\pdfoutput=1
\documentclass[11pt]{amsart}
\input{mathbase.tex}
\usepackage{lmodern,mathrsfs,microtype}
\usepackage[margin=1.05in]{geometry}
\hypersetup{colorlinks=true,linkcolor=blue!45!black,citecolor=blue!45!black,urlcolor=blue!45!black,
  pdftitle={Almost all Markoff numbers are unique},
  pdfauthor={William Y. Chen}}
\newtheorem{theorem}{Theorem}[section]
\newtheorem{proposition}[theorem]{Proposition}
\newtheorem{conjecture}[theorem]{Conjecture}
\newtheorem{lemma}[theorem]{Lemma}
\newtheorem{corollary}[theorem]{Corollary}
\theoremstyle{definition}
\newtheorem{definition}[theorem]{Definition}
\theoremstyle{remark}
\newtheorem{remark}[theorem]{Remark}
\newtheorem{example}[theorem]{Example}
\numberwithin{equation}{section}
% macros specific to this paper (the general ones are in mathbase.tex)
\newcommand{\area}{\operatorname{area}}
\DeclareMathOperator{\hgt}{ht}
\DeclareMathOperator{\arcosh}{arcosh}
\DeclareMathOperator{\covol}{covol}
\newcommand{\sect}{\mathcal S}
\newcommand{\ball}{\mathbb B}
\renewcommand{\angle}{\measuredangle}

\title[Almost all Markoff numbers are unique]{Almost all Markoff numbers are unique}
\date{October 14, 2026}
\author{William Y. Chen}
\begin{document}
\begin{abstract}
A Markoff number is the largest coordinate of a positive integer solution of
$a^2+b^2+c^2=3abc$. Frobenius conjectured in 1913 that every Markoff number
is the largest coordinate of exactly one such solution with $a\le b\le c$.
We prove that the conjecture holds for almost all Markoff numbers: the
proportion of exceptions among the Markoff numbers up to $X$ tends to $0$
as $X\to\infty$.

Two such solutions with the same largest coordinate give simple closed
geodesics of equal length on the modular torus, that is, primitive homology
classes $v,w$ with equal values $N(v)=N(w)$ of the McShane--Rivin stable
norm $N$. Estimates for the boundary of the unit ball of $N$ reduce the
problem, up to an arbitrarily small proportion, to the equations
$N(Gv)=N(v)$ for finitely many rational linear maps $G$. For each $G$ that
is not a symmetry of the torus, we use a zero-one law for the
$3$-adic analytic dynamics of the mapping class group on the Markoff
surface to show that the solutions $v$ have density zero.
\end{abstract}
\maketitle
\markboth{\MakeUppercase{Almost all Markoff numbers are unique}}{\MakeUppercase{Almost all Markoff numbers are unique}}
\setcounter{tocdepth}{1}
\tableofcontents
\input{sections/introduction}
\input{sections/dictionary}
\input{sections/geometry}
\input{sections/fixed}
\input{sections/graphs}
\input{sections/sum}
\input{sections/remarks}
\input{bibliography}
\end{document}