\section{Concluding remarks}\label{sec:remarks}

\subsection*{Quantitative refinements}
Theorem~\ref{thm:main} leaves open the rate at which $D_2(X)/D(X)$ tends
to zero. A rate would follow from a quantitative form of the proof: an
explicit decay of
\[
 \mu_Y\bigl(F\equiv0\pmod{3^n}\bigr)\longrightarrow0
\]
in the proof of Proposition~\ref{prop:fixed}, together with an explicit error term in
Theorem~\ref{thm:cuspcount}, both uniformly over the finite family
$\mathcal G_{\le j_0}$, and a choice of the parameters $A_0$ and $j_0$ as
functions of $L$.

The uniqueness conjecture asks for the pointwise
statement that no collision pair exists. The zero-one law
(Lemma~\ref{lem:zeroone}) permits finitely many zeros of the trace
difference $F$ along each orbit of the $3$-adic flows, and the excision of
\S\ref{sec:excision} removes directions rather than individual
pairs; the pointwise question therefore requires an additional mechanism.

\subsection*{The role of the prime $3$}
The $3$-adic analysis uses the base point $o=(1,1,1)$ of the
Markoff surface $a^2+b^2+c^2=3abc$, on which the $3$-adic unit points form eight
residue balls that are analytically isomorphic to $\bZ_3^2$. Poonen's
theorem requires a congruence to the identity modulo a power of $3$ with
exponent greater than $1/2$; in the normalized coordinates of
Lemma~\ref{lem:interpolation} the congruence is $g(x)\equiv x\pmod3$,
with exponent $1$, which is why fixation modulo $9$ suffices for
interpolation.
The prime $3$ is convenient because the scaled traces of all primitive
classes are integral and the base point is the unit point $(1,1,1)$, so
that the whole orbit of the base point lies in the eight residue balls on
which the analytic coordinates are defined.

\subsection*{Relation to strong approximation}
The finite spaces used in \S\ref{sec:fixed} are the quotients
$Y(3^n)$ of the orbit closures of the paired action, which are transitive
by construction; this transitivity supplies the compatible uniform
measures of Lemma~\ref{lem:zeroone}. Strong approximation, as studied by
Bourgain, Gamburd and Sarnak~\cite{BGS} and by the author~\cite{Chen}, concerns
the orbits of the Nielsen group on the full set of points of the Markoff
surface over finite fields. In the present argument, the reduction to
rational graphs of \S\ref{sec:graphs}--\ref{sec:sum} and the
cusp-orbit asymptotic of Theorem~\ref{thm:cuspcount} convert the
measures $\mu_Y$ into the count of lattice points $v$ with $N(v)\le L$.