\label{main} Let $G$ be a primitive permutation group with socle $\hbox{\rm PSU}(3,q)$, where $q=p^m\ge 7$ for an odd prime $p$. Suppose a point-stabilizer
  $M$ of $G$ contains $\hbox{\rm PSO}(3,q)$. Then $b(G)=2$ and either (i) $G=\hbox{\rm PSU}(3,q)$ and $M=\hbox{\rm SO}(3,q)$; or (ii) $G=\hbox{\rm PSU}(3,q)\ZZ_{m_1}$ and  $M=\hbox{\rm SO}(3,q)\ZZ_{m_1}$ $(m_1\mid 2m$ and  $m_1\ne  1)$. Further, the BG-Conjecture is valid for  case (i);
  and for   $\sqrt{q}\geq 17$ or $45$,  if $3\nmid (q+1)$  and $3\bigm| (q+1)$, respectively in case (ii).

\begin{theorem} \label{main} Let $G$ be a primitive permutation group with socle $\PSU(3,q)$, where $q=p^m\ge 7$ for an odd prime $p$. Suppose a point-stabilizer
  $M$ of $G$ contains $\PSO(3,q)$. Then $b(G)=2$ and either (i) $G=\PSU(3,q)$ and $M=\SO(3,q)$; or (ii) $G=\PSU(3,q)\ZZ_{m_1}$ and  $M=\SO(3,q)\ZZ_{m_1}$ $(m_1\mid 2m$ and  $m_1\ne  1)$. Further, the BG-Conjecture is valid for  case (i);
  and for   $\sqrt{q}\geq 17$ or $45$,  if $3\nmid (q+1)$  and $3\di (q+1)$, respectively in case (ii).
  \end{theorem}

A {\em base} for a finite permutation group $G$ on a set $\Omega$ is a subset $\Delta\subset \Omega$ whose pointwise  stabilizer $G_{(\Delta)}$ is trivial.  This  concept  is a natural generalization of basis for  vector spaces. The {\em base size} $b(G)$ of $G$  is the minimal cardinality $|\Delta|$, where $\Delta$ goes  over all bases for $G$, while  a  base $\Delta '$ with $|\Delta '|=b(G)$  is called a {\em base size set}. With a history dating to the 19th century, these concepts remain vital to contemporary algebra and combinatorics; see \cite{Bailey-Cameron,LS} and \cite[$\S5$]{B} for overviews and \cite{G,S} for applications.

Let $G\leq \hbox{\rm Sym}(\Omega)$ and $\Delta$  a base for $G$. If $x,y\in G$, then $\alpha^x=\alpha^y$ for all $\alpha\in\Delta$ if and only if $x=y$. This implies that each group element is uniquely determined by its action on a base $\Delta$. In particular, if $\Omega$ is finite, then $|G|\leq |\Omega|^{b(G)}$. The problem of bounding $|G|$ in terms of $|\Omega|$, which attracted significant interest in the 19th century, motivated early investigations of bases. In 1992, Blaha \cite{Ba} proved that determining whether or not $G$ has a base size at most a given constant is an NP-complete problem.
 Having established the O'Nan-Scott theorem, which characterizes finite primitive groups through the structure and action of their socle, there has been a surge of research into the base sizes of primitive permutation groups. This surge was catalyzed by the resolution of several highly influential conjectures by Babai, Cameron, Kantor, and Pyber from the 1990s, spurring new research directions as evidenced by \cite{BGS1,BGS2,BLS,BBW,DHM,Fawcett1,Fawcett2,Ha,H,S1}.

Primitive permutation groups with a base of size $1$ are cyclic groups $\ZZ_p$, which makes the family of finite primitive groups with a base of size 2 particularly significant due to their notably small base.
Saxl's ``base-two programme", which aims to classify finite primitive permutation groups with base size $2$, has inspired a great deal of research. To study the bases of permutation groups $G$ with $b(G)=2$, Burness and Giudici \cite{BG} introduced the {\em Saxl graph}, which is defined in terms of these groups. Let $G$ be a permutation group on $\Omega$ with $b(G)=2$. The vertex set of a Saxl graph $\Sigma(G)$ (denoted by $\Sigma$, simply) of $G$ is just
$\Omega$, while two vertices are adjacent if and only if they form a base. Clearly, if $G$ is a transitive permutation group, then the Saxl graph graph $\Sigma(G)$ is vertex-transitive. Moreover, if $G$ is primitive, then $\Sigma(G)$ is connected, but the converse is not true.
  In \cite{BG}, the authors discussed  the valency, connectivity, hamiltonicity and the independence number of $\Sigma(G)$, and   proposed the following conjecture.
\vskip -10pt
\begin{conj}{\rm \cite{BG}} \label{BG}
Let $G$ be a finite primitive permutation group with $b(G)=2$. Then every two vertices in $\Sigma(G)$ have a common neighbour.
\end{conj}
\vskip -5pt

 From now on, we assume that $G\leq \hbox{\rm Sym}(\Omega)$ acts transitively on $\Omega$. Fixing a point $\alpha\in \Omega$, we call the orbits of $G_\alpha$ the {\em suborbits} of $G$ relative to $\alpha$; the trivial suborbit is $\{\alpha\}$ itself.
   Now $\{\alpha,\beta\}$ is a base for $G$ with $b(G)=2$ if and only if $G_{\alpha}$ acts regularly on the suborbit containing $\beta$.
   It follows that the neighborhood $\si_1(\alpha)$ of $\alpha$ in
  $\Sigma(G)$ is the union $\Gamma$ of all regular suborbits of $G$ relative to $\alpha$. Consequently,
  Conjecture~\ref{BG} can be restated in terms of $\Gamma$ as follows, which we refer to as the BG-Conjecture.
\begin{conj}$($BG-Conjecture$)$ \label{BG1}
 Every finite primitive permutation group $G$ with $b(G)=2$ has the property  that
   $\Gamma\cap \Gamma^g\ne \emptyset$ for any $g\in G$, where $\Gamma$ is the union of all regular suborbits of $G$ relative to a point $\alpha$.
\end{conj}
\vskip -5pt

Using a probabilistic approach, Burness and Giudici \cite{BG} verified the conjecture for a range of almost simple groups. For instance, it holds when $G=S_n$ or $A_n$ (with $n>12$) and the point stabilizer is primitive. Their results also cover certain primitive groups of diagonal type and twisted wreath products of sufficiently large order, which have been extended recently by Huang (see Sections 5.8 and 5.9 of \cite{HPHD}).
Furthermore, by employing the Magma database, they verified the conjecture for all primitive groups of degree at most $4095$. The paper \cite{BG} also confirms the conjecture for many sporadic simple groups.
Burness and Huang \cite{BH} later proved the conjecture for almost simple primitive groups with soluble point stabilisers; additionally, they studied it for primitive groups of product type in a separate paper \cite{BH2}.
Lee and Popiel \cite{LP} have recently proved the conjecture for most affine-type groups with sporadic point stabilizers. The theory was subsequently extended with the introduction of generalized Saxl graphs
and Saxl hypergraphs in
\cite{FH, LP1}.

So far this conjecture remains open for all  primitive  groups $G$ whose socle is a simple group of  Lie-type of rank $1$. Our research aimed to tackle this conjecture in case of such groups of rank one.
  This class of groups contains four families, with  respective socles $\hbox{\rm PSL}(2,q)$,   $\hbox{\rm PSU}(3,q)$, $\hbox{\rm Ree}(q)$ and $\hbox{\rm Sz}(q)$.
Having established the conjecture for groups with socle $\hbox{\rm PSL}(2,q)$ in \cite{BH,Chen-Du},
  we now address the case of $\hbox{\rm PSU}(3,q)$ in a two-part series. In this paper (Part I), we prove the conjecture for primitive groups $G$ with this socle, whose point stabilizers contain $\hbox{\rm PSO}(3,q)$, with a few small exceptions (see Therorem~\ref{main}). Part II (i.e.~\cite{Chen-Du-Li1}) will treat the remaining primitive actions for $G$. Part III (i.e., \cite{Chen-Du2}) will present the cases  $\hbox{\rm soc}(G)\in \{\hbox{\rm Ree}(q)$, $\hbox{\rm Sz}(q)$\}.
We now state the main theorem of this paper.

Let $G\leq \hbox{\rm Sym}(\Omega)$ and $\Delta$ a base for $G$. If $x,y\in G$, then $\alpha^x=\alpha^y$ for all $\alpha\in\Delta$ if and only if $x=y$. This implies that each group element is uniquely determined by its action on a base $\Delta$. In particular, if $\Omega$ is finite, then $|G|\leq |\Omega|^{b(G)}$. The problem of bounding $|G|$ in terms of $|\Omega|$, which attracted significant interest in the 19th century, motivated early investigations of bases. In 1992, Blaha \cite{Ba} proved that determining whether or not $G$ has a base size at most a given constant is an NP-complete problem.
 Having established the O'Nan-Scott theorem, which characterizes finite primitive groups through the structure and action of their socle, there has been a surge of research into the base sizes of primitive permutation groups. This surge was catalyzed by the resolution of several highly influential conjectures by Babai, Cameron, Kantor, and Pyber from the 1990s, spurring new research directions as evidenced by \cite{BGS1,BGS2,BLS,BBW,DHM,Fawcett1,Fawcett2,Ha,H,S1}.

