(\epsilon)}}.
\end{equation}
\end{definition}

As "$T\to 0$, the measure concentrates on $V_\epsilon$." For $\epsilon=0$, "symmetry implies $m(0, 0) = 1/(d_{\rm A}-1)$." For $\epsilon\neq 0$, the following theorem implies alignment, $m(\epsilon, 0) = 1$.

\begin{theorem}[Localization Theorem (Gordon-Motzkin)]\label{thm:algebraic_alignment}
Let $\A$ be a real normed division algebra ($\C, \HH, \OO$). Let $P(x) \in \A[x]$. Any isolated root $x_0$ of $P(x)$ belongs to the coefficient subalgebra $\A(P)$.
\end{theorem}

The discontinuity characterizes the phase transition by the following theorem.

\begin{theorem}[Symmetry Breaking and Phase Transition]\label{thm:phase_transition}
Consider a perturbation $P_\epsilon(x)$ breaking the symmetry along the $\nu$-axis. We analyze the behavior in the zero temperature limit ($T \to 0$). (Part 4 of Main Theorem~\ref{thm:main_bifurcation}).
\begin{enumerate}
    \item \textbf{Symmetric Phase (Disordered):} For $\epsilon=0$, the measure concentrates uniformly on $S^{d_{\rm A}-2}$. $m(0, 0) = 1/(d_{\rm A}-1)$.
    \item \textbf{Ordered Phase (Aligned):} For $\epsilon \neq 0$, the measure concentrates on the isolated roots aligned with the $\nu$-axis. $m(\epsilon, 0) = 1$.
\end{enumerate}
\end{theorem}

Question: Let \(\mathbb{A}\in\{\mathbb{H},\mathbb{O}\}\) and let \(d_{\rm A}=\dim(\mathbb{A})\). Suppose \(P_0(x)\) is a central polynomial over \(\mathbb{A}\), meaning its coefficients lie in \(\mathbb{R}\), and let its root variety be \(V_0=Z(P_0)\). Assume \(V_0\) contains a non-real manifold of dimension \(d_{\rm M}>0\). Let \(P_\epsilon(x)\in \mathbb{A}[x]\) be a generically non-central analytic deformation of \(P_0\), meaning that the coefficients of \(P_\epsilon\) depend analytically on \(\epsilon\in\mathbb{R}\), one has \(P_\epsilon=P_0\) at \(\epsilon=0\), and for \(\epsilon\neq 0\) in some punctured neighborhood of \(0\), the automorphism stabilizer \(G_{P_\epsilon}=\{g\in \operatorname{Aut}(\mathbb{A})\mid g(a_k)=a_k\text{ for all coefficients }a_k\}\) is trivial. Write \(V_\epsilon=Z(P_\epsilon)\), and let \(\mathcal{V}_\epsilon(x)=\|P_\epsilon(x)\|^2\). Which statement holds for every such choice of \(\mathbb{A}\), \(P_0\), and deformation \(P_\epsilon\)?

All of the following hold: (1) the Hausdorff dimension of the root variety changes discontinuously at \(\epsilon=0\); (2) the central fiber \(V_0\) is characterized by a Jacobian rank deficiency, namely \(\operatorname{rank}(J_{P_0}(x))=d_{\rm A}-d_{\rm M}\); (3) the gradient flow \(\dot x=-\nabla \mathcal{V}_\epsilon(x)\) realizes the collapse as a deformation retract, and its collapse timescale satisfies \(T_{\rm collapse}\propto \epsilon^{-2}\); and (4) in a statistical ensemble at temperature \(T\), the collapse appears as a phase transition, with an alignment order parameter \(m(\epsilon,T)\) that is discontinuous at \((\epsilon,T)=(0,0)\).

Choices:
B. All of the following hold: (1) the Hausdorff dimension of the root variety changes discontinuously at \(\epsilon=0\); (2) the central fiber \(V_0\) is characterized by a Jacobian rank deficiency, namely \(\operatorname{rank}(J_{P_0}(x))=d_{\rm A}-d_{\rm M}\); (3) the gradient flow \(\dot x=-\nabla \mathcal{V}_\epsilon(x)\) realizes the collapse as a deformation retract, and its collapse timescale satisfies \(T_{\rm collapse}\propto \epsilon^{-2}\) for every analytic deformation; and (4) in a statistical ensemble at temperature \(T\), the collapse appears as a phase transition, with an alignment order parameter \(m(\epsilon,T)\) that is discontinuous at \((\epsilon,T)=(0,0)\).
C. The Hausdorff dimension of the root variety changes discontinuously at \(\epsilon=0\), the central fiber \(V_0\) has Jacobian rank deficiency, and the gradient flow \(\dot x=-\nabla \mathcal{V}_\epsilon(x)\) realizes the collapse as a deformation retract.
D. All of the following hold: (1) the Hausdorff dimension of the root variety changes discontinuously at \(\epsilon=0\); (2) the central fiber \(V_0\) is characterized by a Jacobian rank deficiency, namely \(\operatorname{rank}(J_{P_0}(x))=d_{\rm A}-d_{\rm M}\); (3) the gradient flow \(\dot x=-\nabla \mathcal{V}_\epsilon(x)\) realizes the collapse as a deformation retract, and its collapse timescale satisfies \(T_{\rm collapse}\propto \epsilon^{-2}\); and (4) in a statistical ensemble at temperature \(T\), the collapse appears as a phase transition, with an alignment order parameter \(m(\epsilon,T)\) that is discontinuous at every point of the line \(T=0\).
E. All of the following hold: (1) for \(\epsilon\neq 0\) sufficiently small, the root variety \(V_\epsilon\) still contains a non-real manifold of dimension \(d_{\rm A}-2\), so the Hausdorff dimension varies continuously through \(\epsilon=0\); (2) the central fiber \(V_0\) is characterized by a Jacobian rank deficiency, namely \(\operatorname{rank}(J_{P_0}(x))=d_{\rm A}-d_{\rm M}\); (3) the gradient flow \(\dot x=-\nabla \mathcal{V}_\epsilon(x)\) realizes the collapse as a deformation retract, and its collapse timescale satisfies \(T_{\rm collapse}\propto \epsilon^{-2}\); and (4) in a statistical ensemble at temperature \(T\), the collapse appears as a phase transition, with an alignment order parameter \(m(\epsilon,T)\) that is discontinuous at \((\epsilon,T)=(0,0)\).


Question: Let \(\mathbb{A}\in\{\mathbb{H},\mathbb{O}\}\) be a real division algebra, let \(d_{\rm A}=\dim(\mathbb{A})\), and let \(P_0(x)\in\mathbb{R}[x]\) be a central polynomial whose root variety \(V_0=Z(P_0)\) contains a non-real manifold of dimension \(d_{\rm M}>0\). Let \(P_\epsilon(x)\in\mathbb{A}[x]\) be a generically non-central analytic deformation, meaning that its coefficients depend analytically on \(\epsilon\), \(P_\epsilon(x)=P_0(x)\) at \(\epsilon=0\), and for all \(\epsilon\neq 0\) in a punctured neighborhood of \(0\), the automorphism stabilizer \(G_{P_\epsilon}=\{g\in\operatorname{Aut}(\mathbb{A})\mid g\text{ fixes every coefficient of }P_\epsilon\}\) is trivial. Write \(V_\epsilon=Z(P_\epsilon)\), define \(\mathcal{V}_\epsilon(x)=\|P_\epsilon(x)\|^2\), and let \(J_{P_0}(x)\) denote the Jacobian of \(P_0\). Under these assumptions, as \(\epsilon\to 0\), which limiting/asymptotic statement holds for the root varieties, the gradient-flow collapse, and the statistical order parameter?

The bifurcation satisfies all of the following: (i) the Hausdorff dimension of the root variety \(V_\epsilon\) changes discontinuously at \(\epsilon=0\); (ii) the central fiber \(V_0\) is characterized by a Jacobian rank deficiency, namely \(\operatorname{rank}(J_{P_0}(x))=d_{\rm A}-d_{\rm M}\); (iii) for \(\mathcal{V}_\epsilon(x)=\|P_\epsilon(x)\|^2\), the gradient flow \(\dot x=-\nabla \mathcal{V}_\epsilon(x)\) realizes the collapse as a deformation retract, and its timescale obeys the critical-slowing-down law \(T_{\rm collapse}\propto \epsilon^{-2}\); and (iv) in a statistical ensemble at temperature \(T\), the collapse appears as a phase transition, with the alignment order parameter \(m(\epsilon,T)\) discontinuous at \((\epsilon,T)=(0,0)\).

Choices:
B. The bifurcation satisfies all of the following: (i) the Hausdorff dimension of the root variety \(V_\epsilon\) varies continuously through \(\epsilon=0\), with the non-real component persisting for sufficiently small \(\epsilon\neq 0\); (ii) the central fiber \(V_0\) is characterized by a Jacobian rank deficiency, namely \(\operatorname{rank}(J_{P_0}(x))=d_{\rm A}-d_{\rm M}\); (iii) for \(\mathcal{V}_\epsilon(x)=\|P_\epsilon(x)\|^2\), the gradient flow \(\dot x=-\nabla \mathcal{V}_\epsilon(x)\) realizes the collapse as a deformation retract, and its timescale obeys \(T_{\rm collapse}\propto \epsilon^{-2}\); and (iv) in a statistical ensemble at temperature \(T\), the alignment order parameter \(m(\epsilon,T)\) remains continuous at \((\epsilon,T)=(0,0)\).
C. At minimum, the following hold: (i) the Hausdorff dimension of the root variety \(V_\epsilon\) changes discontinuously at \(\epsilon=0\); (ii) the central fiber \(V_0\) is characterized by a Jacobian rank deficiency, so in particular \(J_{P_0}(x)\) is singular on \(V_0\); (iii) for \(\mathcal{V}_\epsilon(x)=\|P_\epsilon(x)\|^2\), the gradient flow \(\dot x=-\nabla \mathcal{V}_\epsilon(x)\) realizes the collapse as a deformation retract; and (iv) in a statistical ensemble at temperature \(T\), the collapse appears as a phase transition, with the alignment order parameter \(m(\epsilon,T)\) discontinuous at \((\epsilon,T)=(0,0)\).
D. The bifurcation satisfies all of the following: (i) the Hausdorff dimension of the root variety \(V_\epsilon\) changes discontinuously at \(\epsilon=0\); (ii) the central fiber \(V_0\) is characterized by a Jacobian rank deficiency, namely \(\operatorname{rank}(J_{P_0}(x))=d_{\rm A}-d_{\rm M}\); (iii) for \(\mathcal{V}_\epsilon(x)=\|P_\epsilon(x)\|^2\), the gradient flow \(\dot x=-\nabla \mathcal{V}_\epsilon(x)\) realizes the collapse as a deformation retract, and there is a constant \(C>0\), independent of the particular generically non-central analytic deformation, such that \(T_{\rm collapse}\sim C\epsilon^{-2}\) for all sufficiently small \(\epsilon\); and (iv) in a statistical ensemble at temperature \(T\), the collapse appears as a phase transition, with the alignment order parameter \(m(\epsilon,T)\) discontinuous at \((\epsilon,T)=(0,0)\).
E. The bifurcation satisfies all of the following: (i) the Hausdorff dimension of the root variety \(V_\epsilon\) changes discontinuously at \(\epsilon=0\); (ii) the central fiber \(V_0\) is characterized by a Jacobian rank deficiency, namely \(\operatorname{rank}(J_{P_0}(x))=d_{\rm A}-d_{\rm M}\); (iii) for \(\mathcal{V}_\epsilon(x)=\|P_\epsilon(x)\|^2\), the gradient flow \(\dot x=-\nabla \mathcal{V}_\epsilon(x)\) realizes the collapse as a deformation retract, and for every analytic non-central deformation one still has the critical-slowing-down law \(T_{\rm collapse}\propto \epsilon^{-2}\); and (iv) in a statistical ensemble at temperature \(T\), as \(T\to 0\) one has \(m(0,0)=1/(d_{\rm A}-1)\) and, for \(\epsilon\neq 0\), merely \(0<m(\epsilon,0)<1\).

trivial-stabilizer-implies-isolated-roots and resulting dimension jump

dropped the explicit quantitative timescale \(T_{\rm collapse}\propto \epsilon^{-2}\)

deformation-dependent scaling constant replaced by a uniform constant across all deformations

genericity requirement in the Lojasiewicz-exponent step and full alignment at zero temperature


\label{prop:bound_a}
    If $a!b!=c!$ is a solution of class $k$, i.e., $c-b = k$, then $k < a < k + 2 \lceil \log_2 c \rceil$.

