\section{The boundary of the McShane--Rivin unit ball along Farey sectors}
\label{sec:geometry}

\subsection{Supporting functionals and corners}\label{sec:supports}
A \emph{supporting functional} of $N$ at a nonzero vector $x$ is a linear
form $\alpha$ on $\bR^2$ with
\[
 \alpha\le N\quad\text{on }\bR^2\qquad\text{and}\qquad\alpha(x)=N(x).
\]
Geometrically, $\{\alpha=N(x)\}$ is a supporting line of the convex body
$N(x)\ball$ at the boundary point $x$.

The set
\[
 \partial N(x):=\{\alpha\in(\bR^2)^*:\ \alpha\le N,\ \alpha(x)=N(x)\}
\]
of supporting functionals at $x$ (the subdifferential of $N$ at $x$) is a
nonempty compact convex set that depends only on the ray through $x$; it
consists of a single functional exactly when $\partial\ball$ is
differentiable at $x/N(x)$.

We call $\alpha/|\alpha|\in S^1$ the
\emph{normal direction} of a supporting functional $\alpha$ at $x$; it is an outward unit normal of
$\partial\ball$ at $x/N(x)$. For two linear forms or vectors we write
$\angle(\alpha,\beta)\in[0,\pi]$ for the angle between them, and for a set
$\{\beta,-\beta\}$ we write $\angle(\alpha,\{\beta,-\beta\})=\min(\angle(\alpha,\beta),\angle(\alpha,-\beta))$,
the angle between the lines they span.

Since $c|y|\le N(y)\le C|y|$,
every supporting functional satisfies $c\le|\alpha|\le C$. A supporting
functional is determined by its normal direction $n=\alpha/|\alpha|$, as
$\alpha=n/\max_\ball n$, and the support function $n\mapsto\max_\ball n$ of
$\ball$ is positive and Lipschitz on the unit circle; hence
$|\alpha-\alpha'|\le C\,|\alpha/|\alpha|-\alpha'/|\alpha'||$ for
supporting functionals $\alpha,\alpha'$.

Strict convexity of $\ball$ has the following consequence, used
repeatedly. For every nonzero linear form $\beta$ there is a unique point
$x_\beta\in\partial\ball$ with $\beta(x_\beta)=\max_\ball\beta$; we call
$x_\beta$ the \emph{contact point} of $\beta$. Equivalently, $\beta$ is a
positive multiple of a supporting functional at $x_\beta$ and at no other
boundary point.

As $x$ runs once around $\partial\ball$ in the positive
sense, the normal directions of the supporting functionals at $x$ run once
around $S^1$ in the positive sense, monotonically; the normal directions at
a fixed boundary point $x$ form a closed arc of $S^1$, which is a single
point unless $\partial\ball$ has a corner at $x$.

The one-sided directional derivative of the convex function $N$ at $x$
in the direction $h$ is
\[
 D_hN(x)=\lim_{t\downarrow0}\frac{N(x+th)-N(x)}{t}=\max_{\alpha\in\partial N(x)}\alpha(h).
\]
The \emph{corner jump}
\[
 J_N(x;h):=D_hN(x)+D_{-h}N(x)\ \ge0
\]
measures the opening of the corner of the graph of $N$ at $x$ in the
direction $h$; it vanishes for all $h$ if and only if $N$ is differentiable
at $x$. If $\partial N(x)$ is a segment with endpoints $\alpha^+,\alpha^-$, the
\emph{extreme supporting functionals} at $x$, then
$J_N(x;h)=|(\alpha^+-\alpha^-)(h)|$ for every $h$.

\subsection{The corner formula and the exponential estimate}
The next proposition contains the geometric input of the paper. Its first
part says that $\partial\ball$ has a corner at every rational direction
and computes its size; its second part says that between two consecutive
rays of the fan of level $H$ the boundary is flat to within $e^{-\kappa H}$.