Primitive permutation groups with a base of size $1$ are cyclic groups $\ZZ_p$, which makes the family of finite primitive groups with a base of size 2 particularly significant due to their notably small base.
Saxl's ``base-two programme", which aims to classify finite primitive permutation groups with base size $2$, has inspired a great deal of research. To study the bases of permutation groups $G$ with $b(G)=2$, Burness and Giudici \cite{BG} introduced the {\em Saxl graph}, which is defined in terms of these groups. Let $G$ be a permutation group on $\Omega$ with $b(G)=2$. The vertex set of a Saxl graph $\Sigma(G)$ (denoted by $\Sigma$, simply) of $G$ is just
$\Omega$, while two vertices are adjacent if and only if they form a base. Clearly, if $G$ is a transitive permutation group, then the Saxl graph graph $\Sigma(G)$ is vertex-transitive. Moreover, if $G$ is primitive, then $\Sigma(G)$ is connected, but the converse is not true.
 In \cite{BG}, the authors discussed the valency, connectivity, hamiltonicity and the independence number of $\Sigma(G)$, and proposed the following conjecture.
\vskip -10pt
\begin{conj}{\rm \cite{BG}} \label{BG}
Let $G$ be a finite primitive permutation group with $b(G)=2$. Then every two vertices in $\Sigma(G)$ have a common neighbour.
\end{conj}
\vskip -5pt

From now on, we assume that $G\leq \hbox{\rm Sym}(\Omega)$ acts transitively on $\Omega$. Fixing a point $\alpha\in \Omega$, we call the orbits of $G_\alpha$ the {\em suborbits} of $G$ relative to $\alpha$; the trivial suborbit is $\{\alpha\}$ itself.
 Now $\{\alpha,\beta\}$ is a base for $G$ with $b(G)=2$ if and only if $G_{\alpha}$ acts regularly on the suborbit containing $\beta$.
 It follows that the neighborhood $\si_1(\alpha)$ of $\alpha$ in
 $\Sigma(G)$ is the union $\Gamma$ of all regular suborbits of $G$ relative to $\alpha$. Consequently,
 Conjecture~\ref{BG} can be restated in terms of $\Gamma$ as follows, which we refer to as the BG-Conjecture.
\begin{conj}$($BG-Conjecture$)$ \label{BG1}
 Every finite primitive permutation group $G$ with $b(G)=2$ has the property that
 $\Gamma\cap \Gamma^g\ne \emptyset$ for any $g\in G$, where $\Gamma$ is the union of all regular suborbits of $G$ relative to a point $\alpha$.
\end{conj}
\vskip -5pt

Using a probabilistic approach, Burness and Giudici \cite{BG} verified the conjecture for a range of almost simple groups. For instance, it holds when $G=S_n$ or $A_n$ (with $n>12$) and the point stabilizer is primitive. Their results also cover certain primitive groups of diagonal type and twisted wreath products of sufficiently large order, which have been extended recently by Huang (see Sections 5.8 and 5.9 of \cite{HPHD}).
Furthermore, by employing the Magma database, they verified the conjecture for all primitive groups of degree at most $4095$. The paper \cite{BG} also confirms the conjecture for many sporadic simple groups.
Burness and Huang \cite{BH} later proved the conjecture for almost simple primitive groups with soluble point stabilisers; additionally, they studied it for primitive groups of product type in a separate paper \cite{BH2}.
Lee and Popiel \cite{LP} have recently proved the conjecture for most affine-type groups with sporadic point stabilizers. The theory was subsequently extended with the introduction of generalized Saxl graphs
and Saxl hypergraphs in
\cite{FH, LP1}.

So far this conjecture remains open for all primitive groups $G$ whose socle is a simple group of Lie-type of rank $1$. Our research aimed to tackle this conjecture in case of such groups of rank one.
 This class of groups contains four families, with respective socles $\hbox{\rm PSL}(2,q)$, $\hbox{\rm PSU}(3,q)$, $\hbox{\rm Ree}(q)$ and $\hbox{\rm Sz}(q)$.
Having established the conjecture for groups with socle $\hbox{\rm PSL}(2,q)$ in \cite{BH,Chen-Du},
 we now address the case of $\hbox{\rm PSU}(3,q)$ in a two-part series. In this paper (Part I), we prove the conjecture for primitive groups $G$ with this socle, whose point stabilizers contain $\hbox{\rm PSO}(3,q)$, with a few small exceptions (see Therorem~\ref{main}). Part II (i.e.~\cite{Chen-Du-Li1}) will treat the remaining primitive actions for $G$. Part III (i.e., \cite{Chen-Du2}) will present the cases $\hbox{\rm soc}(G)\in \{\hbox{\rm Ree}(q)$, $\hbox{\rm Sz}(q)$\}.
We now state the main theorem of this paper.

Using a probabilistic approach, Burness and Giudici \cite{BG} verified the conjecture for a range of almost simple groups. For instance, it holds when $G=S_n$ or $A_n$ (with $n>12$) and the point stabilizer is primitive. Their results also cover certain primitive groups of diagonal type and twisted wreath products of sufficiently large order, which have been extended recently by Huang (see Sections 5.8 and 5.9 of \cite{HPHD}).
Furthermore, by employing the Magma database, they verified the conjecture for all primitive groups of degree at most $4095$. The paper \cite{BG} also confirms the conjecture for many sporadic simple groups.
Burness and Huang \cite{BH} later proved the conjecture for almost simple primitive groups with soluble point stabilisers; additionally, they studied it for primitive groups of product type in a separate paper \cite{BH2}.
Lee and Popiel \cite{LP} have recently proved the conjecture for most affine-type groups with sporadic point stabilizers. The theory was subsequently extended with the introduction of generalized Saxl graphs
and Saxl hypergraphs in
\cite{FH, LP1}.

Remark that for the reason of the length of the paper, we are not able to improve the lower bound $\sqrt{q}\geq 17$ (resp. $45$) for small exceptions in case (ii) of Theorem~\ref{main}.
 Hopefully, these small cases may be verified directly with a sufficiently powerful computer.

Combining Theorem~\ref{main} and the main results of \cite{BH,Chen-Du,Chen-Du2}, we get
\begin{theorem} \label{main'} Let $G$ be a primitive group whose socle is a simple group of Lie-type of rank $1$ such that $b(G)=2$.
 Then the BG-Conjecture is valid, except for $G=\PSU(3,q)\ZZ_{m_1}$ and $M=\SO(3,q)\ZZ_{m_1}$ $(m_1\mid 2m$ and $m_1\ne 1)$, where
 $\sqrt{q}\le 17$ or $45$, if $3\nmid (q+1)$ and $3\di (q+1)$, respectively.
 \end{theorem}

\section{$\PSL(3,q)\lvertneqq G$ and $\SO(3,q)\lvertneqq M$}\label{ell}
 In this section, we mainly prove the following theorem.
\vskip -7pt \begin{theorem} \label{main2} Let $G=\PSU(3,q)\ZZ_{m_1}$ and
 $\a=M=\PSO(3,q)\ZZ_{m_1}\le G$, with $q=p^m\ge 7$ for an odd prime $p$, $m_1\mid 2m$ and $m_1\ne 1$.
Under the primitive right multiplication action of $G$ on $\Omega=[G:M]$, we have $b(G)=2$ and the BG-Conjecture is true, provided $\sqrt{q}\geq 17$ or $45$ if $3\nmid (q+1)$ and $3\di (q+1)$, respectively.
 \end{theorem}