Erd\H{o}s asks the following problem in \cite{ErdosGraham1980}: for which integers $a_1, \ldots, a_t$ with $t \geq 2$ and $a_i \geq 2$ for each $i$ does there exist an integer $c$ so that $\prod_{i=1}^t a_i! = c!$? This question, which is open even for $t=2$, has been the inspiration for dozens of papers in elementary number theory since at least the 1970s. It is widely believed that there are only finitely many ``nontrivial'' solutions, i.e., those for which $c - \max_i a_i > 1$. In 1991, Erd\H{o}s \cite{Erdos91} showed that for sufficiently large $c$, $k \leq 5 \log\log c$; the constant was subsequently lowered, most recently by \cite{BkhatRamachandra10}, to $(1+o(1))/\log 2 \approx 1.44$. In 2007, Luca \cite{Luca07} showed that there are only finitely many nontrivial solutions if the notorious $abc$ Conjecture holds, and Nair-Shorey \cite{NAIR2016307} refined this to a complete list of solutions conditional on an explicit form of the $abc$ Conjecture due to Baker. In the same paper, Luca showed that the set of $c$'s that arise in solutions is of asymptotic density zero. Here, we focus on the $t=2$ case, i.e., $a!b!=c!$, showing that the possible values of $a$ are also rather sparse.

The following definition will play a key role throughout.

\begin{defn}
A solution to $a! b! = c!$ with $a<b$ in the integers is said to be ``class k'' if $c - b = k$.
\end{defn}

Note that class $0$ solutions must have $a=0$ or $a=1$ and $b=c$; it is easy to see that the solution is class $1$ iff $b=a!-1$ and $c=a!$. Thus, a solution to $a!b!=c!$ is said to be ``trivial'' if $k \leq 1$. It is conjectured that the only non-trivial solution is the triple $(6,7,10)$. Already the conjecture that there are no class $2$ solutions is open (see \cite{BerendHarmse06}). In the definition above, and throughout the sequel, we assume that $a < b$. It is straightforward to see that $a \neq b$ for nontrivial solutions. Indeed, if $a=b$, then $c!=a!^2$. However, writing $\nu_p(n!)$ for the $p$-adic order of $n$, this implies that $\nu_p(c!)=2\nu_p(a!)$ for every prime $p$, contradicting that $\nu_p(c!) = 1$ for every prime in $(c/2,c]$, which exists by Bertrand's Postulate.

In Section \ref{sec:primes}, we show that (Corollary \ref{cor:SerreSet}) for any fixed $k>1$, there exists $1/k$ natural density fraction of primes $p$ such that if $a = p-1$, there are no solutions to $a!b!=c!$. The proof makes use of some interesting elementary number theory: old results of Westland-Fl\"uge on irreducible polynomials which are sparse in the falling-factorial basis, Wilson's Theorem, and Dirichlet's Theorem. Then, in Section \ref{sec:equidistribution}, by analyzing the asymptotics of falling factorials, we show that $c$ falls into a small interval defined by $a$ and $k$, implying that if $\sqrt[k]{a!} \pmod 1$ is sufficiently equidistributed as $a,k \rightarrow \infty$ -- a much stronger version of which we conjecture is true -- then the set of $a$’s that appear in a solution is also of asymptotic density zero. Below we write $x^{\underline{k}}$ for the $k$-th falling factorial of $x$, i.e., $x(x-1)\cdots (x-k+1)$.

Here we collect a few useful bounds on possible solutions, especially, that $k < a < b < c$.

\begin{cor} \label{cor:SerreSet}
    Fix $k > 1$.  For at least a $1/k$ natural density subset of primes $p$, the equation
    $$
    (p-1)! b! = (b+k)!
    $$
    has no solutions for integers $b$.
\end{cor}

Here we collect a few useful bounds on possible solutions, especially, that $k < a < b < c$.

\section{Solutions Modulo a Prime} \label{sec:primes}

\begin{prop} There are no class $2$ solutions to $a!b!=c!$ in positive integers if $a+1$ is a prime congruent to $5 \pmod{6}$.
\end{prop}
\begin{proof}
Note that $c-b=2$ and $a!b!=c!$ implies $a! = c(c-1)$. Suppose $a!=c(c-1)$ where $a+1=p$ is a prime congruent to $5 \pmod{6}$. Then
$$
c^2 - c - a! = 0
$$
so we have $c = \frac{1\pm \sqrt{1-4(-a!)}}{2}$. Thus, $1+4a! = 1 + 4(p-1)!$ is a perfect square. By Wilson's Theorem, $1+4(p-1)! \equiv 1+4(-1) = -3 \pmod{p}$, so $-3$ is a quadratic residue modulo $p$. However, this contradicts Lemma \ref{lem:-3}.
\end{proof}

\begin{prop} \label{thm:pIs4} There are no class $4$ solutions to $a!b!=c!$ in positive integers if $a+1$ is a prime congruent to $2$ or $3 \pmod{5}$.
\end{prop}
\begin{proof}
Suppose $a! = c(c-1)(c-2)(c-3)$, where $a=p-1$ for some prime $p$. Note that $b>0$ implies $c=b+4>4$, so $c!/b! \geq 24 > 3!$, so we may assume $a>3$ and so $p>4$; furthermore, if $b>0$ then $4! = c(c-1)(c-2)(c-3)$ only has the solution $c=4$, but then $b = c-4 = 0$, a contradiction, so $p \neq 5$. Let $r \equiv c/2 \pmod{p}$, and apply Wilson's Theorem to conclude that
$$
2r(2r-1)(2r-2)(2r-3) \equiv -1 \pmod{p}.
$$
Now,
$$
2r(2r-1)(2r-2)(2r-3) = (4r^2-6r+1)^2-1,
$$
so
$$
(4r^2-6r+1)^2 \equiv 0 \pmod{p},
$$
i.e., $p | 4r^2 - 6r + 1$. However, $(2r-3/2)^2 - 5/4 = 4r^2 - 6r + 1$, so $5/4$ is a quadratic residue mod $p$, which happens iff $5$ is a quadratic residue mod $p$. It is known this occurs only for primes $p \equiv 1,4 \pmod{5}$.
\end{proof}

\begin{theorem} \label{thm:PositiveDensityIfIrreducible}Suppose $F_k(x) = x(x-1)\cdots (x-k+1)+1$ is irreducible over $\mathbb{Z}$. Let $P_0$ be the set of primes so that, if $a=p-1$ for some $p \in P_0$, then $a!b!=c!$ has no class $k$ solutions in positive integers. Then the natural density of $P_0$ in the primes is at least $1/k$.
\end{theorem}
\begin{proof}
 Suppose $a!b!=c!$ is class $k$ and $a = p-1$ for some prime $p$. Then $c = b+k$ and $a! = c!/b! = F_k(c)-1$. Then, reducing this equation modulo $p$ yields $c^{\underline{k}} \equiv a! \pmod{p}$, and by Wilson's Theorem, this is equivalent to $F_k(x) \equiv 0 \pmod{p}$. However, by Theorem 1 and 2 of \cite{Serre03}, the set $P_0$ of primes $p$ for which $F_k(x)$ does not have solution modulo $p$ has positive density, and the density is at least $1/k$.
\end{proof}

\begin{prop}\label{thm:WestlundFluge}
 For $k \geq 2$, $F_k(x)$ is reducible over $\mathbb{Z}$ iff $k=4$.
\end{prop}
\begin{proof}
 This is an immediate consequence of a result by Westlund and Fl\"{u}ge, answering a question of Schur, as discussed in \cite{DorwartOre33}.
\end{proof}

\begin{cor} \label{cor:SerreSet}
 Fix $k > 1$. For at least a $1/k$ natural density subset of primes $p$, the equation
 $$
 (p-1)! b! = (b+k)!
 $$
 has no solutions for integers $b$.
\end{cor}
\begin{proof}
 By Theorem \ref{thm:WestlundFluge}, $F_k$ is irreducible if $k \neq 4$, so Theorem \ref{thm:PositiveDensityIfIrreducible} implies that $a!b!=c!$ has no solutions in the integers when $c = b+k$ and $a=p-1$, for an at least $1/k$ density subset of the primes. Proposition \ref{thm:pIs4} yields the remaining case, by Dirichlet's Theorem on primes in arithmetic progressions.
\end{proof}

\begin{prop}
 For $p$ prime, the equation
 $$
 (p-1)!b! = (b+k)!
 $$
 has at most $nk/p + 1$ solutions in the integers with $b \leq n$.
\end{prop}
\begin{proof}
 Suppose $(p-1)! b! = (b+k)!$. Then, $(p-1)! = F_k(b)-1$. Taking this equation modulo $p$ and applying Wilson's Theorem gives
 $$
 -1 \equiv F_k(b) - 1 \pmod{p},
 $$
 i.e., $F_k(b) \equiv 0 \pmod{p}$. However, this equation has at most $k$ solutions $A_k \subseteq \mathbb{Z}/p\mathbb{Z}$. If $b \pmod{p} \not \in A_k$, then $(p-1)!b! \neq (b+k)!$. Since the set of $b \in [1,n]$ so that $b \pmod{p} \in A_k$ has cardinality $\lceil nk/p \rceil \leq nk/p + 1$, the conclusion follows.
\end{proof}

\begin{proof}
    Write $s_q(n)$ for the sum of the base-$q$ digits of $n$. Clearly, $\nu_p(a!)+\nu_p(b!)=\nu_p(c!)$.  Applying Legendre's Formula, we obtain
    $$
    \frac{a-s_p(a)}{p-1} + \frac{b-s_p(b)}{p-1} = \frac{c-s_p(c)}{p-1},
    $$
    from which it follows that $a-(c-b) = s_p(a)+s_p(b)-s_p(c)$ for every prime $p$.
    Since $a < b < c$, in particular
    $$
    | s_2(a)+s_2(b)-s_2(c) | < 2 \lceil \log_2 c \rceil .
    $$
    Thus, $|a-k| < 2 \lfloor \log_2 c \rfloor$.  Now, we argue that $a > k$.  First,
    \begin{align*}
    a! & = \frac{c!}{b!} = c(c-1)\cdots(c-k+1) \\
    & > (c-k+1)^k = (b+1)^k \geq (a+2)^k.
    \end{align*}
    Then taking the log of both sides above and applying Stirling's approximation (that $\log a! < a \log a$) gives
    $$
    a \log a > \log a! > k \log (a+2)
    $$
    so that $k < a \log a / \log(a+2) < a$.
\end{proof}

Using Legendre's formula with digit sums $s_p(n)$, from $\nu_p(a!)+\nu_p(b!)=\nu_p(c!)$ one gets
\[
\frac{a-s_p(a)}{p-1}+\frac{b-s_p(b)}{p-1}=\frac{c-s_p(c)}{p-1},
\]
so for every prime $p$,
\[
a-(c-b)=s_p(a)+s_p(b)-s_p(c).
\]
With $k=c-b$ and $a<b<c$, this yields the base-$2$ bound
\[
|s_2(a)+s_2(b)-s_2(c)|<2\lceil\log_2 c\rceil,
\]
hence $|a-k|<2\lfloor\log_2 c\rfloor$, giving the upper bound $a<k+2\lceil\log_2 c\rceil$.