\begin{proposition}\label{prop:turn}
There are constants $c,C,\kappa>0$ with the following properties.
\begin{enumerate}
\item (Corner formula.) For every primitive $u$ and every $h\in\bR^2$,
\begin{equation}\label{eq:corner}
 J_N(u;h)=4\,|\det(u,h)|\,\log\coth\bigl(N(u)/2\bigr).
\end{equation}
In particular $N$ is not differentiable at any rational direction, and
the two extreme supporting functionals $\alpha^\pm$ at $u$ satisfy
$|\det(\alpha^+,\alpha^-)|=4N(u)\log\coth(N(u)/2)$. If $\hgt(u)\le H$,
then the normal directions of $\alpha^+$ and $\alpha^-$ differ by an
angle of at least $ce^{-CH}$.
\item (Flatness of sectors.) Let $\Sigma$ be a Farey sector of level
$H\ge1$ with boundary vectors $u,v$. Let $\alpha$ be the supporting
functional at $u$ with $\alpha(v)=D_vN(u)$, and $\beta$ the supporting
functional at $v$ with $\beta(u)=D_uN(v)$; by (a) each is unique. We call
$\alpha$ and $\beta$ the \emph{inward} supporting functionals at $u$ and
$v$ (they are the extreme supporting functionals whose normal directions
are closest to the normal directions at points of $\Sigma$). Then
\[
 |\alpha-\beta|\le Ce^{-\kappa H},
\]
and every supporting functional at every point of $\Sigma$ lies within
$Ce^{-\kappa H}$ of $\alpha$ (and of $\beta$).
\item (Total turning.) The sum, over all Farey sectors $\Sigma$ of level
$H$, of the angular width of the arc of normal directions of supporting
functionals at points of $\Sigma$ is at most $Ce^{-\kappa H}$.
\end{enumerate}
\end{proposition}

We first fix notation for the holonomy of a pair of Farey neighbours. Let $u,v$ be Farey neighbours with
$\det(u,v)=1$, and let $U=\rho(w_u)$, $V=\rho(w_v)$ be holonomy matrices of
a free basis $(w_u,w_v)$ lifting $(u,v)$. Their traces
\[
 x:=\tr U=3m(u),\qquad y:=\tr V=3m(v),\qquad z:=\tr UV=3m(u+v)
\]
are positive by Lemma~\ref{lem:dictionary}. Put $\lambda=e^{N(u)/2}$ and $\mu=e^{N(v)/2}$, so that
$x=\lambda+\lambda^{-1}$ and $y=\mu+\mu^{-1}$, and conjugate the pair $(U,V)$ by an element of $\SL_2(\bR)$ so that
$U=\operatorname{diag}(\lambda,\lambda^{-1})$ and
$V=\left(\begin{smallmatrix}a&b_1\\b_2&d\end{smallmatrix}\right)$ with $ad-b_1b_2=1$.

Since $\tr[U,V]=-2$, and $\tr[U,V]=2ad-b_1b_2(\lambda^2+\lambda^{-2})$ by
direct computation, we get $b_1b_2(\lambda-\lambda^{-1})^2=4$, hence
\begin{equation}\label{eq:bc-ad}
 b_1b_2=\frac4{(\lambda-\lambda^{-1})^2},\qquad
 ad=1+b_1b_2=\coth^2\bigl(N(u)/2\bigr).
\end{equation}

As $a+d=y>0$ and $ad>0$, both $a$ and $d$ are positive. The words
$w_u^nw_v$ and $w_u^nw_v^{-1}$ are primitive with homology classes
$nu+v$ and $nu-v$, and
\begin{equation}\label{eq:fan-trace}
 \tr(U^nV)=\lambda^na+\lambda^{-n}d,\qquad
 \tr(U^nV^{-1})=\lambda^nd+\lambda^{-n}a .
\end{equation}

\begin{proof}[Proof of Proposition~\ref{prop:turn}]
\emph{The corner formula.} Let $u$ be primitive and choose $v$ with
$\det(u,v)=1$, with the notation above. Since $N$ is convex and
homogeneous, the one-sided derivative in the direction $v$ is a limit
along the primitive vectors $nu+v$:
\[
 D_vN(u)=\lim_{n\to\infty}\bigl(N(nu+v)-nN(u)\bigr),
\]
because $N(nu+v)-nN(u)=n\bigl(N(u+v/n)-N(u)\bigr)$.