\section{A lower bound of $|\Gamma_{nr}(\a)\cap \Gamma_{nr}(\a')|$}
 Theorem~\ref{main21} is the main result of this section.
\begin{theorem}\label{main21}
Using the notation from Section~\ref{ell},
consider the right multiplication action of $T$ on the cosets $[T:M_0]$, where $d=(3, q+1)$ and $q\geq 7$ is odd. Suppose that $\sqrt{q}\geq 15d$. Then
$|\Gamma_{nr}(\a)\cap \Gamma_{nr}(\a')|\ge \frac{4}{9}q^5-\frac{1}{3}q^4\sqrt{q}-22.1555q^4$ for $d=1$; and
$|\Gamma_{nr}(\a)\cap \Gamma_{nr}(\a')|\ge \frac{4}{27}q^5-\frac{31}{27}q^4\sqrt{q}-16.8372q^4$ for $d=3$.
\end{theorem}
\demo From now on, set
$G=T=\PSU(3,q)$ and $M=M_0=\PSO(3,q)$. Since $M$ is maximal in $G$, we identify each coset $Mg$ with its point stabilizer $M^g$ so that $\O=\{ M^g\mid g\in G\}$, with the conjugacy action. Set $\a=M$.
 Let $\a':=M'=M^{g'}$ for any $g'\in G$. Then we shall get a lower bound of $|\Gamma_{nr}(\a)\cap \Gamma_{nr}(\a')|$.
 \vskip 3mm
The possible isomorphic types for $M\cap M'$ are:
 \vspace{-5pt}\begin{equation}\label{A_i} A_q=\ZZ_p^m.\ZZ_2,\, A_{q\pm 1}=\D_{2(q\pm 1)}, \, A_3=\D_4, A_4=\Alt(4)\, ({\rm if}\, 3\mid q+1), A_2=\ZZ_2 \, {\rm or}\, A_1=1.\end{equation}
All of these groups $A_\imath$ contain at least one involution, except for $A_1$. Therefore, set
\vspace{-5pt}\begin{equation}\label{I} I=\{ g\in M\mid |g|=2\}\quad {\rm and}\quad I'=\{ g\in M'\mid |g|=2\}.\end{equation}
Clearly, $0 \le |I \cap I'| \le q+2$, giving
 \vspace{-5pt}\begin{equation}\label{lw}\Gamma_{nr}(\a)\cap \Gamma_{nr}(\a')=\{M_\imath\in \O \setminus \{M, M'\}\mid t, t'\in M_\imath,\,\, {\rm for \, some}\,\, t\in I, t'\in I' \}.\end{equation}
 For the reasen of a simplicity, we turn to consider a subset ${\cal W}$ of $\Gamma_{nr}(\a)\cap \Gamma_{nr}(\a')$, where
 \vspace{-5pt}\begin{equation}\label{W_1} \cal W=\{ M_\imath\in \O \setminus \{M, M'\}\mid t, t'\in M_\imath\,\, {\rm for \, some}\, \, t\in I, t'\in I' \,\, {\rm but}\,\, t, t'\notin I\cap I'\}.\end{equation}
To determine $|{\cal W}|$, we introduce a set of incident triples (under the inclusion relation)
 \vspace{-5pt}\begin{equation}\label{E} \cal E=\{(t, M_\imath, t')\mid M_\imath\in {\cal W}, t\in M\cap M_\imath, t'\in M'\cap M_\imath, t, t'\notin I\cap I'\},\end{equation}
\f while for any given $ t\in I\setminus I'$ and $M_\imath\in {\cal W}$, we introduce two respective sets of pairs
\vspace{-5pt}\begin{equation}\label{E(t)} \cal E(t)=\{(M_\imath,t')\mid (t, M_\imath, t')\in \cal E\}\,\, {\rm and}\,\, \cal E(M_\imath)=\{(t,t')\mid (t, M_\imath, t')\in \cal E\},\end{equation}
so that we may count $|\cal E|$ by two different ways, that is,
\vspace{-5pt}\begin{equation} \label{2way} \sum_{t\in I\setminus I'}|\cal E(t)|=|\cal E|=\sum_{M_\imath\in {\cal W}}|\cal E(M_\imath)|.\end{equation}
Further, according to $M_\imath\cap M\cong A_j$, $M_\imath\cap M'\cong A_k$, $M_\imath\cap M\cap M'\cong A_l$, where $j, k,l\in \{1,2,3,4,q,q\pm1\}$, as mentioned in Eq(\ref{A_i}),
 the set ${\cal W}$ is the union of the following subsets ${\cal W}_{j, k, l}$, where
\vspace{-5pt}\begin{equation}\label{jkl}\begin{array}{lll}
 {\cal W}_{j,k, l}&=&\{M_\imath\in {\cal W}\mid M_\imath\cap M\cong A_j, M_\imath\cap M'\cong A_k, M_\imath\cap M\cap M'=A_l\}\end{array}\end{equation}
and $j,k,l\in\{1,2,3,4,q,q\pm1\}$,
reminding that $(M_\imath\cap M) \cap (M_\imath\cap M')=M\cap M_\imath\cap M'$. Moreover, set
\vspace{-5pt}\begin{equation}\label{n}
 n_{j, k, l}=|{\cal W}_{j,k,l}|.\end{equation}
 Then by Eq(\ref{2way}) and Eq(\ref{jkl}), we have
 \vspace{-5pt}$$\begin{array}{ll}
 &\sum_{t\in I\setminus I'}|\cal E(t)|=\sum_{j,k, l} n_{j,k, l} |\{\cal E(M_\imath): M_\imath\in {\cal W}_{j,k, l}\}|,
 \end{array}$$
\begin{equation}\label{w1}
\begin{array}{lll}|{\cal W}|&=&\sum\limits_{j,k, l}n_{j,k, l}=\sum\limits_{t\in I\setminus I'}|\cal E(t)|
-(\sum\limits_{t\in I\setminus I'}|\cal E(t)|-\sum\limits_{j,k, l}n_{j,k,l})\\
&=&\sum\limits_{t\in I\setminus I'}|\cal E(t)|
-\sum\limits_{j,k, l} n_{j,k,l}(|\{\cal E(M_\imath): M_\imath\in {\cal W}_{j,k, l}\}|-1)\\
&=&{\hm A}-{\hm B}-{\hm C}-{\hm D},\quad {\rm where} \end{array}
\end{equation}
$$\begin{array}{ll}
&{\hm A}=\sum\limits_{t\in I\setminus I'}|\cal E(t)|,\hskip 2cm
{\hm B}=\sum\limits_{4, q\not\in \{j, k\}}n_{j, k, 1}(|\{\cal E(M_\imath): M_\imath\in{\cal W}_{j,k,1}\}|-1),\\
&{\hm C}=\sum\limits_{4, q\not\in \{j, k\},l\ne 1}n_{j, k, l}(|\{\cal E(M_\imath): M_\imath\in{\cal W}_{j,k,l}\}|-1),\\
&{\hm D}=\sum\limits_{\{4, q\}\cap \{j, k\}\ne \emptyset} n_{j, k, l}(|\{\cal E(M_\imath): M_\imath\in{\cal W}_{j,k, l}\}|-1).\\
\end{array}$$
\f In the following Lemmas~\ref{AB},~\ref{C} and~\ref{D} and~\ref{DD}, it will be respectively shown that
\vspace{-5pt}$$\begin{array}{lll}
{\hm A}&\geq &q^5-1.0355 q^4,\, \hbox{$d=1;$}\,\, \frac 13q^5-0.3342 q^4, \, \hbox{$d=3;$}\,\\
{\hm B} &\leq & \frac{5}{9}q^5+\frac{1}{3}q^4\sqrt{q}+15.25 q^4, \, \hbox{$d=1$;} \,
 \frac{5}{27}q^5+\frac{31}{27}q^4\sqrt{q}+10.74 q^4, \, \hbox{$d=3$;}\\
 {\hm C}&\leq& 3.85q^4, \hbox{$d=1$;} \, 3.26 q^4, \hbox{$d=3$};\\
 {\hm D}&\leq& 2.02 q^4, \, \hbox{$d=1$;} \, 2.503 q^4,\, \hbox{$d=3$}. \\
\end{array}$$
\f By using Eq(\ref{lw}), Eq(\ref{W_1}) and Eq(\ref{w1}), we get $|\Gamma_{nr}(\a)\cap \Gamma_{nr}(\a')|\ge|{\cal W}|= {\hm A}-{\hm B}-{\hm C}-{\hm D}$. Inserting these bounds of ${\hm A}$, ${\hm B}$, ${\hm C}$, ${\hm D}$, we get a lower bound for $|\Gamma_{nr}(\a)\cap \Gamma_{nr}(\a')|$ as shown in Theorem~\ref{main21}, omitting the details.
\qed

Remark that for the reason of the length of the paper, we are not able to  improve the lower bound  $\sqrt{q}\geq 17$ (resp. $45$) for small exceptions in case (ii) of Theorem~\ref{main}.
 Hopefully,  these  small cases may be verified directly with a sufficiently powerful computer.

Combining Theorem~\ref{main} and the main results of \cite{BH,Chen-Du,Chen-Du2}, we get
\begin{theorem} \label{main'} Let $G$ be  a primitive  group whose socle is a simple group of  Lie-type of rank $1$ such that $b(G)=2$.
 Then  the BG-Conjecture is valid, except for $G=\hbox{\rm PSU}(3,q)\ZZ_{m_1}$ and  $M=\hbox{\rm SO}(3,q)\ZZ_{m_1}$ $(m_1\mid 2m$ and  $m_1\ne  1)$, where
    $\sqrt{q}\le 17$ or $45$,  if $3\nmid (q+1)$  and $3\bigm| (q+1)$, respectively.
  \end{theorem}

\vskip 3mm
\noindent {\bf  Outline  of the proof of Theorem~\ref{main}}: From now on, let $G$ be an almost simple group with the socle $T=\hbox{\rm PSU}(3,q)$, so that $T\leq G\leq \hbox{\rm Aut\,}(T)$, where $q=p^m$.
 Let $\alpha=M$ be a maximal subgroup of $G$ containing $M_0=\hbox{\rm PSO}(3,q)$, up to conjugacy. Consider the primitive permutation representations of $G$ on the right cosets $[G:M]$.
 Then $q\ge7$ and either: (i) $G=T$ and $M=M_0$; or (ii) $G=T\ZZ_{m_1}$ and  $M=M_0\ZZ_{m_1}$, where $m_1\mid 2m$ and $m_1\ne 1$.
For the first case, we exhibit a common  neighbor of $\alpha $ and $\alpha^g$ in the Saxl graph for any $g\in G$ via a direct geometric construction in Section 2.
However, we are not able to extend  this geometric construction   to the second case, which necessitates a new approach developed in Section 3.
Consequently, the proof of Theorem~\ref{main} is obtained by combining Theorems~\ref{main1} and \ref{main2}.
Moreover, to prove Theorem~\ref{main2} crucially depends on estimates of $|\Gamma_r(\alpha)|$ and $|\Gamma_{nr}(\alpha) \cap \Gamma_{nr}(\alpha')|$, developed in Sections 4 and 5 respectively. Recall that $\Gamma_r(\alpha)$ (resp. $\Gamma_{nr}(\alpha)$) is the union of all regular (resp. non-regular) suborbits of $G$ relative to $\alpha$.
\vskip -7pt

Outline of the proof of Theorem~\ref{main}: Let $G$ be almost simple with socle $T=\hbox{\rm PSU}(3,q)$, so $T\le G\le \hbox{\rm Aut}(T)$, $q=p^m$. Let $\alpha=M$ be a maximal subgroup of $G$ containing $M_0=\hbox{\rm PSO}(3,q)$ (up to conjugacy), and consider the primitive action of $G$ on the right cosets $[G\!:\!M]$. Then $q\ge 7$ and either (i) $G=T$ and $M=M_0$, or (ii) $G=T\ZZ_{m_1}$ and $M=M_0\ZZ_{m_1}$ with $m_1\mid 2m$ and $m_1\ne 1$. In case (i), a common neighbor of $\alpha$ and $\alpha^g$ in the Saxl graph (for any $g\in G$) is exhibited by a direct geometric construction in Section 2. This geometric construction does not extend to case (ii), so a new approach is developed in Section 3. Theorem~\ref{main} then follows by combining Theorems~\ref{main1} and~\ref{main2}. The proof of Theorem~\ref{main2} depends crucially on estimates for $|\Gamma_r(\alpha)|$ and $|\Gamma_{nr}(\alpha)\cap \Gamma_{nr}(\alpha')|$, developed in Sections 4 and 5, where $\Gamma_r(\alpha)$ (resp. $\Gamma_{nr}(\alpha)$) is the union of regular (resp. non-regular) suborbits of $G$ relative to $\alpha$.