To show $a>k$, rewrite
\[
a!=\frac{c!}{b!}=c(c-1)\cdots(c-k+1)>(c-k+1)^k=(b+1)^k\ge (a+2)^k.
\]
Taking logs and using Stirling's approximation (namely $\log a!<a\log a$) gives
\[
a\log a>\log a!>k\log(a+2),
\]
so $k<a\log a/\log(a+2)<a$, i.e. $k<a$.

Question: Let $a,b,c$ be nonnegative integers with $a<b$ such that $a!\,b!=c!$. Such a solution is said to be of class $k$ if $c-b=k$. Under these assumptions, which quantitative estimate for $a$ holds?

If $a!\,b!=c!$ is a class $k$ solution (that is, $c-b=k$), then \[k<a<k+2\left\lceil \log_2 c\right\rceil.\]

Choices:
B. If $a!\,b!=c!$ is a class $k$ solution (that is, $c-b=k$), then \[k\le a<k+2\left\lceil \log_2 c\right\rceil.\]
C. If $a!\,b!=c!$ is a class $k$ solution (that is, $c-b=k$), then \[a<k+2\left\lceil \log_2 c\right\rceil.\]
D. If $a!\,b!=c!$ is a class $k$ solution (that is, $c-b=k$), then \[k<a<k+2\left\lceil \log_2 a\right\rceil.\]
E. If $a!\,b!=c!$ is a class $k$ solution (that is, $c-b=k$), then \[k<a<k+\left\lceil \log_2 c\right\rceil.\]


Question: Let $a,b,c$ be nonnegative integers with $a<b$ such that $a!\,b!=c!$. A solution is said to be of class $k$ when $c-b=k$. Under these assumptions, which statement about $a$ must hold? (Here $\lceil \cdot \rceil$ denotes the ceiling function and $\log_2$ is the base-$2$ logarithm.)

If $a!\,b!=c!$ and $c-b=k$, then $k<a<k+2\lceil \log_2 c\rceil$.

Choices:
B. If $a!\,b!=c!$ and $c-b=k$, then $k\le a<k+2\lceil \log_2 c\rceil$.
C. If $a!\,b!=c!$ and $c-b=k$, then $a<k+2\lceil \log_2 c\rceil$.
D. If $a!\,b!=c!$ and $c-b=k$, then $k<a<k+2\lceil \log_2 a\rceil$.
E. If $a!\,b!=c!$ and $c-b=k$, then for every prime $p$ one has $k<a<k+2\lceil \log_p c\rceil$.


A collection $H$ of $k$-element subsets of $[n]$  is {\itshape shifted} if for every $i<j$, $j\in S\in H$, $i\notin S$, also $(S\setminus\{j\})\cup \{i\}\in H$. After fixing a field, a canonical operation of {\itshape algebraic shifting}, which associates with every simplicial complex $K$ a shifted simplicial complex $\Delta(K)$,
was introduced by Kalai \cite{K ch, K Alg Sh}. This operation preserves some important properties of $K$, such as its $f$-vector and $\beta$-vector, while loses others, such as the homotopy type of $K$.
Various properties of $K$ are described in terms of its shifting, such as its homology and Cohen-Macaulyness over the fixed field.

Two different variations of algebraic shifting were introduced
by Kalai: {\itshape exterior
shifting} and {\itshape symmetric
shifting}. Given a simplicial complex $K$ we denote by $\Delta^e(K)$ and $\Delta^s(K)$ its exterior and symmetric shifting respectively, and sometimes omit the superscript for  statements that hold for both types of algebraic shifting. Throughout the paper all results on exterior shifting are satisfied over any field, and all results on symmetric shifting are satisfied over any field of characteristic zero.

We start our exposition with a simple known example. Let $n$ be a positive integer. The {\itshape graphic matroid} of the complete graph on $n$ vertices is a collection of all spanning trees on the vertex set $[n]$. Let $G$ be such a spanning tree. Note that since algebraic shifting preserves Betti numbers, for every edge $e\in\Delta(G)$, we have $1\in e$. Note also that since algebraic shifting preserves $f$-vector, it then follows that $\Delta(G)=\{12,13,\ldots,1n\}$. Conversely, one can similarly verify that every $G\in\Delta^{-1}(\{12,13,\ldots,1n\})$ is a spanning tree on $[n]$. Thus, we conclude that $\Delta^{-1}(\{12,13,\ldots,1n\})$ coincides with the graphic matroid of the complete graph on $n$ vertices. The main goal of this paper is to study this phenomenon in a general setting.

Let $H$ be a shifted $k$-uniform hypergraph on $[n]$ (equivalently, the collection of facets of a pure simplicial complex).
Denote by
$$M(H)=\left\{G\subseteq \binom{[n]}k|\ \Delta^e(G)=H\right\}$$
all $k$-uniform hypergraphs whose exterior shifting is $H$.
We say that
$H$ is {\itshape matroidal} if $M(H)$ is the set of bases of a matroid; we abuse notation and also denote that matroid by $M(H)$.

For example, if $H$ is the collection of $k$-subsets of $[n]$ containing $1$, then $M(H)$ is the Crapo-Rota simplicial matroid \cite{CR1, CR2},
whose independent sets are the $(k-1)$-acyclic collections of $k$-subsets of $[n]$, namely those that do not support any $(k-1)$-homology cycle over a fixed
field; see e.g.
~\cite[Chapter 6]{White} and \cite[Section 4, Remark (2)]{K Symm} for further background. If $H=\{ij:\ 1\le i<j\le n, \ i\le k\}$ then $M(H)$ is Kalai's $k$-hyperconnectivity matroid~\cite{K Hyper}; see e.g. \cite[Section 4, Remark (2)]{K Symm} and \cite[Section 2.7]{K Alg Sh} for further background. If $H=\{1ij:\ 2\le i<j\le n, \ i\le 3\}$ then $M(H)$ is the area-rigidity matroid, as recently proved in~\cite[Theorem 1.2]{BNP}.

We conjecture a combinatorial characterization of the algebraic property of being \emph{matroidal}:
\begin{conjecture}\label{conj:any_k}
For every natural numbers $1\le k\le n$, a shifted $k$-uniform hypergraph $H\subseteq\binom{[n]}{k}$ is matroidal if and only if the hyperedges of $H$ form an initial segment w.r.t. the lexicogrpaphic order
(initial lex-segment for short).
\end{conjecture}

The "if" part follows easily from a result of Kalai~\cite[Section 4]{K Symm}.
We prove the "only if" part for $k=2$ (i.e. for graphs) and for $k=3$:

A collection $H$ of $k$-element subsets of $[n]$ is {\itshape shifted} if for every $i<j$, $j\in S\in H$, $i\notin S$, also $(S\setminus\{j\})\cup \{i\}\in H$. After fixing a field, a canonical operation of {\itshape algebraic shifting}, which associates with every simplicial complex $K$ a shifted simplicial complex $\Delta(K)$,
was introduced by Kalai \cite{K ch, K Alg Sh}. This operation preserves some important properties of $K$, such as its $f$-vector and $\beta$-vector, while loses others, such as the homotopy type of $K$.
Various properties of $K$ are described in terms of its shifting, such as its homology and Cohen-Macaulyness over the fixed field.

Two different variations of algebraic shifting were introduced
by Kalai: {\itshape exterior
shifting} and {\itshape symmetric
shifting}. Given a simplicial complex $K$ we denote by $\Delta^e(K)$ and $\Delta^s(K)$ its exterior and symmetric shifting respectively, and sometimes omit the superscript for statements that hold for both types of algebraic shifting. Throughout the paper all results on exterior shifting are satisfied over any field, and all results on symmetric shifting are satisfied over any field of characteristic zero.

Let $H$ be a shifted $k$-uniform hypergraph on $[n]$ (equivalently, the collection of facets of a pure simplicial complex).
Denote by
$$M(H)=\left\{G\subseteq \binom{[n]}k|\ \Delta^e(G)=H\right\}$$
all $k$-uniform hypergraphs whose exterior shifting is $H$.
We say that
$H$ is {\itshape matroidal} if $M(H)$ is the set of bases of a matroid; we abuse notation and also denote that matroid by $M(H)$.

For example, if $H$ is the collection of $k$-subsets of $[n]$ containing $1$, then $M(H)$ is the Crapo-Rota simplicial matroid \cite{CR1, CR2},
whose independent sets are the $(k-1)$-acyclic collections of $k$-subsets of $[n]$, namely those that do not support any $(k-1)$-homology cycle over a fixed
field; see e.g.
~\cite[Chapter 6]{White} and \cite[Section 4, Remark (2)]{K Symm} for further background. If $H=\{ij:\ 1\le i<j\le n, \ i\le k\}$ then $M(H)$ is Kalai's $k$-hyperconnectivity matroid~\cite{K Hyper}; see e.g. \cite[Section 4, Remark (2)]{K Symm} and \cite[Section 2.7]{K Alg Sh} for further background. If $H=\{1ij:\ 2\le i<j\le n, \ i\le 3\}$ then $M(H)$ is the area-rigidity matroid, as recently proved in~\cite[Theorem 1.2]{BNP}.

We conjecture a combinatorial characterization of the algebraic property of being \emph{matroidal}:
\begin{conjecture}\label{conj:any_k}
For every natural numbers $1\le k\le n$, a shifted $k$-uniform hypergraph $H\subseteq\binom{[n]}{k}$ is matroidal if and only if the hyperedges of $H$ form an initial segment w.r.t. the lexicogrpaphic order
(initial lex-segment for short).
\end{conjecture}

The "if" part follows easily from a result of Kalai~\cite[Section 4]{K Symm}.
We prove the "only if" part for $k=2$ (i.e. for graphs) and for $k=3$:

We conjecture a combinatorial characterization of the algebraic property of being \emph{matroidal}:
\begin{conjecture}\label{conj:any_k}
For every natural numbers $1\le k\le n$, a shifted $k$-uniform hypergraph $H\subseteq\binom{[n]}{k}$ is matroidal if and only if the hyperedges of $H$ form an initial segment w.r.t. the lexicogrpaphic order
(initial lex-segment for short).
\end{conjecture}

For $k\ge 4$ we prove the "only if" part for $k$-uniform hypergraphs satisfying some technical condition on their structure.

\begin{theorem} \label{main theorem}
Let $k\ge 4$ and
let $H\subseteq\binom{[n]}{k}$ be a shifted $k$-uniform hypergraph such that $s_H\notin H$, where $s_H$ is defined in terms of the lex-maximum element of $H$ by \eqref{S for case 1}-\eqref{S for case 2}. Then $H$ is matroidal if and only if the hyperedges of $H$ form an initial lex-segment.
\end{theorem}

Let $X=(x_{ij})_{1\le i,j\le n}$ be an $n\times n$-matrix over $\mathbb L$ which is generic
over $\mathbb F$. For a $k$-uniform hypergraph $H\subseteq\binom{[n]}k$ define a matroid $M'(H)$ on $\binom{[n]}k$ as follows: $B\subseteq\binom{[n]}k$ is independent in $M'(H)$ iff the rows of $X(B,H)$ are linearly independent.
\begin{remark} \cite[Section 4, Remark (2)]{K Symm} \label{Remark: Kalai's construction for hypperconnectivity}
 For $H=\{S\subseteq\binom{[n]}2:S\cap[k]\ne\emptyset\},\ M'(H)$ is the $k$-hyperconnectivity matroid.
\end{remark}
In fact, a more general statement is straightforward from the construction above. We call a subset $H\subseteq\binom{[n]}{k}$ an {\itshape initial lex-segment}
if $H=\left\{T'\in\binom{[n]}{k}|\ T'\le_{\text{lex}}T\right\}$ for some $T\in\binom{[n]}{k}$.
\begin{corollary}
\label{Initial segment is a matroid}
 Let $H\subseteq\binom{[n]}k$ be a $k$-uniform hypergraph whose hyperedges form an initial lex-segment. Then $H$ is matroidal.
\end{corollary}
\begin{proof}
We will verify that $M(H)$ is the set of bases of the matroid $M(H')$, or $M(H)=M'(H)$ for short.
Let $B\in M(H)$, then $H$ is a column basis chosen in the greedy way for the matrix $X(B,\binom{[n]}{k})$. Thus, the columns of the matrix $X(B,H)$ are linearly independent, hence $B\in M'(H)$.