By \eqref{eq:fan-trace} and $N=2\arcosh(\tr/2)=2\log\tr+o(1)$,
\[
 N(nu+v)=2\log(\lambda^na+\lambda^{-n}d)+o(1)=nN(u)+2\log a+o(1),
\]
so $D_vN(u)=2\log a$, and likewise $D_{-v}N(u)=2\log d$. Hence, by
\eqref{eq:bc-ad},
\[
 J_N(u;v)=2\log(ad)=4\log\coth\bigl(N(u)/2\bigr).
\]

For general $h$, write $h=\det(u,h)\,v+tu$; adding a multiple of $u$ does not change
$J_N(u;\cdot)$, since $\alpha(u)=N(u)$ for all $\alpha\in\partial N(u)$, and
$J_N(u;sh)=|s|J_N(u;h)$. This proves \eqref{eq:corner}.

Since $J_N(u;v)>0$, $N$ is not differentiable at $u$, so $\partial N(u)$ is a
nondegenerate segment with endpoints $\alpha^\pm$, and
$(\alpha^+-\alpha^-)(u)=0$, $|(\alpha^+-\alpha^-)(v)|=J_N(u;v)$ give
$\alpha^+-\alpha^-=\pm J_N(u;v)\det(u,\cdot)$. Identifying linear forms
with vectors, we get $\det(\alpha^+,\alpha^-)=\det(\alpha^+,\alpha^--\alpha^+)
=\mp J_N(u;v)\,\alpha^+(u)$, whose absolute value is
$4N(u)\log\coth(N(u)/2)$. As $c\le|\alpha^\pm|\le C$, the angle $\theta$
between $\alpha^+$ and $\alpha^-$ satisfies
\[
 \sin\theta\ \ge\ cN(u)\log\coth(N(u)/2)\ \ge\ cN(u)e^{-N(u)},
\]
using $\log\coth(t/2)\ge2e^{-t}$. If $\hgt(u)\le H$ then
$N(u)\le C|u|\le CH$, and $N(u)\ge2\arcosh(3/2)>1$, so $\theta\ge ce^{-CH}$.

\emph{The larger root.} Let $\Sigma$ be a Farey sector of level $H\ge1$
with boundary vectors $u,v$, $\det(u,v)=1$. We claim that
\begin{equation}\label{eq:larger-root}
 m(u-v)\le\max\bigl(m(u),m(v)\bigr),\qquad\text{hence}\qquad
 m(u+v)\ge2\max\bigl(m(u),m(v)\bigr)>m(u-v).
\end{equation}
The numbers $3m(u+v)$ and $3m(u-v)$ are the two roots of
$Z^2-xyZ+x^2+y^2=0$, since the triples $(m(u),m(v),m(u\pm v))$
satisfy the Markoff relation \eqref{eq:fricke-labels}, applied to $u,v$ and to
$u,-v$; so the claim says that $z=3m(u+v)$ is the
\emph{larger} root.

The second statement follows from the first by
\eqref{eq:fareytrace}, since
\[
 m(u+v)=3m(u)m(v)-m(u-v)\ge3\max\bigl(m(u),m(v)\bigr)-\max\bigl(m(u),m(v)\bigr).
\]

The first statement holds for the eight sectors of level $1$. Their
boundary vectors are consecutive among the eight primitive vectors of
height one, $\pm e_1,\pm(e_1+e_2),\pm e_2,\pm(e_1-e_2)$, so the difference
of the two boundary vectors of each sector is one of
$\pm e_1,\pm e_2$, of label $1$, while every label is at
least $1$.

If it holds for a sector with boundary $u,v$, it holds for the two sectors with
boundaries $u,u+v$ and $u+v,v$ obtained by inserting the mediant: for the
first, $m(u-(u+v))=m(v)\le\max(m(u),m(u+v))$ by the second statement for
$(u,v)$. Since every sector of level $H\ge1$ arises from a sector of level
$1$ by successive mediant insertions, the claim follows by induction.

(The restriction to sectors of fans of level
$H\ge1$ matters: $e_1$ and $e_2$ are Farey neighbours, but the ray through
$e_1+e_2$ separates them in every fan, and for this pair the mediant
$e_1+e_2$ carries the smaller root, since $m(e_1+e_2)=1<2=m(e_1-e_2)$.)