Outline of the proof of the main theorem: Let $G$ be almost simple with socle $T=\hbox{\rm PSU}(3,q)$, so $T\le G\le \hbox{\rm Aut}(T)$, $q=p^m$. Let $\alpha=M$ be a maximal subgroup of $G$ containing $M_0=\hbox{\rm PSO}(3,q)$ (up to conjugacy), and consider the primitive action of $G$ on the right cosets $[G\!:\!M]$. Then $q\ge 7$ and either (i) $G=T$ and $M=M_0$, or (ii) $G=T\ZZ_{m_1}$ and $M=M_0\ZZ_{m_1}$ with $m_1\mid 2m$ and $m_1\ne 1$. In case (i), a common neighbor of $\alpha$ and $\alpha^g$ in the Saxl graph (for any $g\in G$) is exhibited by a direct geometric construction developed earlier. This geometric construction does not extend to case (ii), so a new approach is developed later.

We first prove the following theorem.
\begin{theorem} \label{main1}  Let $G=\PSU(3,q)$ and $M=\PSO(3,q),$ a maximal subgroup of $G$, where $q=p^m\ge 7$ for an odd prime $p$.
Consider the primitive right multiplication action  of $G$ on the set of right cosets $\Omega =[G:M]$.  Then $b(G)=2$ and the BG-Conjecture holds.
  \end{theorem}

We also prove the following theorem.
\begin{theorem} \label{main2}  Let $G=\PSU(3,q)\ZZ_{m_1}$ and
    $\a=M=\PSO(3,q)\ZZ_{m_1}\le G$, with  $q=p^m\ge 7$ for an odd prime $p$,  $m_1\mid 2m$ and $m_1\ne 1$.
Under the primitive right multiplication action of $G$ on $\Omega=[G:M]$, we have $b(G)=2$ and the BG-Conjecture is true, provided $\sqrt{q}\geq 17$ or $45$ if $3\nmid (q+1)$ and $3\di (q+1)$, respectively.
  \end{theorem}

In establishing the main theorem, we combine Theorem~\ref{main1} and Theorem~\ref{main2}. The proof of Theorem~\ref{main2} depends crucially on estimates for $|\Gamma_r(\alpha)|$ and $|\Gamma_{nr}(\alpha)\cap \Gamma_{nr}(\alpha')|$, developed later, where $\Gamma_r(\alpha)$ (resp. $\Gamma_{nr}(\alpha)$) is the union of regular (resp. non-regular) suborbits of $G$ relative to $\alpha$.

Question: Let $G\leq \mathrm{Sym}(\Omega)$ be a finite primitive permutation group with socle $\mathrm{PSU}(3,q)$, where $q=p^m\ge 7$ for an odd prime $p$, and let $M=G_\alpha$ be a point-stabilizer containing $\mathrm{PSO}(3,q)$. Here $b(G)$ is the minimal size of a base, i.e. a subset $\Delta\subseteq \Omega$ whose pointwise stabilizer in $G$ is trivial. When $b(G)=2$, the Saxl graph $\Sigma(G)$ is the graph with vertex set $\Omega$ in which two vertices are adjacent exactly when they form a base; the BG-Conjecture asserts that every two vertices of $\Sigma(G)$ have a common neighbour. Under these assumptions, which statement about $G$, $M$, $b(G)$, and the BG-Conjecture holds?

One has $b(G)=2$, and either (i) $G=\mathrm{PSU}(3,q)$ and $M=\mathrm{SO}(3,q)$, or (ii) $G=\mathrm{PSU}(3,q)\mathbb Z_{m_1}$ and $M=\mathrm{SO}(3,q)\mathbb Z_{m_1}$ for some $m_1\mid 2m$ with $m_1\ne 1$. Moreover, the BG-Conjecture holds in case (i); and in case (ii) it holds whenever $\sqrt q\ge 17$ if $3\nmid(q+1)$, and whenever $\sqrt q\ge 45$ if $3\mid(q+1)$.

Choices:
B. One has $b(G)=2$, and either (i) $G=\mathrm{PSU}(3,q)$ and $M=\mathrm{SO}(3,q)$, or (ii) $G=\mathrm{PSU}(3,q)\mathbb Z_{m_1}$ and $M=\mathrm{SO}(3,q)\mathbb Z_{m_1}$ for some $m_1\mid 2m$ with $m_1\ne 1$. Moreover, the BG-Conjecture holds in case (i); and in case (ii) it holds whenever $q\ge 17$ if $3\nmid(q+1)$, and whenever $q\ge 45$ if $3\mid(q+1)$.
C. One has $b(G)=2$, and either (i) $G=\mathrm{PSU}(3,q)$ and $M=\mathrm{SO}(3,q)$, or (ii) $G=\mathrm{PSU}(3,q)\mathbb Z_{m_1}$ and $M=\mathrm{SO}(3,q)\mathbb Z_{m_1}$ for some $m_1\mid 2m$ with $m_1\ne 1$.
D. One has $b(G)=2$, and either (i) $G=\mathrm{PSU}(3,q)$ and $M=\mathrm{SO}(3,q)$, or (ii) $G=\mathrm{PSU}(3,q)\mathbb Z_{m_1}$ and $M=\mathrm{SO}(3,q)\mathbb Z_{m_1}$ for some $m_1\mid 2m$. Moreover, the BG-Conjecture holds in both cases (i) and (ii).
E. One has $b(G)=2$, and either (i) $G=\mathrm{PSU}(3,q)$ and $M=\mathrm{SO}(3,q)$, or (ii) $G=\mathrm{PSU}(3,q)\mathbb Z_{m_1}$ and $M=\mathrm{SO}(3,q)\mathbb Z_{m_1}$ for some $m_1\mid m$ with $m_1\ne 1$. Moreover, the BG-Conjecture holds in case (i); and in case (ii) it holds whenever $\sqrt q\ge 17$ if $3\nmid(q+1)$, and whenever $\sqrt q\ge 45$ if $3\mid(q+1)$.


\label{thm:disper}
		Assume $\varepsilon_0<\varepsilon_*$, with $\varepsilon_*$ as in Lemma \ref{lem:disper-log}. Then for any given $0<\tau<\frac13$, there exists $K_1=K_1(\varepsilon_0,\tau)$ such that for any $\theta\in\mathbb{T}^d$ and any $t\in\mathbb{R}$,
\begin{equation}\label{linest}
			\norm{u\left( \cdot ,t\right)}_{\ell^\infty\left(\mathbb{Z}\right)}\leq K_1\JP{t}^{-\tau}(\norm{\varphi}_{\ell^1\left(\mathbb{Z}\right)}+\norm{\psi}_{\ell^1\left(\mathbb{Z}\right)}) \ , \quad \forall \ \varphi, \psi \in\ell^1(\mathbb{Z}) \ .
		\end{equation}

\begin{theorem}\label{thm:nonlinear}
		Let $p>7$. There exist $\varepsilon > 0$ and a constant $C$, so that whenever
		$\| \varphi \|_{\ell^2\left(\Z\right)}$ and $\| \psi\|_{\ell^2\left(\Z\right)} \leq \varepsilon
		$, there exists a unique global solution to \eqref{eq: nonlinear}, satisfying
		\begin{equation*}
		\|u\|_{L^q_t(\mathbb R;\ell^r(\mathbb Z))}\leq C\varepsilon,
		\end{equation*}
		for all admissible pairs $(q, r)$. In particular, $\| u(n,t) \|_{\ell^r(\Z)} \to 0$ as $t \to \infty$ for every $r > 2$.
	\end{theorem}

\begin{theorem}\label{thm:disper}
		Assume $\varepsilon_0<\varepsilon_*$, with $\varepsilon_*$ as in Lemma \ref{lem:disper-log}. Then for any given $0<\tau<\frac13$, there exists $K_1=K_1(\varepsilon_0,\tau)$ such that for any $\theta\in\T^d$ and any $t\in\R$,
\begin{equation}\label{linest}
			\norm{u\left( \cdot ,t\right)}_{\ell^\infty\left(\Z\right)}\leq K_1\JP{t}^{-\tau}(\norm{\varphi}_{\ell^1\left(\Z\right)}+\norm{\psi}_{\ell^1\left(\Z\right)}) \ , \quad \forall \ \varphi, \psi \in\ell^1(\Z) \ .
		\end{equation}
	\end{theorem}