\begin{lemma}[Isolated vertex] \label{lemma: partial star with an additional edge 2,3 is not a matroid}
 Let $H\subseteq\binom{[n]}{k}$ be a shifted $k$-uniform hypergraph such that $n$ is an isolated vertex and $\{2,3,\ldots,k+1\}\in H$.
 Then $H$ is not matroidal.
\end{lemma}
\begin{proof}
 Consider two $k$-uniform hypergraphs $H_1=H$ and $H_2$ obtained from $H$ by flipping the vertex $1$ with the vertex $n$ (see Fig. \ref{graphs which do not satisfy the exchange exiom} for illustration). By its structure $H_1$ contains the facets of the boundary complex $\partial \sigma$ of the simplex $\sigma=\{1,2,\ldots,k+1\}$. In order to show that the pair $H_1, H_2$ violates the exchange axiom we have to find $e_1\in H_1\setminus H_2$ such that for every $e_2\in H_2\setminus H_1$ we have that
 $(H_1\setminus e_1)\cup e_2\notin M(H)$. Let us take as $e_1$ an arbitrary facet of
 $\partial \sigma$
 which contains $1$. Such $e_1$ indeed lies in $H_1\setminus H_2$ since $1$ is an isolated vertex in $H_2$. Note that $\beta_{k-1}(H_1\setminus e_1)<\beta_{k-1}(H_1)$. As $e_2$ we can take only hyperedges containing the vertex $n$, which is an isolated vertex in $H_1$. Thus, $e_2$
 does not belong to any $(k-1)$-homology cycle in $(H_1\setminus e_1)\cup e_2$. Thus, $\beta_{k-1}((H_1\setminus e_1)\cup e_2)=\beta_{k-1}(H_1\setminus e_1)<\beta_{k-1}(H_1)$. However, since algebraic shifting preserves Betti numbers, see Theorem \ref{Basic properties of algebraic shifting}(4), we have $\beta_{k-1}(\Delta((H_1\setminus e_1)\cup e_2))=\beta_{k-1}((H_1\setminus e_1)\cup e_2)=\beta_{k-1}(H_1\setminus e_1)<\beta_{k-1}(H_1)=\beta_{k-1}(H)$. Thus, $((H_1\setminus e_1)\cup e_2)\notin M(H)$.
\end{proof}

\begin{example} \label{example: of not matroidal graph}
Consider the $2$-uniform hypergraph $H=\{12,13,23\}$ on $[5]$. Note that the (hyper)edges of $H$ do not form an initial lex-segment. Thus, by Theorem \ref{main theorem} $H$ is not matroidal. In this case one can verify this directly. Since algebraic shifting preserves $f$-vector and $\beta$-vector (see Theorem \ref{Basic properties of algebraic shifting}(3,4)) and the only shifted $2$-uniform hypergraph on $[5]$ with $3$ (hyper)edges and an one cycle is $H$, we conclude that $M(H)$ consists of all $2$-uniform hypergraph on $[5]$ with
$3$ edges forming a triangle boundary.
However, the pair of elements
$\{12,13,23\},\ \{14,15,45\}\in M(H)$ does not satisfy the exchange axiom.
\end{example}

\begin{definition}
Let $H$ be a shifted $k$-uniform hypergraph on $[n]$. $H$ is {\itshape s-matroidal} if $M^s(H)$ is the set of bases of a matroid on $\binom{[n]}{k}$.
\end{definition}
\begin{proposition} \label{thm: symmetric case for graphs}
 A simple graph $H\subseteq\binom{[n]}{2}$ is s-matroidal if and only if the edges of $H$ form an initial lex-segment.
\end{proposition}

\begin{proposition} \label{thm: lex-matroidal}
 A shifted $k$-uniform hypergraph $H\subseteq\binom{[n]}{k}$ is lex-matroidal iff the hyperedges of $H$ form an initial lex-segment.
\end{proposition}
\begin{proof}
 By Corollary \ref{Initial segment is a matroid} if the hyperedges of $H$ form an initial lex-segment, then $H$ is matroidal. For every hyperedge $S$ of $H$ the set $H(S)$ is an initial lex-segment, and thus it is matroidal.

\begin{equation} \label{S for case 1}
s_H=\{a,\ldots,b,c-1,n-(l+1),n-l,\ldots,n-1\}.
\end{equation}

\begin{equation} \label{S for case 2}
s_H=\{a,\ldots,b,c-1,n-m-l,n-m-l+1,\ldots,n-m-1,n-m,\ldots,n\}.
\end{equation}

For $k\ge 4$ we prove the "only if" part for $k$-uniform hypergraphs satisfying some technical condition on their structure.

\begin{theorem} \label{main theorem}
Let $k\ge 4$ and
let $H\subseteq\binom{[n]}{k}$ be a shifted $k$-uniform hypergraph such that $s_H\notin H$, where $s_H$ is defined in terms of the lex-maximum element of $H$ by \eqref{S for case 1}-\eqref{S for case 2}. Then $H$ is matroidal if and only if the hyperedges of $H$ form an initial lex-segment.
\end{theorem}

For the "only if" part of Theorem \ref{thm: main thm 2 and 3} we first make a reduction of the general problem to a certain restricted subset of hypergraphs; and for each of them we find a suitable permutation $\pi:[n]\rightarrow [n]$ such that the pair $(H, \pi(H))$, where $\pi(H)=\{\pi(T):\ T\in H\}$, violates the exchange axiom among the elements of $M(H)$. For the "only if" part of Theorem \ref{main theorem} we find a suitable such permutation $\pi=\pi(s_M)$ for each $H$.

The analog of Theorem~\ref{thm: main thm 2 and 3}
for symmetric algebraic shifting is proved
for $k=2$ in Proposition~\ref{thm: symmetric case for graphs}.
For $H=\{ij:\ 1\le i\le j \le n, i\le k\}$ the corresponding matroid is the (generic) $k$-rigidity matroid, see e.g. \cite{Lee} and  \cite[Section 2.7]{K Alg Sh}, \cite[Section 3.2]{Nevo phd} for further background, references and for a discussion of this correspondence.
\

This paper is organized as follows. In Section \ref{sec: preliminaries} we give a background on algebraic shifting and matroids. In Section \ref{sec: Preparatory lemmas} we prove lemmas needed for proofs of Theorem \ref{thm: main thm 2 and 3} and Theorem \ref{main theorem}, and in Section \ref{sec: Proofs of main results} we prove these theorems. In Section \ref{sec: Matroids arising from symmetric shifting} we consider matroids arising from symmetric shifting, and in Proposition \ref{thm: symmetric case for graphs} classify all matroidal
simple graphs
w.r.t. symmetric shifting. In Section \ref{sec: Some partial results} we replace the matroidal property by a stronger algebraic property in Conjecture~\ref{conj:any_k} that allows us to prove the resulted "iff" combinatorial characterization.
In Section \ref{sec: Concluding remarks} we conclude with related open problems.

For the ``only if'' part of Theorem~\ref{thm: main thm 2 and 3} the approach is: (i) ``first make a reduction of the general problem to a certain restricted subset of hypergraphs''; (ii) for each hypergraph in this restricted subset, ``find a suitable permutation $\pi:[n]\rightarrow [n]$ such that the pair $(H, \pi(H))$, where $\pi(H)=\{\pi(T):\ T\in H\}$, violates the exchange axiom among the elements of $M(H)$.''

For the ``only if'' part of Theorem~\ref{main theorem} (the $k\ge 4$ theorem under the condition $s_H\notin H$), the method is similarly to ``find a suitable such permutation $\pi=\pi(s_M)$ for each $H$.''

For the ``only if'' part of \begin{theorem} \label{thm: main thm 2 and 3}
For $k=2,3$,
a shifted
$k$-uniform
hypergraph $H\subseteq\binom{[n]}{k}$ is matroidal if and only if the (hyper)edges of $H$ form an
initial lex-segment.
\end{theorem}
the approach is: (i) ``first make a reduction of the general problem to a certain restricted subset of hypergraphs''; (ii) for each hypergraph in this restricted subset, ``find a suitable permutation $\pi:[n]\rightarrow [n]$ such that the pair $(H, \pi(H))$, where $\pi(H)=\{\pi(T):\ T\in H\}$, violates the exchange axiom among the elements of $M(H)$.''

For the ``only if'' part of the main theorem (the $k\ge 4$ theorem under the condition $s_H\notin H$), the method is similarly to ``find a suitable such permutation $\pi=\pi(s_M)$ for each $H$.''

Question:
Let k\in\{2,3\} and let H\subseteq \binom{[n]}{k} be a shifted k-uniform hypergraph, meaning that whenever i<j, j\in S\in H, and i\notin S, one also has (S\setminus\{j\})\cup\{i\}\in H. Define
\[
M(H)=\{G\subseteq \binom{[n]}{k}:\ \Delta^e(G)=H\},
\]
where \Delta^e denotes exterior algebraic shifting, and call H matroidal if M(H) is the set of bases of a matroid on \binom{[n]}{k}. Which of the following statements is equivalent to H being matroidal?

The hyperedges of H form an initial lex-segment; equivalently, there exists T\in \binom{[n]}{k} such that
\[
H=\{T'\in \binom{[n]}{k}: T'\le_{\mathrm{lex}} T\}.
\]

Choices:
B.
The hyperedges of H form an initial segment with respect to the colexicographic order; equivalently, there exists T\in \binom{[n]}{k} such that
\[
H=\{T'\in \binom{[n]}{k}: T'\le_{\mathrm{colex}} T\}.
\]
C.
The hyperedges of H are contained in an initial lex-segment; equivalently, there exists T\in \binom{[n]}{k} such that
\[
H\subseteq\{T'\in \binom{[n]}{k}: T'\le_{\mathrm{lex}} T\}.
\]
D. There exists a permutation \pi:[n]\to[n] such that the hyperedges of \pi(H)=\{\pi(T):T\in H\} form an initial lex-segment; equivalently, H is matroidal exactly when it is isomorphic, via relabeling of [n], to an initial lex-segment.
E.
The hyperedges of H form an initial lex-segment after deleting at most one hyperedge; equivalently, there exists T\in \binom{[n]}{k} and e\in \binom{[n]}{k} such that
\[
H\setminus\{e\}=\{T'\in \binom{[n]}{k}: T'\le_{\mathrm{lex}} T\}.
\]


\begin{theorem}\label{Thm: Hypercontractivity on tower of n}
  For $n=3\cdot 2^k$ and $n=2^k$ with $k\ge 1$, we have
  \[
    \norm{P_{t}f}_q\le \norm{f}_p\quad \Leftrightarrow \quad t\ge \frac{1}{2}\log(\frac{q-1}{p-1})
    \]
    for $1<p\le q<\infty$.
\end{theorem}