\emph{The expanding entry.} Keep $u,v,\Sigma$ as above and put
\[
 P:=(\lambda-\lambda^{-1})(\mu-\mu^{-1})=\sqrt{(x^2-4)(y^2-4)} .
\]
Since $x,y\ge3$ we have $\lambda,\mu\ge(3+\sqrt5)/2$ and $P\ge5$. The two
roots of $Z^2-xyZ+x^2+y^2$ are $(xy\pm\sqrt{x^2y^2-4x^2-4y^2})/2$, and
$x^2y^2-4x^2-4y^2=P^2-16$; by the larger root claim,
\[
 z=\frac{xy+\sqrt{P^2-16}}2 .
\]

Compare $z$ with $z_0:=\lambda\mu+(\lambda\mu)^{-1}=(xy+P)/2$. The
difference
\[
 \xi:=z_0-z=\frac{P-\sqrt{P^2-16}}2=\frac8{P+\sqrt{P^2-16}}
\]
satisfies $0<\xi\le\min(1,5/P)$ for $P\ge5$.

Solving $a+d=y$ and
$\lambda a+\lambda^{-1}d=z$ for the entry $a$ of $V$ gives
\[
 a=\frac{z-\lambda^{-1}y}{\lambda-\lambda^{-1}}
  =\mu-\frac\xi{\lambda-\lambda^{-1}}=\mu(1-\delta),\qquad
 \delta=\frac\xi{\mu(\lambda-\lambda^{-1})}\le\frac5{P\mu(\lambda-\lambda^{-1})}.
\]
A direct evaluation gives $0<\delta<1/5$, and
$\delta\le C\lambda^{-2}\mu^{-2}=Ce^{-N(u)-N(v)}$. Therefore
\begin{equation}\label{eq:expanding}
 |2\log a-N(v)|=2|\log(1-\delta)|\le Ce^{-N(u)-N(v)} .
\end{equation}

\emph{Flatness of the sector.} By (a), $\partial N(u)$ is a segment whose
endpoints take values at $v$ differing by $J_N(u;v)>0$; so $\alpha$ is the
endpoint at which the value at $v$ is larger, and
$\alpha(v)=D_vN(u)=2\log a$, $\alpha(u)=N(u)$. Interchanging the roles
of $u$ and $v$ (the computation of the expanding entry used only the
traces $x,y,z$ and the trace identities, which are symmetric in $u$ and
$v$) gives $\beta(v)=N(v)$ and $|\beta(u)-N(u)|\le Ce^{-N(u)-N(v)}$. Hence
\[
 |(\alpha-\beta)(u)|+|(\alpha-\beta)(v)|\le Ce^{-N(u)-N(v)} .
\]
The dual basis of $(u,v)$ consists of the forms $\det(\cdot,v)$ and
$\det(u,\cdot)$, of Euclidean norm $|v|,|u|\le\sqrt2H$, so
$|\alpha-\beta|\le CHe^{-N(u)-N(v)}$. Finally $N(u)+N(v)\ge N(u+v)\ge
c|u+v|\ge c\,\hgt(u+v)\ge cH$, because the mediant has height $>H$. Thus
$|\alpha-\beta|\le CHe^{-cH}\le Ce^{-\kappa H}$.

Let $\mathcal A_\Sigma\subset S^1$ be the set of normal directions of
supporting functionals at points of $\Sigma$, together with the normal
directions of $\alpha$ and $\beta$. Supporting functionals $\gamma_t$ at the
points $u+tv$ of $\Sigma$ converge to $\alpha$ as $t\downarrow0$. Since
$\gamma_t\le N$ and $\gamma_t(u+tv)=N(u+tv)$, every limit point of
$\gamma_t$ lies in $\partial N(u)$ and takes at $v$ a value at least
$\lim_{t\downarrow0}(N(u+tv)-N(u))/t=D_vN(u)$, so it is $\alpha$. Likewise,
supporting functionals at the points $v+tu$ converge to $\beta$. By the
monotonicity of normal directions, $\mathcal A_\Sigma$ is therefore the
closed arc from $\alpha/|\alpha|$ to $\beta/|\beta|$ traversed in the
positive sense.