We consider the Klein--Gordon equation with quasi-periodic potentials on the one-dimensional lattice $\mathbb{Z}$:
 \begin{equation}\label{eq:DKG}
 \begin{cases}
 \bigl(\partial_{tt}+G_{\theta}+m^{2}\bigr)u(n,t)=0,\qquad\qquad\quad(n,t)\in\mathbb{Z}\times\left(0,\infty\right),\\
 u(\cdot,0)=\varphi(\cdot)\in\ell^{1}\left(\mathbb{Z}\right),\quad \partial_{t}u(\cdot,0)=\psi(\cdot)\in\ell^{1}\left(\mathbb{Z}\right),\quad n\in\mathbb{Z},\\
 \end{cases}
 \end{equation}
 where $G_{\theta}$ is defined as
 \begin{equation*}
 (G_{\theta}u)(n)
 :=-\Delta_{\text{disc}}u(n)+V(\theta+n\omega)u(n),
 \qquad n\in\mathbb{Z},
 \end{equation*}
 and the discrete Laplacian is given by
 $\Delta_{\text{disc}}u(n):=u(n+1)+u(n-1)-2u(n)$.
 Here $\theta \in \mathbb{T}^d$, the potential $V\colon\mathbb{T}^{d}\to\mathbb{R}$ is a real-analytic with $d\geq 1$, and the frequency vector $\omega\in\mathbb{R}^{d}$ is \textbf{Diophantine}; i.e., there exist constants $\gamma>0$ and $\eta>d-1$ such that
 \begin{equation}\label{dio}
 \inf_{j\in\mathbb{Z}}\bigl|\langle k,\omega\rangle_{\mathbb{R}^{d}}-j\pi\bigr|
 >\frac{\gamma}{|k|^{\eta}},
 \qquad \forall\,k\in\mathbb{Z}^{d}\setminus\{0\},
 \end{equation}
 where $\langle\cdot{,}\cdot\rangle_{\mathbb{R}^{d}}$ denotes the standard scalar product on $\mathbb{R}^{d}$. We assume that $V$ admits a bounded analytic complex extension to $\left\{z\in\mathbb{C}^{d}:\left|{\rm Im}z\right|<r\right\}$. We also denote
 \begin{equation*}
 \varepsilon_{0}:=\left|V\right|_{r}=\sup_{\left\{z\in\mathbb{C}^{d}:\left|{\rm Im}z\right|<r\right\}}\left|V\left(z\right)\right|.
 \end{equation*}
 In the literature, one often sets $H_{\theta}=G_{\theta}-2$ to be the \textbf{quasi-periodic Schr\"odinger operator}. We are interested in the dispersive estimates for the equation \eqref{eq:DKG}, namely the $\ell^{1}\!\to\!\ell^{\infty}$ estimates of the form
 \begin{equation}\label{eq:dispersive}
 \norm{u(\cdot,t)}_{\ell^{\infty}(\mathbb{Z})}
 \leq C\JP{t}^{-\tau}\bigl(\norm{\varphi}_{\ell^{1}(\mathbb{Z})}+\norm{\psi}_{\ell^{1}(\mathbb{Z})}\bigr),
 \qquad \forall\,\varphi,\psi\in\ell^{1}(\mathbb{Z}),
 \end{equation}
 where $\JP{t}:=\sqrt{1+t^{2}}$ is the Japanese bracket, $C>0$ is a constant independent of $t$ and $\tau$ is so called dispersive decay rate.

\textbf{Motivation} The goal of this work is to understand to what extent dispersive properties of lattice Klein–Gordon dynamics are stable under introducing a quasi-periodic medium. On the one hand, the free discrete Klein–Gordon equation on $\mathbb Z$ is known to exhibit a precise dispersive decay rate (see \cite{SK05}), which plays a central role in both the linear and nonlinear theory. On the other hand, quasi-periodic Schr\"odinger operators $H_\theta$ are canonical models for wave propagation in quasicrystals and are known, in the small analytic and Diophantine regime, to have purely absolutely continuous spectrum of Cantor type (see for example \cite{AD08, Avi08}). From a dispersive PDE viewpoint, this spectral picture is highly singular and lies completely outside the standard setting of the fine continuous spectrum given by a finite union of intervals, so it is a priori unclear whether any robust $\ell^1\to\ell^\infty$ decay should survive.

Our main result states as follows.

\begin{equation}\label{eq:DKG}
		\begin{cases}
			\bigl(\partial_{tt}+G_{\theta}+m^{2}\bigr)u(n,t)=0,\qquad\qquad\quad(n,t)\in\mathbb{Z}\times\left(0,\infty\right),\\
            u(\cdot,0)=\varphi(\cdot)\in\ell^{1}\left(\Z\right),\quad \partial_{t}u(\cdot,0)=\psi(\cdot)\in\ell^{1}\left(\Z\right),\quad n\in\mathbb{Z},\\
		\end{cases}
	\end{equation}

\begin{lemma}\label{lem:disper-log}
		Assume that $\left|V\right|_{r}=\varepsilon_{0}\leq \varepsilon_{\ast}$ with $\varepsilon_{\ast}$ defined in Proposition \ref{propsana}. For any $k,k_{\Delta},n,n_{\Delta}$,
		\begin{equation*}
			\begin{aligned}
				\left|\mathbb{I}\left(\Sigma\right)\right|\leq\frac{C\left(m\right)\left|\ln\varepsilon_{0}\right|^{82000d\left(\ln\ln\left(2+\JP{t}\right)\right)^{2}}}{\JP{t}^{\frac{1}{3}}},
			\end{aligned}
		\end{equation*}
		for any $t\in\R$ where $m>0$ and the constant $C\left(m\right)$ only depends on $m$.
	\end{lemma}

By \eqref{infinito}, we have
 \begin{equation}\label{eq:u(t)_int}
 |u(n,t)|\leq\frac{1}{\pi}\left|\int_\Sigma \left(g_1(E,t){\cK}_n(E)+g_2(E,t){\cJ}_n(E)\right) \rho^{\prime}\d E\right|+\varepsilon_0^{\frac{\sigma^2}{10}}\|u(\cdot,t)\|_{\ell^\infty\left(\Z\right)} \ .
 \end{equation}
 To estimate $\|u(\cdot,t)\|_{\ell^\infty\left(\Z\right)}$, it suffices to control the integral term by taking $\varepsilon_{0}$ to be small. Substituting the expression of $\mathcal{S}(u(t))$ into the integral, and recalling the definitions of $g^{\varphi}$ and $g^{\psi}$, we obtain
 \begin{equation}\label{eq:appro_u}
 \begin{aligned}
 &\int_\Sigma \left(g_1(E,t){\cK}_n(E)+g_2(E,t){\cJ}_n(E)\right)\, \rho^{\prime}\d E\\
 =&\frac{1}{2} \int_\Sigma e^{{\rm i}\Omega(E)t} \sum_{k\in\Z} \left(\varphi(k) + \frac{\psi(k)}{{\rm i}\Omega(E)}\right) \left({\cK}_k(E){\cK}_n(E)+{\cJ}_k(E){\cJ}_n(E)\right)\, \rho^{\prime}\d E \\
 &+\frac{1}{2} \int_\Sigma e^{-{\rm i}\Omega(E)t} \sum_{k\in\Z} \left(\varphi(k) - \frac{\psi(k)}{{\rm i}\Omega(E)}\right) \left({\cK}_k(E){\cK}_n(E)+{\cJ}_k(E){\cJ}_n(E)\right)\, \rho^{\prime}\d E \ .
 \end{aligned}
 \end{equation}
 Note that the term $\left({\cK}_k(E){\cK}_n(E)+{\cJ}_k(E){\cJ}_n(E)\right)$ can be expanded as
 \begin{equation}\label{eq:K,J expansions}
 \sum_{k_{\Delta}, n_{\Delta}} \beta_{k,k_{\Delta}} \beta_{n,n_{\Delta}} \cos(k_{\Delta}-n_{\Delta})\rho(E) \ .
 \end{equation}
 Since $m>0$, the factor $\frac{1}{\Omega(E)}= \frac{1}{\sqrt{2+m^2+E}}$ is smooth and bounded on the spectrum of $G_{\theta}+m^2$. By the symmetry of the complex conjugate terms in \eqref{eq:appro_u}, the dispersive decay estimates for both integrals depend on the oscillatory behavior of the phase function. Thus, the problem reduces to establishing the estimates for the oscillatory integral associated with the kernel
 $e^{-{\rm i} t \sqrt{2+m^2+E}}$. For simplicity, we denote
 $$\mathbb{I}(S;k_{\Delta},n_{\Delta},\rho):=\int_{S} \beta_{k,k_{\Delta}}\beta_{n,n_{\Delta}}\cos
 \big((k_{\Delta}-n_{\Delta})\rho\left(E\right)\big)e^{-\mathrm{i}t\Omega\left(E\right)}\cdot\rho^{\prime}\d E, \quad \text{for}\ S\subset\R,$$
 and
 $$\mathbb{I}^{J}(S;k_{\Delta},n_{\Delta},\rho):=\int_{S} \beta^J_{k,k_{\Delta}} \beta^J_{n,n_{\Delta}}\cos
 \big((k_{\Delta}-n_{\Delta})\rho\left(E\right)\big)e^{-\mathrm{i}t\Omega\left(E\right)}\cdot\rho^{\prime} \d E, \quad \text{for}\ S\subset\R.$$
 Whenever no confusion can arise, we simply write $\mathbb{I}(E)$ and $\mathbb{I}^{J}(E)$ respectively.