The hypercontractivity of the Poisson-like semigroup on the cyclic group $\mathbb{Z}_n$ is a long-standing open problem for all $n$ except $n=2$ and $n=4$. More precisely, by equipping $\mathbb{Z}_n$ with the normalized counting measure $\mu_n$, the Poisson-like semigroup $(P_t)_{t\in \mathbb{R}_+}$ is defined as the family of maps $P_t:L_\infty(\mathbb{Z}_n,\mu_n)\to L_\infty(\mathbb{Z}_n,\mu_n)$, which acts on Fourier series by
\[
 P_t:\sum_{k=0}^{n-1} a_k\chi_k(x) \mapsto \sum_{k=0}^{n-1} e^{-t\psi_n(k)}a_k\chi_k(x),
\]
where $\psi_n(k)=\min(k,n-k)$ is the word-length function on $\mathbb{Z}_n$, and $\chi_k(x)=e^{\frac{2\pi ikx}{n}}\in L_\infty(\mathbb{Z}_n)$. The hypercontractivity problem asks for the optimal time $t_{p,q}$ with $1< p\leq q<\infty$ such that
\[
 \norm{P_t f}_{q}\le \norm{f}_{p}\qquad \text{for all } t\ge t_{p,q}.
\]
For the case $n=2$, the optimal time $t_{p,q}=\tfrac{1}{2}\log\big(\tfrac{q-1}{p-1}\big)$ follows by applying the classical two-point inequality. That inequality was first proved by Bonami~\cite{MR283496}, later rediscovered by Gross~\cite{MR420249}, and was also used by Beckner~\cite{MR385456} to obtain the best constants for the Hausdorff--Young inequality. For $n=4$, Beckner, Janson, and Jerison~\cite{MR730056} obtained the same optimal time $t_{p,q}=\tfrac{1}{2}\log\big(\tfrac{q-1}{p-1}\big)$ by a clever reduction from $\mathbb{Z}_2$ to $\mathbb{Z}_4$, though the approach does not extend to other $\mathbb{Z}_{m\times n}$. For $n=3$, by Wolff's reduction \cite[Corollary 3.1]{MR2314078}, the optimal time $t_{2,q}$ of $P_t$ on $\mathbb{Z}_3$ coincides the optimal time
\[
t_{2,q}=\frac{1}{2}\log(\frac{\frac{2}{3}(\frac{1}{3})^{\frac{2}{q}-1}-\frac{1}{3}(\frac{2}{3})^{\frac{2}{q}-1}}{(\frac{2}{3})^{\frac{2}{q}}-(\frac{1}{3})^{\frac{2}{q}}})
\]
of the simple semigroup $T_t=e^{-t(\mathrm{id}-\mathbb{E}_{\delta})}$ on the weighted two-point space $L_\infty(\{-1,1\},\tfrac{1}{3}\delta_{-1}+\tfrac{2}{3}\delta_{1})$, which was computed by Oleszkiewicz and Lata\l a \cite{MR1796718,MR2073432}. For the general optimal time $t_{p,q}$ of $(P_t)_{t\in \mathbb{R}_+}$ on $\mathbb{Z}_3$, only asymptotic information is available; see \cite[Theorem 2.1]{MR2314078}. For $n=5$, Andersson~\cite{MR1883499} determined $t_{2,q}=\tfrac{1}{2}\log(q-1)$ for $q\in 2\mathbb{Z}_+$. For $n\ge 6$, Junge, Palazuelos, Parcet, and Perrin~\cite{MR3709719} proved partial results: for even $n$, $t_{2,q}=\tfrac{1}{2}\log(q-1)$ for $q\in 2\mathbb{Z}_+$; for odd $n$, the same holds when $n\ge q$, via a rather involved combinatorial method. Combining Stein's interpolation method \cite{MR82586} with Gross's extrapolation technique \cite{MR420249,MR372613,MR2325763}, they further obtained $t_{p,q}\le \tfrac{\log 3}{2}\log\big(\frac{q-1}{p-1}\big)$ in those regimes.\par

Hypercontractivity is also widely studied for semigroups on manifolds. Among the most classical examples, Weissler~\cite{MR578933} proved optimal hypercontractivity for the heat and Poisson semigroups on the circle $\mathbb{S}^1$, while Rothaus~\cite{MR593787} independently treated the heat case. For the sphere $\mathbb{S}^n$ ($n\ge 2$), Mueller and Weissler~\cite{MR674060} established optimal hypercontractivity for the heat semigroup. For Poisson-type semigroups on $\mathbb{S}^n$, see Janson~\cite{MR706641}, Beckner~\cite{MR1164616}, and Frank--Ivanisvili~\cite{MR4278128}.\par

Now we present our main result for the Poisson-like semigroup $(P_t)_{t\in\mathbb{R}_+}$ on $\mathbb{Z}_n$. Note that the standard argument, based on a simple series expansion of $\|P_t(1+\epsilon f)\|_q$ and $\|1+\epsilon f\|_p$ at $\epsilon=0$, shows the universal lower bound $t_{p,q}\ge \tfrac{1}{2}\log\big(\tfrac{q-1}{p-1}\big)$. The theorem below records the optimal times $t_{p,q}$ along a dyadic tower of $n$.

Without loss of generality, suppose $\lambda_{j}<\lambda_{j+3}$. The function $F$ strictly increases on $(0,1)$ and strictly decreases on $(1,\infty)$, so from \eqref{eq: opposite F} with $\nu_{j}=\nu_{j+3}=0$ proved in Step 1, we get
\begin{equation}
 F(\lambda_{j}) = F(\lambda_{j+3}).
\end{equation}
And hence
\[
 \lambda_{j}<1<\lambda_{j+3}.
\]
Define $\Theta(x)\coloneqq F(x)-F(2-x)$. Then $\Theta(1)=0$ and $\Theta'(x)=-\frac{2}{3}\log\big(x(2-x)\big)>0$ on $(0,1)$, so $\Theta(x)<0$ on $(0,1)$. Applying this to $x=\lambda_{j}$ gives $F(\lambda_{j})<F(2-\lambda_{j})$. Since $F$ decreases on $(1,\infty)$ and $F(\lambda_{j+3})=F(\lambda_{j})$, we deduce
\begin{equation}\label{Ineq: opposite sum lower bound}
 \lambda_{j+3}>2-\lambda_{j}
 \quad\Rightarrow\quad
 \lambda_{j}+\lambda_{j+3}>2.
\end{equation}\par
\medskip
\textbf{Step 3.} We show that for $\ell,\ell'\in\{0,1,2\}\setminus\{j_0\}$, we have $\lambda_{\ell}=\lambda_{\ell+3}=\lambda_{\ell'}=\lambda_{\ell'+3}$.\par
By the assumption $\lambda_{j_0}\neq \lambda_{j_0+3}$, we have $\lambda_{j_0}+\lambda_{j_0+3}>2$ from Step 2. Hence
\[
 \lambda_{j_0}^2+\lambda_{j_0+3}^2 \ge \frac{(\lambda_{j_0}+\lambda_{j_0+3})^2}{2} > 2.
\]
Consequently, if $\lambda_{j}\neq \lambda_{j+3}$ for all $0\leq j\le 2$, then $\sum_{k=0}^{5}\lambda_k^2>6$, contradicting $\sum_{k=0}^{5}\lambda_k^2<6$ in \eqref{Eqns: Lagrange for n=6, (0,1,2,1,2,1)}. Therefore, there exists at least one $\ell$ with $\lambda_{\ell}=\lambda_{\ell+3}$. Let $\ell$ be an index in $\{0,1,2\}\setminus\{j_0\}$ such that $\lambda_\ell=\lambda_{\ell+3}$ and $\lambda_\ell$ is minimal among all pairs $\{\lambda_j=\lambda_{j+3}\}$ with $j\in\{0,1,2\}\setminus\{j_0\}$. If $\lambda_\ell=\lambda_{\ell+3}\ge 1$, then $\sum_{k=0}^{5}\lambda_k^2>6$, which again contradicts $\sum_{k=0}^{5}\lambda_k^2<6$. So there at least one $\ell$ with $\lambda_{\ell}=\lambda_{\ell+3}<1$. Denote by $\ell'$ the remaining index in $\{0,1,2\}\setminus\{j_0,\ell\}$. \par

\begin{proof}
Recall the case $n=2$ for Theorem~\ref{Thm: Log Sobolev inequality n=6 times 2^k and n=8 times 2^k} is known in \cite{MR420249}, i.e.,
\begin{equation}\label{Ineq: 2-LSI}
 \frac{1}{4}\left( x^2\log\frac{2x^2}{x^2+y^2}+y^2\log\frac{2y^2}{x^2+y^2}\right)
\le \left( \frac{x-y}{2} \right)^2\qquad x,y\geq 0.
\end{equation}
Given $\lambda=(a_0,b_0,a_1,b_1,\ldots,a_{n-1},b_{n-1})$ with $a=(a_0,\ldots,a_{n-1})$ and $b=(b_0,\ldots,b_{n-1})$, the $2n$--entropy splits as
\[
\begin{aligned}
\mathrm{H}_{2n}[\lambda]&=\frac{1}{2n}\sum_{i=0}^{n-1} a_i^2 \log(\frac{2na_i^2}{\sum_{i=0}^{n-1}(a_i^2+b_i^2)})+\frac{1}{2n}\sum_{i=0}^{n-1} b_i^2 \log(\frac{2nb_i^2}{\sum_{i=0}^{n-1}(a_i^2+b_i^2)})\\
&=\frac{1}{2n}\sum_{i=0}^{n-1} a_i^2\log(\frac{n a_i^2}{\sum_{i=0}^{n-1}a_i^2})+\frac{1}{2n}\sum_{i=0}^{n-1} b_i^2\log(\frac{n b_i^2}{\sum_{i=0}^{n-1}b_i^2})\\
 &\quad +\frac{1}{2n}\left(\sum_{i=0}^{n-1} a_i^2\log(\frac{2 \sum_{i=0}^{n-1} a_i^2}{\sum_{i=0}^{n-1}(a_i^2+b_i^2)})+\sum_{i=0}^{n-1} b_i^2\log(\frac{2 \sum_{i=0}^{n-1} b_i^2}{\sum_{i=0}^{n-1}(a_i^2+b_i^2)})\right).
\end{aligned}
\]
Applying the $n$--LSI to the first two terms, and $2$--LSI \eqref{Ineq: 2-LSI} with $x=\norm{a}_2$ and $y=\norm{b}_2$ to the last term, yields
\[
\mathrm{H}_{2n}[\lambda]\le \inner{a,\Gamma(n)a}+\inner{b,\Gamma(n)b}
+\frac{1}{2n}\big(\norm{a}_2-\norm{b}_2\big)^2.
\]
Finally, by Proposition~\ref{Prop: Compare Dirichlet form for n and 2n},
\begin{equation}\label{Eqn: induction from n-LSI to 2n-LSI}
 \mathrm{H}_{2n}[\lambda]\le \inner{a,\Gamma(n)a}+\inner{b,\Gamma(n)b}
+\frac{1}{2n}\big(\norm{a}_2-\norm{b}_2\big)^2
\le 2\inner{\lambda,\Gamma(2n)\lambda}.
\end{equation}\par

For odd $n_0$, define the weight function on $\mathbb{Z}_n$ for even $n\geq n_0$
\begin{equation}\label{Eqn: definition of induced gamma_n for odd base case}
 \gamma_n(k)=
 \begin{cases}
 \psi_n(k), & k\ne \tfrac{n}{2},\\
 1, & k=\tfrac{n}{2}.
 \end{cases}
\end{equation}
The pair $(\psi_{n_0},\gamma_{2n_0})$ satisfies \eqref{Cond: gamma pair condition} and \eqref{Ineq: Quadratic inequality from n to 2n}. For even $n$, the pair $(\gamma_n,\gamma_{2n})$ also satisfies \eqref{Cond: gamma pair condition}, and the desired inequality \eqref{Ineq: Quadratic inequality from n to 2n} becomes
\begin{equation}\label{Ineq: quadratic for odd n}
 f(x)\coloneqq r_b (n-3) x^2+x \left(2 \sqrt{r_a+1} \sqrt{r_b+1}-2\right)+r_a (n-3)\geq 0
\end{equation}
for all $x\ge 0$ and $0\le r_a,r_b\le 1$. Since $n-3>0$, the minimum of $f$ is attained at $x=-\frac{2 \sqrt{r_a+1} \sqrt{r_b+1}-2}{2r_b(n-3)}\leq 0$ and the value of the minimum of $f$ on $[0,\infty)$ is $f(0)=r_a(n-3)\geq 0$. Hence \eqref{Ineq: quadratic for odd n} holds. Therefore Proposition~\ref{Prop: Compare Dirichlet form for n and 2n} applies to the pairs $(\gamma_n,\gamma_{2n})$ defined by \eqref{Eqn: definition of induced gamma_n for odd base case}. Iterating \eqref{Eqn: induction from n-LSI to 2n-LSI} along $\big((\psi_{n_0},\gamma_{2n_0}),(\gamma_{2n_0},\gamma_{4n_0}),\ldots\big)$, we obtain for all $m\ge 0$,
\[
 \mathrm{H}_{n_0\cdot 2^{m+1}}[\lambda]
 \le 2\inner{\lambda,\Gamma(n_0\cdot 2^{m+1})\lambda}.
\]
Let $\Psi(n)=\frac{1}{n}F_n\diag(\psi_n)F_n^{-1}$. Since $\gamma_n\leq \psi_n$ pointwise, we have
\[
 \inner{\lambda,\Gamma(n_0\cdot 2^{m+1})\lambda}
 \le \inner{\lambda,\Psi(n_0\cdot 2^{m+1})\lambda}.
\]
Therefore,
\[
 \mathrm{H}_{n_0\cdot 2^{m+1}}[\lambda]
 \le 2\inner{\lambda,\Psi(n_0\cdot 2^{m+1})\lambda}.
\]
By Lemma~\ref{Lem: Explicit form of Log Sobolev ineq}, the \(n_0\cdot 2^{m+1}\)--LSI holds for
\(\psi_{n_0\cdot 2^{m+1}}\) for all \(m\ge 0\), which proves the claim.
\end{proof}