The arc $\mathcal A_{-\Sigma}$ of the opposite sector is $-\mathcal A_\Sigma$, and
the two arcs have disjoint interiors. Two antipodal arcs with disjoint
interiors have length at most $\pi$. An arc of length at most $\pi$ whose
endpoints are at chord distance $\varepsilon$ has length at most
$\pi\varepsilon/2$, and here $\varepsilon\le2|\alpha-\beta|/|\alpha|\le Ce^{-\kappa H}$.

Hence every normal direction at a point of $\Sigma$ is within angle
$Ce^{-\kappa H}$ of that of $\alpha$, and by the Lipschitz dependence of a
supporting functional on its normal direction (\S\ref{sec:supports}) every
supporting functional at a point of $\Sigma$ is within $Ce^{-\kappa H}$ of
$\alpha$. This proves (b).

\emph{Total turning.} The fan of level $H$ has $O(H^2)$ sectors, so the
sum of the lengths of the arcs $\mathcal A_\Sigma$ is $O(H^2e^{-\kappa H})$,
which is $O(e^{-\kappa'H})$ for a smaller $\kappa'$. This proves (c).
\end{proof}

\subsection{Contact points of integral forms, and the excision of sectors}
\label{sec:excision}
A linear form $\beta=(\beta_1,\beta_2)$ on $\bR^2$, $\beta(x)=\beta_1x_1+\beta_2x_2$,
is \emph{integral} if $\beta_1,\beta_2\in\bZ$, and its size is
$|\beta|=(\beta_1^2+\beta_2^2)^{1/2}$.

\begin{lemma}\label{lem:contacts}
The following statements hold.
\begin{enumerate}
\item For $M\ge1$, the contact points $x_\beta$ of the nonzero integral
forms $\beta$ with $|\beta|\le M$ lie on at most $C\log^2(2+M)$ rays.
\item Let $H\ge1$ and $k\ge1$ with $\sqrt k\le cH$ for a sufficiently small
constant $c$. The union of any $k$ Farey sectors of level $H$ has angular
width at most $C\sqrt k/H$.
\end{enumerate}
\end{lemma}

\begin{proof}
(a) Let $\beta,\beta'$ be integral forms of size at most $M$ that are not
positive multiples of each other. The angle between them is at least
$M^{-2}$: it is $\pi$ if $\beta'$ is a negative multiple of $\beta$, and
otherwise $|\det(\beta,\beta')|\ge1$ gives
$\angle(\beta,\beta')\ge\sin\angle(\beta,\beta')\ge M^{-2}$. Choose $H_0=\lceil C\log(2+M)\rceil$ so large that the total
turning bound of Proposition~\ref{prop:turn}(c) at level $H_0$ is smaller
than $M^{-2}$. The normal directions at the points of one open sector of
level $H_0$ then form an arc of length $<M^{-2}$, so the integral forms of
size at most $M$ whose contact points lie in that sector are positive
multiples of one another, and therefore have one and the same contact
point. Hence the contact points lying in open sectors lie on at most
$O(H_0^2)$ rays, and the contact points lying on rays of the fan lie on at
most $O(H_0^2)$ rays, namely the rays of $\mathcal F_{H_0}$.

(b) By the symmetries of the fan it suffices to treat sectors in the
octant $0\le p\le q$, where a sector of level $H$ consists of the vectors
$(p,q)$ with $p/q$ strictly between consecutive terms $a/b<c/d$ of the
Farey sequence of order $H$;
here $bc-ad=1$, $b,d\le H$, $b+d>H$, and the sector has slope width
$1/(bd)$, which is comparable to its angular width. Fix an integer $n$
with $1\le n\le H/2$. A
sector having an endpoint of denominator $b\le n$ has its other
denominator $d>H-b\ge H/2$, hence slope width $O(1/(bH))$; since there are
at most $b$ fractions of denominator $b$ in $[0,1]$, each bounding two
sectors, the sectors with an endpoint of denominator at most $n$ have
total width $O(n/H)$. Every other sector has both denominators $>n$ and
their sum $>H$, so $bd\ge n(H-n)\ge nH/2$ and its width is $O(1/(nH))$.
Thus $k$ sectors have total width $O(n/H+k/(nH))$; taking $n\asymp\sqrt k$
gives the claim, the hypothesis on $k$ ensuring $n\le H/2$.
\end{proof}