\begin{proof}[Proof of Theorem \ref{thm:disper} assuming Lemma \ref{lem:disper-log}]
 By the expansions \eqref{eq:K,J expansions}, we apply Lemma \ref{lem:disper-log} to equation \eqref{eq:appro_u} and derive
 \begin{equation}\label{eq:L infty estimate}
 \begin{aligned}
 &\left|\int_\Sigma \left(g_1(E,t){\cK}_n(E)+g_2(E,t){\cJ}_n(E)\right)\, \rho^{\prime}\d E\right|\\
 \leq&C\left(m\right)\cdot\frac{\left|\ln\varepsilon_{0}\right|^{82000\left(\ln\ln\left(2+\JP{t}\right)\right)^{2}d}}{\JP{t}^{\frac{1}{3}}}\left(\sum_{k\in\Z}\left|\varphi\left(k\right)\right|+\frac{1}{\inf_{E\in\Sigma}\left|\Omega\left(E\right)\right|}\sum_{k\in\Z}\left|\psi\left(k\right)\right|\right),
 \end{aligned}
 \end{equation}
 where the absolute constant $C_{1}$ relates to the estimate of $e^{{\rm i}\Omega\left(\lambda\right)t}$ and so does $C_{2}$ for the estimate related to $e^{-{\rm i}\Omega\left(\lambda\right)t}$. Since $\epsilon_{0}<1$ is small, and by \eqref{eq:u(t)_int}, \eqref{eq:L infty estimate} we deduce
 \begin{equation*}
 \norm{u\left(t\right)}_{\ell^{\infty}\left(\Z\right)}\leq\frac{C\left(m\right)}{1-\varepsilon_0^{\frac{\sigma^2}{10}}}\frac{\left|\ln\varepsilon_{0}\right|^{82000\left(\ln\ln\left(2+\JP{t}\right)\right)^{2}d}}{\JP{t}^{\frac{1}{3}}}\left(\norm{\varphi}_{\ell^{1}\left(\Z\right)}+\frac{1}{\inf_{E\in\Sigma}\left|\Omega\left(E\right)\right|}\norm{\psi}_{\ell^{1}\left(\Z\right)}\right).
 \end{equation*}
 This completes the proof.
 \end{proof}

In the case of $|k_{\Delta}-n_{\Delta}|$ is large, we derive the following bound.
\begin{lemma}\label{lem:large}
For any $t \in \R$, the following estimate holds,
\begin{equation*}
 \left|\mathbb{I}^{J}(\Sigma)\right|\leq\frac{18}{\left|k_{\Delta}-n_{\Delta}\right|}\left|\ln \varepsilon_{0}\right|^{2J^{2}d}+\frac{3\Theta_{1}\JP{t}}{\left|k_{\Delta}-n_{\Delta}\right|}\left(\left(1+\varepsilon_{0}^{\frac{1}{4}}\right)^{2}+\varepsilon_{j}^{6\sigma}\right)\left(1+\frac{1}{\Theta_{2}}\right).\\
\end{equation*}
\end{lemma}
\begin{proof}
By Proposition~\ref{propsana1}, for each $0\leq j\leq J$ we can write
\begin{equation*}
 \Gamma_{j}^{(J)}
 =\mathcal{R}_{j}\cup\Bigl(\,\bigsqcup_{s^{J}_{j}}I_{s^{J}_{j}}\Bigr),
\end{equation*}
where $\mathcal{R}_{j}$ is a finite set of points (possibly empty) and each $I_{s^{J}_{j}}$ is an open interval in $\Gamma_{j}^{(J)}$. Recall that $\rho$ is absolutely continuous. Then
 \begin{equation}\label{eq:Ij-M large}
 \begin{aligned}
 \mathbb{I}^{J}(\Sigma)=&\int_{\inf\Sigma}^{\sup\Sigma}\beta^J_{k,k_{\Delta}} \beta^J_{n,n_{\Delta}}\cos\big((k_{\Delta}-n_{\Delta})\rho\left(E\right)\big)e^{-{\rm i}t\Omega\left(E\right)}\cdot\rho^{\prime} \d E\\
 =&\frac{1}{k_{\Delta}-n_{\Delta}}\sum_{j=0}^{J}\sum_{s_{j}^{J}}\big(\beta^J_{k,k_{\Delta}} \beta^J_{n,n_{\Delta}}\sin\big((k_{\Delta}-n_{\Delta})\rho\left(E\right)\big)e^{-{\rm i}t\Omega\left(E\right)}\big)\Big|_{\inf I_{s_{j}^{J}}}^{\sup I_{s_{j}^{J}}}\\
 &\quad-\frac{1}{k_{\Delta}-n_{\Delta}}\sum_{j=0}^{J}\sum_{s_{j}^{J}}\int_{I_{s_{j}^{J}}}\big(\beta^J_{k,k_{\Delta}} \beta^J_{n,n_{\Delta}}e^{-{\rm i}t\Omega\left(E\right)}\big)^{\prime}\sin\big((k_{\Delta}-n_{\Delta})\rho\left(E\right)\big)\d E.\\
 \end{aligned}
 \end{equation}
 Using Proposition \ref{propsana1} again, we derive the upper bound for connected components of $\bigcup \Gamma_{j}^{(J)}$:
 \begin{equation*}
 \sum_{j=0}^{J}\#\left(s_{j}^{J}\right)\leq \left|\ln \varepsilon_{0}\right|^{2J^{2}d}.
 \end{equation*}
 For the first term in equation \eqref{eq:Ij-M large}, by the $L^{\infty}$-estimate and inequality \eqref{esti_beta_0} that
 \begin{equation}\label{eq:first term}
 \begin{aligned}
 &\left|\frac{1}{k_{\Delta}-n_{\Delta}}\sum_{j=0}^{J}\sum_{s_{j}^{J}}\big(\beta^J_{k,k_{\Delta}} \beta^J_{n,n_{\Delta}}\sin\big((k_{\Delta}-n_{\Delta})\rho\left(E\right)\big)e^{-{\rm i}t\Omega\left(E\right)}\big)\Big|_{\inf I_{s_{j}^{J}}}^{\sup I_{s_{j}^{J}}}\right|\\
 \leq&\frac{2}{\left|k_{\Delta}-n_{\Delta}\right|}\left|\ln \varepsilon_{0}\right|^{2J^{2}d}\cdot\left(1+\varepsilon_{0}^{\frac{1}{4}}+\varepsilon_{0}^{3\sigma}\right)^{2}\\
 \leq&\frac{18}{\left|k_{\Delta}-n_{\Delta}\right|}\left|\ln \varepsilon_{0}\right|^{2J^{2}d}.
 \end{aligned}
 \end{equation}
For the second term of the result in equation \eqref{eq:Ij-M large}, we need the following direct estimate
\begin{equation}\label{eq:betac1}
 \left|\beta^J_{k,k_{\Delta}} \beta^J_{n,n_{\Delta}}\right|_{\mathcal{C}^{1}\left(\Gamma^{\left(J\right)}_{j}\right)}\leq
 \begin{cases}
 3\left(1+\varepsilon_{0}^{\frac{1}{4}}\right)^{2}\;{\rm when}\;j=0;\\
 3\varepsilon_{j}^{6\sigma}\qquad{\rm when}\;1\leq j\leq J.\\
 \end{cases}
\end{equation}
 Using inequality
 \eqref{eq:betac1}, we deduce

For our Theorem \ref{thm:disper}, we have the following remarks.

	\begin{itemize}
		\item[($\mathbf{R}_1$)] Stefanov and Kevrekidis~\cite{SK05} first established a sharp dispersive decay rate
		$
		\frac{1}{3}
		$
		for the free discrete Klein-Gordon equation (i.e., equation \eqref{eq:DKG} with $V=0$) on $\mathbb Z$ via Van der Corput's lemma.
		Recently, Borovyk and Goldberg \cite{BG17} derived the corresponding rate on $\mathbb Z^{2}$, namely  $\langle t\rangle^{-\frac{3}{4}}$. Later, Cuenin and Ikromov~\cite{CI21} derived the rates on~$\mathbb Z^{d}$ for $d=3,4$, namely $
		\langle t\rangle^{-\frac{7}{6}}\ (d=3)$, and
		$\langle t\rangle^{-\frac{3}{2}}\log \bigl(2+|t|\bigr)\ (d=4)$. The obstacle to extending Cuenin-Ikromov to higher dimensions is that the finite classification of stable phase singularities (Thom's catastrophes), which is essential for deriving sharp decay rates, breaks down in dimensions $d \ge 5$. Our result can be seen as a perturbation result for Stefanov and Kevrekidis, which shows that the decay rate $(\frac{1}{3})^{-}$ persists under small quasi-periodic perturbations of the potentials.

		\item[($\mathbf{R}_2$)] While several studies (e.g., \cite{CT09, GPP13, PS08, KPS09, KKK06}) have addressed dispersive equations with potentials whose continuous spectrum is a union of disjoint intervals, generic behavior of the operator $H_{\theta}$ is markedly different: its spectrum is purely absolutely continuous and forms a Cantor set; see, e.g., \cite{AD08, Avi08, E92}. Our approach builds heavily on recent advances in the spectral theory of quasi-periodic Schr\"odinger operators; see, for example, \cite{Z16}. Specifically, we employ a KAM-based perturbative construction to approximate the rotation number and decompose the spectrum into many small intervals. Using a modified  spectral transform, we represent the solution as an oscillatory integral and then apply Van der Corput's lemma on each interval to establish dispersive decay. We emphasize that a similar strategy appears in the study of ballistic motion and dispersive estimates for discrete Schr\"odinger equations with quasi-periodic potentials; see, e.g., \cite{BZ20, Z16}.