\textbf{(2) Case $n=2^m$ with $m\geq 1$.} The case $n=2$ is classical and the case $n=4$ is due to the work \cite{MR730056} and Gross's extrapolation technique~\cite{MR420249}. Note that the pair $(\phi_4,\gamma_8)$ satisfies \eqref{Cond: gamma pair condition}, and \eqref{Ineq: Quadratic inequality from n to 2n} becomes
\[
 f(x)\coloneqq \left(2-\frac{r_b}{5}\right)x^2+2x\left(\sqrt{r_a+1}\sqrt{r_b+1}-3\right)
 -\frac{r_a}{5}+2\ \ge 0,
\]
with $0\le r_a,r_b\le 1$ and $x\ge0$. As $2-\frac{r_b}{5}>0$, the minimum of $f$ on $[0,\infty)$ is attained at $x=-\tfrac{2 \left(\sqrt{r_a^2+1} \sqrt{r_b^2+1}-3\right)}{2\left( 2-r_b^2/5 \right)}$ and the value of the minimum is
\[
 \frac{r_a(24r_b+35)+5\left(-30\sqrt{r_a+1}\sqrt{r_b+1}+7r_b+30\right)}
 {5(r_b-10)}
 =\frac{h(r_a,r_b)}{5(r_b-10)}.
\]
Since $r_b\in[0,1]$, the denominator is negative. In order to show \eqref{Ineq: Quadratic inequality from n to 2n} hold, it suffices to show $h(r_a,r_b)\le 0$ on $[0,1]^2$. A direct computation gives
\[
 \pdv[2]{h}{r_a}(r_a,r_b)=\frac{75\sqrt{r_b+1}}{2(r_a+1)^{3/2}}>0,
 \qquad
 \pdv[2]{h}{r_b}(r_a,r_b)=\frac{75\sqrt{r_a+1}}{2(r_b+1)^{3/2}}>0.
\]
Thus $h$ is convex in each variable separately. So the maximum of $h$ on $[0,1]^2$ is attained at a corner. Evaluating,
\[
 h(0,0)=0,\quad h(0,1)=h(1,0)=5\big(37-30\sqrt{2}\big)<0,\quad h(1,1)=-56<0.
\]
Hence $\max_{[0,1]^2} h\le 0$, and since $5(r_b-10)<0$ we conclude
\[
 \frac{r_a(24r_b+35)+5\left(-30\sqrt{r_a+1}\sqrt{r_b+1}+7r_b+30\right)}
 {5(r_b-10)} \ge 0.
\]
Therefore $(\phi_4,\gamma_8)$ satisfies \eqref{Ineq: Quadratic inequality from n to 2n}. Recall that $(\gamma_{8\cdot 2^m},\gamma_{8\cdot 2^{m+1}})$ also satisfies \eqref{Ineq: Quadratic inequality from n to 2n}. Hence by Theorem~\ref{Thm: Log Sobolev inequality n=4} and Theorem~\ref{Thm: induction from n-LSI to 2n-LSI} the $8\cdot 2^m$--LSI holds for $\gamma_{8\cdot 2^m}$ for all $m\ge 0$, and therefore for $\psi_{8\cdot 2^m}$.
\end{proof}

A standard route to hypercontractivity proceeds through Log--Sobolev inequalities (LSI) by Gross's celebrated work~\cite{MR420249}: $\norm{P_{t}f}_q\le \norm{f}_p$ holds whenever $t\geq \frac{C}{4}\log\big(\frac{q-1}{p-1}\big)$ if and only if the corresponding LSI holds with constant $C$. Thus, to prove Theorem~\ref{Thm: Hypercontractivity on tower of n}, it suffices to establish the following $n$--LSI with the optimal constant $2$ along the above dyadic tower of $n$. Denote by $A_{\psi_n}$ the generator of semigroup $(P_t)_{t\in\mathbb{R}_+}$, that is
\begin{equation}
  A_{\psi_n}: \sum_{k=0}^{n-1} a_k\chi_k(x)\mapsto \sum_{k=0}^{n-1} \psi_n(k)a_k\chi_k(x).
\end{equation}
 The case $n=4$ in the following theorem is due to the work \cite{MR730056} and Gross's extrapolation technique~\cite{MR420249}.
\begin{theorem}\label{Thm: Log Sobolev inequality n=6 times 2^k and n=8 times 2^k}
For $n=3\cdot 2^k$ and $n=2^k$ with $k\ge 1$, we have the following LSI with the optimal constant $2$:
\[
  \int_{\mathbb{Z}_n} f^2\log f^2\dd\mu_n-\norm{f}_2^2\log\norm{f}_2^2\le 2\inner{f,A_{\psi_n}f}_{L_2(\mathbb{Z}_n,\mu_n)},\qquad f\in L_2^+(\mathbb{Z}_n,\mu_n).
\]
\end{theorem}

Our proof of Theorem~\ref{Thm: Log Sobolev inequality n=6 times 2^k and n=8 times 2^k} is based on a new induction scheme with three key ingredients:
\begin{enumerate}
  \item Auxiliary weights $\phi_4$ on $\mathbb{Z}_4$ and $\phi_6$ on $\mathbb{Z}_6$ together with their corresponding LSIs. These LSIs are tighter than those for the word-lengths $\psi_n$; this refinement is new even when $n=4$ and plays a key role in the analysis of LSIs.
  \item Karush--Kuhn--Tucker (KKT) analysis for the $4$-- and $6$--LSI.
We develop an efficient way to handle LSIs on $\mathbb{Z}_n$ via KKT analysis, combined with the aforementioned manipulation of the length functions. The specific structure of $\phi_6$ introduces a symmetry in the KKT system that makes the analysis tractable.
  \item An induction from the $n$--LSI to the $2n$--LSI under a crucial compatibility condition. We use a Cooley--Tukey factorization of the $2n$-point discrete Fourier transform (DFT), which expresses a large DFT as a combination of smaller DFTs and yields a comparison of Dirichlet forms at the scales $n$ and $2n$. The choice of the new weights $\phi_4$ and $\phi_6$ mentioned in (1) is crucial for the base steps $4\to 8$ and $6\to 12$.
\end{enumerate}
Finally, we remark that the above ideas are also useful for studying LSIs along other towers of the form $m\cdot n^k$. The KKT analysis and the compatibility condition vary in their technical details, depending on the specific value of $n$ and on the particular choice of weights on $\mathbb{Z}_n$, and we will not carry out this analysis in this paper.\par

The article is organized as follows. After a brief introduction to the LSI formulation and the $n$-dimensional KKT framework in Section~\ref{Sec: DFT and KKT analysis}, we analyze the KKT systems associated with the LSIs for $n=6$ and $n=4$ in Section~\ref{Sec: Log Sobolev inequality n=4 and n=6}. Section~\ref{Sec: Induction from n to 2n} presents a comparison criterion for a pair of weights that allows us to compare the Dirichlet forms at the scales $n$ and $2n$. We then apply it to the transition $n\to 2n$ of LSIs and establish Theorem~\ref{Thm: Log Sobolev inequality n=6 times 2^k and n=8 times 2^k}.

To prove Theorem~\ref{Thm: Hypercontractivity on tower of n}, the text follows the standard route via Log--Sobolev inequalities (LSI): by Gross's equivalence, $\|P_t f\|_q\le \|f\|_p$ holds for $t\ge \frac{C}{4}\log\big(\frac{q-1}{p-1}\big)$ iff the corresponding LSI holds with constant $C$. Hence, it “suffices to establish” an $n$--LSI “with the optimal constant $2$” along the dyadic towers $n=2^k$ and $n=3\cdot 2^k$, namely Theorem~\ref{Thm: Log Sobolev inequality n=6 times 2^k and n=8 times 2^k}.

The proof of Theorem~\ref{Thm: Log Sobolev inequality n=6 times 2^k and n=8 times 2^k} is described as a “new induction scheme” with three ingredients: (1) introduce auxiliary weights $\phi_4$ on $\mathbb{Z}_4$ and $\phi_6$ on $\mathbb{Z}_6$ and prove their LSIs, which are “tighter than those for the word-lengths $\psi_n$,” and are “crucial for the base steps $4\to 8$ and $6\to 12$;” (2) perform “Karush--Kuhn--Tucker (KKT) analysis for the $4$-- and $6$--LSI,” using an “efficient way” via KKT combined with “manipulation of the length functions,” where “the specific structure of $\phi_6$ introduces a symmetry in the KKT system” that makes it tractable; (3) carry out an “induction from the $n$--LSI to the $2n$--LSI under a crucial compatibility condition,” using a “Cooley--Tukey factorization of the $2n$-point discrete Fourier transform (DFT)” to express the large DFT via smaller ones and obtain a “comparison of Dirichlet forms at the scales $n$ and $2n$.” Together these steps yield the LSI theorem with constant $2$, and by the Gross equivalence this gives the stated hypercontractivity threshold for Theorem~\ref{Thm: Hypercontractivity on tower of n}.

Biconditional or Equivalence
Inequality or Bound

Question:
Let \(\mathbb Z_n\) be the cyclic group with normalized counting measure \(\mu_n\), and let the Poisson-like semigroup \((P_t)_{t\ge 0}\) act on Fourier series by
\[
P_t\!:\sum_{k=0}^{n-1} a_k\chi_k(x)\mapsto \sum_{k=0}^{n-1} e^{-t\psi_n(k)}a_k\chi_k(x),
\]
where \(\psi_n(k)=\min(k,n-k)\) and \(\chi_k(x)=e^{2\pi i kx/n}\). For \(n\) of the form \(2^k\) or \(3\cdot 2^k\) with integer \(k\ge 1\), and for \(1<p\le q<\infty\), which quantitative estimate gives exactly when the \(L_p(\mathbb Z_n,\mu_n)\to L_q(\mathbb Z_n,\mu_n)\) hypercontractive bound holds?