\begin{definition}[Excision]\label{def:excision}
Let $M\ge1$ and let $H$ be an integer with $H\ge C_{\mathrm{exc}}\log(2+M)$,
where $C_{\mathrm{exc}}$ is a sufficiently large constant fixed once and
for all. The \emph{$(M,H)$-excision} removes from the plane:
\begin{itemize}
\item every ray of the fan $\mathcal F_H$;
\item every Farey sector of level $H$ whose closure contains the contact
point $x_\beta$ of an integral form $\beta$ with $0<|\beta|\le M$.
\end{itemize}
(When $x_\beta$ lies on a ray of $\mathcal F_H$, both sectors adjacent to
that ray are excised.)
The sectors removed are called \emph{excised}; the remaining sectors of
level $H$ are \emph{retained}. A vector is \emph{retained} if it lies in a
retained sector, and \emph{excised} otherwise.
\end{definition}

\begin{lemma}\label{lem:excision}
Let $M,H$ be as in Definition~\ref{def:excision} and let $L\ge1$.
\begin{enumerate}
\item At most $C\log^2(2+M)$ sectors are excised, and their total angular
width is at most $C\log(2+M)/H$.
\item The number of excised primitive vectors $v$ with $N(v)\le L$ is at
most
\[
 C\Bigl(\frac{\log(2+M)}H\,L^2+L\log^2(2+M)+H^2\Bigr).
\]
\item (Separation.) There is a constant $C_{\mathrm{sep}}$, depending only on $N$, such
that for every retained primitive vector $v$, every supporting
functional $\gamma$ at $v$, and every integral form $\beta$ with
$0<|\beta|\le M$,
\[
 \angle\bigl(\gamma,\{\beta,-\beta\}\bigr)\ge ce^{-C_{\mathrm{sep}}H}.
\]
\end{enumerate}
\end{lemma}

\begin{proof}
(a) By Lemma~\ref{lem:contacts}(a) the contact points in question lie on
$k_0\le C\log^2(2+M)$ rays, and each such ray meets at most two closed
sectors of level $H$; so at most $2k_0$ sectors are excised. By
Lemma~\ref{lem:contacts}(b), which applies because $\sqrt{2k_0}\le cH$
when $C_{\mathrm{exc}}$ is large, their total width is at most
$C\sqrt{2k_0}/H\le C\log(2+M)/H$.

(b) The intersection of a sector of angular width $\theta$ with the ball
$L\ball$ is convex, of area $O(\theta L^2)$ and perimeter $O(L)$, so it
contains $O(\theta L^2+L+1)$ lattice points. Summing over the $O(\log^2(2+M))$
excised sectors, and adding one primitive vector for each of the $O(H^2)$
rays of $\mathcal F_H$, gives the bound.

(c) Let $v$ lie in a retained sector $\Sigma$, with boundary rays
$R_1,R_2$, and let $\beta$ be integral with $0<|\beta|\le M$. Both
$x_\beta$ and $x_{-\beta}=-x_\beta$ are contact points of integral forms of
size at most $M$, so neither lies in the closure of $\Sigma$.

Hence, moving from $v$ to $x_\beta$ in the positive direction around the origin
one crosses one of the rays $R_1,R_2$ before reaching $x_\beta$, and moving
in the negative direction one crosses the other ray first; the same holds
for $-x_\beta$.

Since $\gamma$ is a supporting functional at a point of
the open sector $\Sigma$ and $\beta$ is a positive multiple of a
supporting functional at $x_\beta$, the monotonicity of normal directions
along $\partial\ball$ shows that the normal directions of $\gamma$ and of
$\beta$ are separated, in each of the two directions around $S^1$, by
the full closed arc of normal directions at the primitive vector spanning
the ray crossed. This primitive vector has height at most $H$, so by
Proposition~\ref{prop:turn}(a) the arc has length at least $ce^{-CH}$;
by strict convexity, the normal direction of $\gamma$, a supporting
functional at a point of the open sector, lies outside the arc. Hence
$\angle(\gamma,\beta)\ge ce^{-CH}$, and likewise
$\angle(\gamma,-\beta)\ge ce^{-CH}$; this is the claim, with
$C_{\mathrm{sep}}$ the constant $C$ of Proposition~\ref{prop:turn}(a).
\end{proof}