        \item[($\mathbf{R}_3$)] For discrete wave equation with quasi-periodic potentials (i.e., equation \eqref{eq:DKG} with $m=0$), our method does not work. In fact, there does not exist $\ell^1 \to \ell^{\infty}$ estimates for fundamental solution of the free wave equation. We point out the following development regarding the dispersive estimate for the free discrete wave equation. This phenomenon also holds on $\mathbb{R}^1$. Schultz \cite{Sch98} proved dispersive estimates for the wave equation on lattice graphs $\mathbb{Z}^d$ for $d=2,3$, namely  $\langle t\rangle^{-\frac{3}{4}}\ (d=2)$ and $\langle t\rangle^{-\frac{7}{6}}\ (d=3)$. Combining the singularity theory
		with results in uniform estimates of oscillatory integrals, Bi-Chen-Hua \cite{BCH1} extended this result  to $d=4$: they proved that the optimal
		time decay rate is $|t|^{-\frac{3}{2}} \log |t|$. By tools from Newton polyhedra, Bi-Chen-Hua \cite{BCH2} further extended the result to $d=5$: the sharp decay rate of the fundamental solution of the wave equation on $\mathbb{Z}^5$ is $|t|^{-\frac{6}{11}}$.
    \end{itemize}

	A typical application for dispersive estimates is the derivation of Strichartz inequalities. For the inhomogeneous discrete Klein-Gordon equation,
	\begin{equation}\label{eq: inhomogegeneous}
		\begin{cases}
			\bigl(\partial_{tt}+G_{\theta}+m^{2}\bigr)u(n,t)=F(n,t),
			&(n,t)\in\mathbb{Z}\times\left(0,\infty\right),\\[2mm]
			u(n,0)=\varphi(n),\quad \partial_{t}u(n,0)=\psi(n),
			&n\in\mathbb{Z},
		\end{cases}
	\end{equation}
	we have the following mixed space-time estimates via the standard Keel-Tao argument. For convenience, we say
	a pair of exponents $(q,r)$ with
	$
	2\le q,r\le\infty
	$
	is called \emph{$\tau$--admissible} if
	$
	\frac{2}{q}
	=\tau\Bigl(1-\frac{2}{r}\Bigr).
	$ Here $0<\tau<\tfrac13$ is fixed as in Theorem~\ref{thm:disper}.

	\begin{theorem}\label{thm:Strichartz}

        Fix
		$0<\tau<\tfrac13$ and $F\in L^{\tilde q'}(\mathbb R;\ell^{\tilde r'}(\mathbb Z))$. Let $(q,r)$ and $(\tilde q,\tilde r)$ be two $\tau$--admissible pairs with $q,\tilde q<\infty.$  Then the Cauchy
		problem \eqref{eq: inhomogegeneous} has a unique global solution
		$
		u\in C(\mathbb R;\ell^2(\mathbb Z))
		$
		and \begin{equation}\label{eq:inhom-Strichartz}
			\|u\|_{L^q_t(\mathbb R;\ell^r(\mathbb Z))}
			\;\le\; C\Bigl(
			\|\varphi\|_{\ell^2(\mathbb Z)}
			+\|\psi\|_{\ell^2(\mathbb Z)}
			+\|F\|_{L^{\tilde q'}_t(\mathbb R;\ell^{\tilde r'}(\mathbb Z))}
			\Bigr),
		\end{equation}
		where $\tilde q'$ and $\tilde r'$ denote the Hölder conjugates of
		$\tilde q$ and $\tilde r$, respectively, and the constant $C$ is
		independent of $\theta$, $\varphi$, $\psi$ and $F$.

	\end{theorem}

	Using Strichartz estimates, we establish the global well-posedness of the nonlinear Klein-Gordon equation  \begin{equation}\label{eq: nonlinear}
		\begin{cases}
			\bigl(\partial_{tt}+G_{\theta}+m^{2}\bigr)u(n,t)= \pm |u|^{p-1}u,
			&(n,t)\in\mathbb{Z}\times\mathbb{R}_{>0},\\[2mm]
			u(n,0)=\varphi(n),\quad \partial_{t}u(n,0)=\psi(n),
			&n\in\mathbb{Z},
		\end{cases}
	\end{equation}
with small initial data.
	\begin{theorem}\label{thm:nonlinear}
		Let $p>7$. There exist $\varepsilon > 0$ and a constant $C$, so that whenever
		$\| \varphi \|_{\ell^2\left(\mathbb{Z}\right)}$ and $\| \psi\|_{\ell^2\left(\mathbb{Z}\right)} \leq \varepsilon
		$, there exists a unique global solution to \eqref{eq: nonlinear}, satisfying
		\begin{equation*}
		\|u\|_{L^q_t(\mathbb R;\ell^r(\mathbb Z))}\leq C\varepsilon,
		\end{equation*}
		for all admissible pairs $(q, r)$. In particular, $\| u(n,t) \|_{\ell^r(\mathbb{Z})} \to 0$ as $t \to \infty$ for every $r > 2$.
	\end{theorem}

The remainder of the paper is organized as follows. Section \ref{sec:2} reviews the structure of the spectrum of the quasi-periodic Schr\"odinger operator and presents a spectral transform that will be used throughout. Section \ref{sec:3} establishes Theorem \ref{thm:disper} by analysing a specific oscillatory integral on the spectrum. In Section \ref{sec:4}, we prove Theorems \ref{thm:Strichartz} and \ref{thm:nonlinear} as applications of the dispersive estimate.

A proof strategy for Theorem~\ref{thm:disper} is outlined in Remark $\mathbf{R}_2$ and the paper organization: the authors "employ a KAM-based perturbative construction to approximate the rotation number and decompose the spectrum into many small intervals." Then, "[u]sing a modified spectral transform, we represent the solution as an oscillatory integral and then apply Van der Corput's lemma on each interval to establish dispersive decay." Concretely, Section~\ref{sec:3} "establishes Theorem~\ref{thm:disper} by analysing a specific oscillatory integral on the spectrum."

A proof strategy for the main theorem is outlined in Remark $\mathbf{R}_2$ and the paper organization: the authors "employ a KAM-based perturbative construction to approximate the rotation number and decompose the spectrum into many small intervals." Then, "[u]sing a modified spectral transform, we represent the solution as an oscillatory integral and then apply Van der Corput's lemma on each interval to establish dispersive decay." Concretely, next the authors establish the main theorem by analysing a specific oscillatory integral on the spectrum.

\label{thm:disper}
		Assume $\varepsilon_0<\varepsilon_*$, with $\varepsilon_*$ as in the following lemma.

\begin{lemma}\label{lem:disper-log}
		Assume that $\left|V\right|_{r}=\varepsilon_{0}\leq \varepsilon_{\ast}$ with $\varepsilon_{\ast}$ defined in Proposition \ref{propsana}. For any $k,k_{\Delta},n,n_{\Delta}$,
		\begin{equation*}
			\begin{aligned}
				\left|\mathbb{I}\left(\Sigma\right)\right|\leq\frac{C\left(m\right)\left|\ln\varepsilon_{0}\right|^{82000d\left(\ln\ln\left(2+\JP{t}\right)\right)^{2}}}{\JP{t}^{\frac{1}{3}}},
			\end{aligned}
		\end{equation*}
		for any $t\in\R$ where $m>0$ and the constant $C\left(m\right)$ only depends on $m$.
	\end{lemma}

		Then for any given $0<\tau<\frac13$, there exists $K_1=K_1(\varepsilon_0,\tau)$ such that for any $\theta\in\mathbb{T}^d$ and any $t\in\mathbb{R}$,
\begin{equation}\label{linest}
			\norm{u\left( \cdot ,t\right)}_{\ell^\infty\left(\mathbb{Z}\right)}\leq K_1\JP{t}^{-\tau}(\norm{\varphi}_{\ell^1\left(\mathbb{Z}\right)}+\norm{\psi}_{\ell^1\left(\mathbb{Z}\right)}) \ , \quad \forall \ \varphi, \psi \in\ell^1(\mathbb{Z}) \ .
		\end{equation}

Existential–Universal
Inequality or Bound

Question:
Consider the discrete Klein--Gordon equation on \(\mathbb Z\)
\[
\bigl(\partial_{tt}+G_{\theta}+m^{2}\bigr)u(n,t)=0,\qquad (n,t)\in \mathbb Z\times (0,\infty),
\]
with initial data
\[
u(\cdot,0)=\varphi\in \ell^{1}(\mathbb Z),\qquad \partial_t u(\cdot,0)=\psi\in \ell^{1}(\mathbb Z),
\]
where
\[
(G_{\theta}u)(n)=-\Delta_{\mathrm{disc}}u(n)+V(\theta+n\omega)u(n),
\qquad
\Delta_{\mathrm{disc}}u(n)=u(n+1)+u(n-1)-2u(n).
\]
Assume \(\theta\in \mathbb T^d\), \(d\ge 1\), \(m>0\), \(\omega\in\mathbb R^d\) is Diophantine, and \(V:\mathbb T^d\to\mathbb R\) extends to a bounded analytic function on \(\{z\in\mathbb C^d:|\operatorname{Im} z|<r\}\). Let
\[
\varepsilon_0:=|V|_r=
\sup_{\{z\in\mathbb C^d:|\operatorname{Im} z|<r\}} |V(z)|,
\]
and assume \(\varepsilon_0<\varepsilon_*\). Also write \(\langle t\rangle:=\sqrt{1+t^2}\). Under these assumptions, which quantitative \(\ell^1\to\ell^\infty\) dispersive estimate is valid for the solution \(u\)?