For every \(f\in L_p(\mathbb Z_n,\mu_n)\),
\[
\|P_t f\|_{L_q(\mathbb Z_n,\mu_n)}\le \|f\|_{L_p(\mathbb Z_n,\mu_n)}
\]
holds if and only if
\[
t\ge \frac12\log\!\left(\frac{q-1}{p-1}\right).
\]

Choices:
B.
For every \(f\in L_p(\mathbb Z_n,\mu_n)\),
\[
\|P_t f\|_{L_q(\mathbb Z_n,\mu_n)}\le \|f\|_{L_p(\mathbb Z_n,\mu_n)}
\]
holds if and only if
\[
t\ge \log\!\left(\frac{q-1}{p-1}\right).
\]
C.
For every \(f\in L_p(\mathbb Z_n,\mu_n)\), if
\[
t\ge \frac12\log\!\left(\frac{q-1}{p-1}\right),
\]
then
\[
\|P_t f\|_{L_q(\mathbb Z_n,\mu_n)}\le \|f\|_{L_p(\mathbb Z_n,\mu_n)}.
\]
D.
For every \(f\in L_p(\mathbb Z_n,\mu_n)\),
\[
\|P_t f\|_{L_q(\mathbb Z_n,\mu_n)}\le \|f\|_{L_p(\mathbb Z_n,\mu_n)}
\]
holds if and only if there exists a constant \(C_n>0\), depending only on \(n\), such that
\[
t\ge C_n\log\!\left(\frac{q-1}{p-1}\right).
\]
E.
For every \(f\in L_p(\mathbb Z_n,\mu_n)\),
\[
\|P_t f\|_{L_q(\mathbb Z_n,\mu_n)}\le \|f\|_{L_p(\mathbb Z_n,\mu_n)}
\]
holds if and only if
\[
t\ge \frac12\log\!\left(\frac{q}{p}\right).
\]

dropped the 'if and only if' optimality/necessity direction

uniform exact threshold replaced by an unspecified n-dependent constant

replaced the Gross-type ratio (q-1)/(p-1) by q/p


Question:
Let n be of the form 2^k or 3\cdot 2^k with k\ge 1, and let \mu_n be the normalized counting measure on \mathbb{Z}_n. Consider the Poisson-like semigroup (P_t)_{t\ge 0} on \mathbb{Z}_n defined on Fourier series by
\[
P_t\!\left(\sum_{k=0}^{n-1} a_k\chi_k\right)=\sum_{k=0}^{n-1} e^{-t\psi_n(k)}a_k\chi_k,
\]
where \psi_n(k)=\min(k,n-k) and \chi_k(x)=e^{2\pi i kx/n}. Fix 1<p\le q<\infty. Which statement is equivalent to the estimate
\[
\|P_t f\|_{L_q(\mathbb{Z}_n,\mu_n)}\le \|f\|_{L_p(\mathbb{Z}_n,\mu_n)}\quad\text{for all }f?
\]

The time parameter satisfies
\[
t\ge \frac12\log\!\left(\frac{q-1}{p-1}\right).
\]

Choices:
B.
The time parameter satisfies
\[
t\ge \log\!\left(\frac{q-1}{p-1}\right).
\]
C.
The estimate certainly holds whenever
\[
t\ge \log\!\left(\frac{q-1}{p-1}\right).
\]
D.
The time parameter satisfies
\[
t\ge \frac12\log\!\left(\frac{q}{p}\right).
\]
E.
The time parameter satisfies
\[
t\ge \frac{\log 3}{2}\log\!\left(\frac{q-1}{p-1}\right).
\]

equivalence replaced by a sufficient but non-optimal bound

replacement of the Gross-type factor (q-1)/(p-1) by q/p

upper-bound constant from partial results substituted for the optimal constant


\label{thm-main}
Let $a$ and $b$ be integers with $b>a>1$. Suppose that there exist positive integers $N$ and $D,$ and nonnegative integers $x_i, y_i$ such that
$$
N+iD = a^{x_i}+b^{y_i}, \; \; \mbox{ for } \; i \in \{ 0, 1, 2, 3, 4 \}.
$$
Then we have either
$$
(a,b,N,D)=(2,2^k+1,2^k+1,2^k)  \; \mbox{ or } \;  (3,4 \cdot 3^{k-1}+1,3^{k-1}+1,2 \cdot 3^{k-1}) \; \mbox{ for } \; k \in \mathbb{N},
$$
or
$$
\begin{array}{c}
(a,b,N,D) \in \left\{ (2,3,5,2),  (2,3,7,6), (2,3,9,8),  (2,3,17,24),  \right. \\
\left. (2,3,41,24), (2,5,5,8),  (2,9,17,24),  (2,9,41,24), (3,4,7,6) \right\}.
\end{array}
$$

Let $a$ and $b$ be integers with $b>a>1$, and define the set $S_{a,b}$ to be the sumset of the geometric progressions
$$
1, a, a^2, a^3, \ldots \; \mbox{ and } \; 1, b, b^2, b^3, \ldots,
$$
i.e.
$$
S_{a,b} = \{ n \in \mathbb{N} \; : \; n = a^x+b^y, \; \mbox{ for } x, y \in \mathbb{Z}, \; x, y \geq 0 \}.
$$
The main goal of this paper is to study the arithmetic progressions in the sets $S_{a,b}$. These sets are sufficiently thin that, were they random, our expectation for a fixed pair $(a,b)$ would be that $S_{a,b}$ should contain at most finitely many three-term arithmetic progressions, and indeed this is the case ``most'' of the time. By way of example, $S_{2,7}$ contains precisely $22$ such progressions, while, in contrast, $S_{2,9}$ contains infinitely many and $S_{2,8}$ even contains infinitely many four-term arithmetic progressions. It is likely that $S_{a,b}$ contains infinitely many
three-term arithmetic progressions precisely when either $a=2$ and $b=2^k+1$, for $k$ a positive integer, or when $a$ and $b$ are multiplicatively dependent, i.e. there exist positive integers $c$ and $d$ such that $a^c=b^d$. We will discuss this further in Section \ref{Sec9}.

In this paper, we will focus our attention on the problem of characterizing longer arithmetic progressions in the sets $S_{a,b}$.
Our starting point is recent work of Chen, Huang and Zhang \cite{ChHuZh} who showed, via a completely elementary argument, that the longest arithmetic progression in the set
$S_{2,3}$
has length $6$. Regarding such maximal progressions, we prove the following broad generalization of this, characterizing all $5$-term progressions in the sets $S_{a,b}$.

\noindent An almost immediate corollary of this is the following.
\begin{cor} \label{cor-main}
If $a$ and $b$ are integers with $b>a>1$, then the only $6$-term arithmetic progressions in $S_{a,b}$ are
$$
3,5,7,9,11,13 \; \; \mbox{ and } \; \; 17, 41, 65, 89, 113, 137,
$$
in $S_{2,3}$ and in both $S_{2,3}$ and $S_{2,9}$, respectively.
\end{cor}

Suppose that $a$ and $b$ are integers with $b>a \geq 2$ and that we have a $5$-term arithmetic progression in integers of the shape $a^x+b^y$, i.e. that there exist positive integers $N$ and $D,$ and nonnegative integers $x_i, y_i$ such that
$$
N+iD = a^{x_i}+b^{y_i}, \; \; \mbox{ for } 0 \leq i \leq 4.
$$
We will write
$$
n = \left[ \frac{\log (N+4D)}{\log a} \right] \; \mbox{ and } \; m = \left[ \frac{\log (N+4D)}{\log b} \right],
$$
where by $[x]$ we mean the greatest integer $\leq x$.
Let us call a term $N+iD$ in our arithmetic progression {\it $a$-large} if there is some representation of $N+iD = a^{x_i}+b^{y_i}$ with $x_i=n$, {\it $b$-large} if there is a representation of $N+iD = a^{x_i}+b^{y_i}$ with $y_i=m$, and {\it large}, if $N+iD$ is either $a$-large or $b$-large.

\begin{prop} \label{equality}
Let $a$ and $b$ be integers with $b>a>1$. Suppose that there exist positive integers $N$ and $D,$ and nonnegative integers $x_i, y_i$ such that
$$
N+iD = a^{x_i}+b^{y_i}, \; \; \mbox{ for } \; i \in \{ 0, 1, 2, 3, 4 \}.
$$
Suppose further that here exist integers $i, j$ and $k$ with $0 \leq i < j < k \leq 4$ and either
\begin{equation} \label{x}
x_i=x_j=x_k,
\end{equation}
or
\begin{equation} \label{y}
y_i=y_j=y_k.
\end{equation}
Then either
$$
(a,b,N,D)=(2,3,5,2), (2,3,7,6), (2,3,9,8), (3,4,7,6),
$$
\begin{equation} \label{family}
(a,b,N,D)=(2,2^{x_0}+1,2^{x_0}+1,2^{x_0})
\end{equation}
or
\begin{equation} \label{family2}
(a,b,N,D)=(3,4 \cdot 3^{x_0}+1,3^{x_0}+1,2 \cdot 3^{x_0}).
\end{equation}
\end{prop}

Suppose next that precisely three of $N+iD$, $i \in \{ 0, 1, 2, 3, 4 \}$ are large. From our previous work, we may assume that either
\begin{equation} \label{Case1}
\mbox{there exist } 0 \leq i < j \leq 4 \mbox{ and } k \not\in \{ i, j \} \mbox{ with } x_i=x_j=n, \; y_k=m,
\end{equation}
or
\begin{equation} \label{Case2}
\mbox{there exist } 0 \leq i < j \leq 4 \mbox{ and } k \not\in \{ i, j \} \mbox{ with } y_i=y_j=m, \; x_k=n.
\end{equation}
Since two of the terms $N+iD$, $i \in \{ 0, 1, 2, 3, 4 \}$ are not large, in all cases we have inequality (\ref{starty}) with $\kappa=1$.
In particular,
$$
N+D \leq a^{n-1}+b^{m-1} < \left( \frac{1}{a} + \frac{1}{b} \right) (N+4D)
$$
and hence, from $4(N+D)>N+4D$, $\frac{1}{a} + \frac{1}{b} > \frac{1}{4}$,
i.e.
\begin{equation} \label{special2}
a \in \{ 2, 3, 4 \}, \; a=5, \; 6 \leq b \leq 19, \; \; a=6, \; 7 \leq b \leq 11 \; \mbox{ or } \; a=7, \; 8 \leq b \leq 9.
\end{equation}