For every \(\tau\) with \(0<\tau<\tfrac13\), there exists a constant \(K_1=K_1(\varepsilon_0,\tau)\) such that, for every \(\theta\in\mathbb T^d\), every \(t\in\mathbb R\), and all \(\varphi,\psi\in \ell^1(\mathbb Z)\), the corresponding solution satisfies
\[
\|u(\cdot,t)\|_{\ell^\infty(\mathbb Z)}
\le K_1\langle t\rangle^{-\tau}
\bigl(\|\varphi\|_{\ell^1(\mathbb Z)}+\|\psi\|_{\ell^1(\mathbb Z)}\bigr).
\]

Choices:
B.
There exists a constant \(K_1=K_1(\varepsilon_0)\) such that, for every \(\theta\in\mathbb T^d\), every \(t\in\mathbb R\), and all \(\varphi,\psi\in \ell^1(\mathbb Z)\), the corresponding solution satisfies
\[
\|u(\cdot,t)\|_{\ell^\infty(\mathbb Z)}
\le K_1\langle t\rangle^{-1/3}
\bigl(\|\varphi\|_{\ell^1(\mathbb Z)}+\|\psi\|_{\ell^1(\mathbb Z)}\bigr).
\]
C.
There exists a constant \(K_1=K_1(\varepsilon_0)\) such that, for every \(\theta\in\mathbb T^d\), every \(t\in\mathbb R\), and all \(\varphi,\psi\in \ell^1(\mathbb Z)\), the corresponding solution satisfies
\[
\|u(\cdot,t)\|_{\ell^\infty(\mathbb Z)}
\le K_1
\bigl(\|\varphi\|_{\ell^1(\mathbb Z)}+\|\psi\|_{\ell^1(\mathbb Z)}\bigr).
\]
D.
For every \(\tau\) with \(0<\tau<\tfrac13\), there exists a constant \(K_1=K_1(\varepsilon_0,\tau,\theta)\) such that, for every \(t\in\mathbb R\) and all \(\varphi,\psi\in \ell^1(\mathbb Z)\), the corresponding solution satisfies
\[
\|u(\cdot,t)\|_{\ell^\infty(\mathbb Z)}
\le K_1\langle t\rangle^{-\tau}
\bigl(\|\varphi\|_{\ell^1(\mathbb Z)}+\|\psi\|_{\ell^1(\mathbb Z)}\bigr).
\]
E.
For every \(\tau\) with \(0<\tau<\tfrac12\), there exists a constant \(K_1=K_1(\varepsilon_0,\tau)\) such that, for every \(\theta\in\mathbb T^d\), every \(t\in\mathbb R\), and all \(\varphi,\psi\in \ell^1(\mathbb Z)\), the corresponding solution satisfies
\[
\|u(\cdot,t)\|_{\ell^\infty(\mathbb Z)}
\le K_1\langle t\rangle^{-\tau}
\bigl(\|\varphi\|_{\ell^1(\mathbb Z)}+\|\psi\|_{\ell^1(\mathbb Z)}\bigr).
\]

endpoint decay exponent from oscillatory-integral bound

dropped time-decay factor \(\langle t\rangle^{-\tau}\)

uniformity in phase parameter \(\theta\)

allowed range of admissible decay exponents


Question:
Let \(u(n,t)\) solve the one-dimensional lattice Klein--Gordon equation
\[
(\partial_{tt}+G_{\theta}+m^2)u(n,t)=0,\qquad (n,t)\in \mathbb Z\times(0,\infty),
\]
with initial data \(u(\cdot,0)=\varphi\in \ell^1(\mathbb Z)\) and \(\partial_tu(\cdot,0)=\psi\in \ell^1(\mathbb Z)\), where
\[
(G_{\theta}u)(n)=-\Delta_{\mathrm{disc}}u(n)+V(\theta+n\omega)u(n),
\qquad
\Delta_{\mathrm{disc}}u(n)=u(n+1)+u(n-1)-2u(n).
\]
Assume \(m>0\), \(\theta\in \mathbb T^d\), the frequency \(\omega\in\mathbb R^d\) is Diophantine in the sense that there exist \(\gamma>0\) and \(\eta>d-1\) such that
\[
\inf_{j\in\mathbb Z}|\langle k,\omega\rangle-j\pi|>\frac{\gamma}{|k|^{\eta}}
\qquad \forall k\in\mathbb Z^d\setminus\{0\},
\]
and the real-analytic potential \(V:\mathbb T^d\to\mathbb R\) extends bounded analytically to \(\{z\in\mathbb C^d:|\operatorname{Im} z|<r\}\). Let
\[
\varepsilon_0=|V|_r:=\sup_{|\operatorname{Im} z|<r}|V(z)|,
\]
and assume the smallness condition \(\varepsilon_0<\varepsilon_*\). Also write \(\langle t\rangle=\sqrt{1+t^2}\). Which uniform dispersive statement holds?

For every \(\tau\) with \(0<\tau<\tfrac13\), there exists a constant \(K_1=K_1(\varepsilon_0,\tau)\) such that for every phase \(\theta\in\mathbb T^d\), every time \(t\in\mathbb R\), and all initial data \(\varphi,\psi\in\ell^1(\mathbb Z)\), the corresponding solution \(u\) satisfies
\[
\|u(\cdot,t)\|_{\ell^\infty(\mathbb Z)}\le K_1\langle t\rangle^{-\tau}\bigl(\|\varphi\|_{\ell^1(\mathbb Z)}+\|\psi\|_{\ell^1(\mathbb Z)}\bigr).
\]
In particular, the constant is uniform in \(\theta\), \(t\), \(\varphi\), and \(\psi\).

Choices:
B.
For every \(\tau\) with \(0<\tau\le \tfrac13\), there exists a constant \(K_1=K_1(\varepsilon_0,\tau)\) such that for every phase \(\theta\in\mathbb T^d\), every time \(t\in\mathbb R\), and all initial data \(\varphi,\psi\in\ell^1(\mathbb Z)\), the corresponding solution \(u\) satisfies
\[
\|u(\cdot,t)\|_{\ell^\infty(\mathbb Z)}\le K_1\langle t\rangle^{-\tau}\bigl(\|\varphi\|_{\ell^1(\mathbb Z)}+\|\psi\|_{\ell^1(\mathbb Z)}\bigr).
\]
In particular, the estimate is uniform in \(\theta\), \(t\), \(\varphi\), and \(\psi\).
C.
There exists some exponent \(\tau>0\) and a constant \(K_1=K_1(\varepsilon_0,\tau)\) such that for every phase \(\theta\in\mathbb T^d\), every time \(t\in\mathbb R\), and all initial data \(\varphi,\psi\in\ell^1(\mathbb Z)\), the corresponding solution \(u\) satisfies
\[
\|u(\cdot,t)\|_{\ell^\infty(\mathbb Z)}\le K_1\langle t\rangle^{-\tau}\bigl(\|\varphi\|_{\ell^1(\mathbb Z)}+\|\psi\|_{\ell^1(\mathbb Z)}\bigr).
\]
Thus one has a uniform \(\ell^1\!\to\!\ell^\infty\) dispersive bound for at least one positive decay rate.
D.
For every \(\tau\) with \(0<\tau<\tfrac13\), there exists a constant \(K_1=K_1(\varepsilon_0,\tau,\theta)\) such that for every fixed phase \(\theta\in\mathbb T^d\), every time \(t\in\mathbb R\), and all initial data \(\varphi,\psi\in\ell^1(\mathbb Z)\), the corresponding solution \(u\) satisfies
\[
\|u(\cdot,t)\|_{\ell^\infty(\mathbb Z)}\le K_1\langle t\rangle^{-\tau}\bigl(\|\varphi\|_{\ell^1(\mathbb Z)}+\|\psi\|_{\ell^1(\mathbb Z)}\bigr).
\]
In particular, the constant is uniform in \(t\), \(\varphi\), and \(\psi\), but may depend on the phase \(\theta\).
E.
There exists a constant \(K_1=K_1(\varepsilon_0)\) such that for every phase \(\theta\in\mathbb T^d\), every time \(t\in\mathbb R\), and all initial data \(\varphi,\psi\in\ell^1(\mathbb Z)\), the corresponding solution \(u\) satisfies the endpoint estimate
\[
\|u(\cdot,t)\|_{\ell^\infty(\mathbb Z)}\le K_1\langle t\rangle^{-1/3}\bigl(\|\varphi\|_{\ell^1(\mathbb Z)}+\|\psi\|_{\ell^1(\mathbb Z)}\bigr).
\]
In particular, one obtains a uniform \(\ell^1\!\to\!\ell^\infty\) decay rate exactly equal to \(\tfrac13\).

replaced the full range \(0<\tau<1/3\) by existence of one positive decay exponent

uniformity in the phase parameter \(\theta\)

log-loss/interval decomposition prevents a stated endpoint \(t^{-1/3}\) theorem