We therefore may suppose that $a=2$. Our goal is to show, in this case, that we have only $5$-term arithmetic progressions corresponding to
 $$
(a,b,N,D) \in \{ (2,3,17,24), (2,3,41,24), (2,9,17,24), (2,9,41,24) \}.
$$
 Let us assume first that $\delta_1=1$ and that $x_j=n-1$, whence (\ref{awesome}) and (\ref{spooky2}) imply that
 $$
 (i,j,k,\kappa) \in
 \{ (2,3,4,1), (2,4,3,1), (3,4,2,1), (1,3,4,2) \}.
 $$
 We will treat each of these four cases in turn. Suppose first that $(i,j,k,\kappa)=(2,3,4,1)$. Then
 $$
 \left( b^{\delta_2} -1 \right) b^{m-\delta_2} = 2^n-2^{x_2+1}
 $$
 and
 $$
2^n = 3 \cdot b^m - 2 b^{y_4} - b^{m-\delta_2},
 $$
 from (\ref{gaewA}) and (\ref{gaewB}), respectively. The second of these implies that
 $$
 2^n \geq 3 (b^m-b^{m-1}) > b^m
 $$
 and hence we can apply Theorem 1.6 of \cite{Be-Pill} to the first equation to conclude that $b=3$ and that
 $$
 (n,m,x_2,\delta_2) \in \{ (5,3,2,2), (8,5,3,4) \}.
 $$
 Neither coincides with a $5$-term progression (in each case $N+4D \not\in S_{2,3}$). If $(i,j,k,\kappa)=(2,4,3,1)$ or $(3,4,2,1)$, from the identity $(N+D)+(N+4D)=(N+2D)+(N+3D)$, we find that
 $$
 b^{m-\delta_2} - b^{y_k} = 2^{x_i},
 $$
 and so Lemma \ref{tech-lem} implies that
 we have one of
 \begin{equation} \label{ann1}
 b=2^{x_i}+1, \; m-\delta_2=1, \; y_k=0,
\end{equation}
\begin{equation} \label{ann3}
b=3, \; m-\delta_2=2, \; y_k=0, \; x_i=3,
\end{equation}
or
\begin{equation} \label{ann4}
b=9, \; m-\delta_2=1, \; y_k=0, \; x_i=3.
\end{equation}
In the first case, if $(i,j,k,\kappa)=(2,4,3,1)$, we have
$$
D = 2^{n-2}-2^{x_2-1} = \frac{1}{3} 2^{x_2} ( 2^{x_2}+1)^{m-1},
$$
a contradiction modulo $2^{x_2}$. If, on the other hand, we have (\ref{ann1}) and $(i,j,k,\kappa)=(3,4,2,1)$,
$$
D = 2^{n-1}-2^{x_3} = \frac{1}{3} 2^{x_3} ( 2^{x_3}+1)^{m-1},
$$
so that
$$
2^{n-1-x_3}-1 = \frac{1}{3} ( 2^{x_3}+1)^{m-1}.
$$
Modulo $3$, $x_3$ is odd and $m \geq 2$. Thus, modulo $8$, we find that either $x_3=1$, $m=2$, $n=3$, or $x_3=1$, $m=3$, $n=4$, or $x_3=3$, $m=2$, $n=4$. Only the first of these corresponds to a $5$-term arithmetic progression in $S_{a,b}$, namely $(a,b,N,D)=(2,3,5,2)$, which we have previously encountered. If $(i,j,k,\kappa)=(2,4,3,1)$, (\ref{gaewA}) becomes
$$
\left( b^{\delta_2} -1 \right) b^{m-\delta_2} = 3 \cdot (2^{n-2}-2^{x_2-1}),
$$
while $(i,j,k,\kappa)=(3,4,2,1)$ yields
$$
\left( b^{\delta_2} -1 \right) b^{m-\delta_2} = 3 \cdot (2^{n-1} - 2^{x_3}).
$$
In cases (\ref{ann3}) and (\ref{ann4}), we may thus apply Proposition \ref{DT} to these equations, concluding after a short computation with the $5$-term arithmetic progressions corresponding to
$$
(a,b,N,D) = (2,3,17,24) \; \mbox{ and } \; (2,9,17,24).
$$
Finally,
 if $(i,j,k,\kappa)=(1,3,4,2)$, then, from $(N+D)+(N+4D)=(N+2D)+(N+3D)$, we have
 $$
 b^{m-\delta_2}-b^{y_4}=2^{x_1},
 $$
 while (\ref{gaewB}) gives
 $$
 2^{n-1} = 2 b^m -b^{y_4} - b^{m-\delta_2}.
 $$
 Lemma \ref{tech-lem} thus implies that $y_4=0$ and so combining the preceding two equations,
 $$
 b^m =2^{n-2} +2^{x_1-1}+1.
 $$
 Since necessarily $m \geq 2$ (else we have three equal $y_i$), we may appeal to Theorem 1 of \cite{BeBuMi} to conclude that
 $$
 b^m \in \{ 7^2, 23^2, 3^4 \},
 $$
 so that
 $$
 (b,m,n,x_1) \in \{ (7,2,7,5), (23,2,11,5), (3,4,8,5), (9,2,8,5) \};
 $$
 none of these correspond to a $5$-term arithmetic progression in $S_{a,b}$.

If $S_{a,b}$ contains a $6$-term arithmetic progression, then, from Theorem \ref{thm-main}, we require $N+5D \in S_{a,b}$ for
$$
(a,b,N,D)=(2,2^k+1,2^k+1,2^k) \; \mbox{ or } \; (3,4 \cdot 3^{k-1}+1,3^{k-1}+1,2 \cdot 3^{k-1}) \; \mbox{ for } \; k \in \mathbb{N},
$$
or
$$
\begin{array}{c}
(a,b,N,D) \in \left\{ (2,3,5,2), (2,3,7,6), (2,3,9,8), (2,3,17,24), \right. \\
\left. (2,3,41,24), (2,5,5,8), (2,9,17,24), (2,9,41,24), (3,4,7,6) \right\}.
\end{array}
$$
A short check of the latter $9$ sporadic cases, reveals the $6$-term progressions corresponding to
$$
(a,b,N,D) = (2,3,17,24) \; \mbox{ and } \; (2,9,17,24),
$$
and no $7$-term progressions. Suppose that
$(a,b,N,D)=(2,2^k+1,2^k+1,2^k)$, for $k$ a positive integer, and that
$$
N+5D = 2^k+1 + 5 \cdot 2^k =3 \cdot 2^{k+1}+1= 2^{x_5} + (2^k+1)^{y_5},
$$
for $x_5$ and $y_5$ nonnegative integers. If $k=1$, then $x_5=y_5=2$, so that $(a,b,N,D) = (2,3,3,2)$. If $k \geq 2$, then
$$
(2^k+1)^2 \geq 3 \cdot 2^{k+1}+1,
$$
whence necessarily $y_5 \in \{ 0, 1 \}$. Thus
$$
3 \cdot 2^{k+1} = 2^{x_5} \; \mbox{ or } \; 2^{k+2}+1= 2^{x_5},
$$
each an immediate contradiction.
Finally, suppose that $(a,b,N,D)=(3,4 \cdot 3^{k-1}+1,3^k+1,2 \cdot 3^k)$ for $k$ a positive integer, and that
$$
N+5D = 3^{k-1}+1+ 10 \cdot 3^{k-1} = 11 \cdot 3^{k-1}+1=3^{x_5}+(4 \cdot 3^{k-1}+1)^{y_5},
$$
with $x_5$ and $y_5$ nonnegative integers. For each positive integer $k$, we have
$$
(4 \cdot 3^{k-1}+1)^2 > 11 \cdot 3^{k-1}+1
$$
and so $y_5 \in \{ 0, 1 \}$, corresponding to
$$
11 \cdot 3^{k-1}=3^{x_5} \; \mbox{ and } \;
7 \cdot 3^{k-1}=3^{x_5},
$$
respectively. These contradictions complete the proof of
Corollary \ref{cor-main}.

\noindent An almost immediate corollary of this is the following.
\begin{cor} \label{cor-main}
If $a$ and $b$ are integers with $b>a>1$,  then the only $6$-term arithmetic progressions in $S_{a,b}$ are
$$
3,5,7,9,11,13 \; \; \mbox{ and } \; \; 17, 41, 65, 89, 113, 137,
$$
in $S_{2,3}$ and in both $S_{2,3}$ and $S_{2,9}$, respectively.
\end{cor}

Our arguments, in contrast to those of  \cite{ChHuZh}, rely, implicitly or explicitly, upon various results from Diophantine approximation, including bounds for linear forms in logarithms, both complex and $p$-adic, and upon various Diophantine consequences of the modularity of Frey-Hellegouarch curves. The outline of this paper is the following.
In Section \ref{Sec2}, we present the various results we will need for explicitly solving certain polynomial-exponential and $3, 4$ and $5$-term $S$-unit equations. Sections \ref{Sec3}--\ref{Sec7} are devoted to dealing with $5$-term arithmetic progressions in $S_{a,b}$ with fixed numbers of ``large'' terms, proving Theorem \ref{thm-main}. In Section \ref{Sec8}, we prove Corollary \ref{cor-main}. Finally, in Section \ref{Sec9}, we discuss various families of pairs $(a,b)$ for which the $S_{a,b}$ are known to have $3$ or $4$-term progressions.

The post-theorem introduction gives only a high-level outline for proving Theorem~\ref{thm-main}: the arguments (unlike \cite{ChHuZh}) "rely, implicitly or explicitly, upon various results from Diophantine approximation, including bounds for linear forms in logarithms, both complex and $p$-adic," and "upon various Diophantine consequences of the modularity of Frey-Hellegouarch curves." In Section~\ref{Sec2} the paper "present[s] the various results" needed to explicitly solve certain polynomial-exponential and $3,4,5$-term $S$-unit equations, and Sections~\ref{Sec3}--\ref{Sec7} "deal[] with $5$-term arithmetic progressions in $S_{a,b}$ with fixed numbers of \`\`large\'\' terms," thereby proving Theorem~\ref{thm-main}.

The post-theorem introduction gives only a high-level outline for proving the main theorem: the arguments (unlike \cite{ChHuZh}) "rely, implicitly or explicitly, upon various results from Diophantine approximation, including bounds for linear forms in logarithms, both complex and $p$-adic," and "upon various Diophantine consequences of the modularity of Frey-Hellegouarch curves." Next the paper "present[s] the various results" needed to explicitly solve certain polynomial-exponential and $3,4,5$-term $S$-unit equations, and later sections "deal[] with $5$-term arithmetic progressions in $S_{a,b}$ with fixed numbers of ``large'' terms," thereby establishing the main theorem.

Question:
Let $a$ and $b$ be integers with $b>a>1$. Suppose there exist positive integers $N$ and $D$, and nonnegative integers $x_i,y_i$ for $i=0,1,2,3,4$, such that
$$
N+iD=a^{x_i}+b^{y_i}
$$
for each $i\in\{0,1,2,3,4\}$. In other words, the five terms $N,N+D,N+2D,N+3D,N+4D$ of an arithmetic progression are all representable in the form $a^x+b^y$ with $x,y\ge 0$. Which of the following conclusions about the quadruple $(a,b,N,D)$ holds?

One must have either
$$
(a,b,N,D)=(2,2^k+1,2^k+1,2^k)
\quad\text{or}\quad
(a,b,N,D)=(3,4\cdot 3^{k-1}+1,3^{k-1}+1,2\cdot 3^{k-1})
$$
for some positive integer $k$, or else
$$
(a,b,N,D)\in\{(2,3,5,2),(2,3,7,6),(2,3,9,8),(2,3,17,24),(2,3,41,24),(2,5,5,8),(2,9,17,24),(2,9,41,24),(3,4,7,6)\}.
$$

Choices:
B.
One must have either
$$
(a,b,N,D)=(2,2^k+1,2^k+1,2^k)
\quad\text{or}\quad
(a,b,N,D)=(3,4\cdot 3^{k-1}+1,3^{k-1}+1,2\cdot 3^{k-1})
$$
for some nonnegative integer $k$, or else
$$
(a,b,N,D)\in\{(2,3,5,2),(2,3,7,6),(2,3,9,8),(2,3,17,24),(2,3,41,24),(2,5,5,8),(2,9,17,24),(2,9,41,24),(3,4,7,6)\}.
$$
C.
One must have either
$$
(a,b,N,D)=(2,2^k+1,2^k+1,2^k)
$$
for some positive integer $k$, or else
$$
(a,b,N,D)\in\{(2,3,5,2),(2,3,7,6),(2,3,9,8),(2,3,17,24),(2,3,41,24),(2,5,5,8),(2,9,17,24),(2,9,41,24),(3,4,7,6),(3,4\cdot 3^{k-1}+1,3^{k-1}+1,2\cdot 3^{k-1})\text{ for some }k\in\mathbb N\}.
$$
D.
One must have either
$$
(a,b,N,D)=(2,2^k+1,2^k+1,2^k)
\quad\text{or}\quad
(a,b,N,D)=(3,4\cdot 3^{k-1}+1,3^{k}+1,2\cdot 3^{k-1})
$$
for some positive integer $k$, or else
$$
(a,b,N,D)\in\{(2,3,5,2),(2,3,7,6),(2,3,9,8),(2,3,17,24),(2,3,41,24),(2,5,5,8),(2,9,17,24),(2,9,41,24),(3,4,7,6)\}.
$$
E.
One must have either
$$
(a,b,N,D)=(2,2^k+1,2^k+1,2^k)
\quad\text{or}\quad
(a,b,N,D)=(3,4\cdot 3^{k-1}+1,3^{k-1}+1,2\cdot 3^{k-1})
$$
for some positive integer $k$, and in addition one necessarily has $a\in\{2,3\}$; equivalently, outside these two infinite families one must have
$$
(a,b,N,D)\in\{(2,3,5,2),(2,3,7,6),(2,3,9,8),(2,3,17,24),(2,3,41,24),(2,5,5,8),(2,9,17,24),(2,9,41,24)\}.
$$