math.NT — Number Theory

Almost all Markoff numbers are unique.

Contributed by William Y. Chen

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 to the equations $N(Gv)=N(v)$ with $G$ a rational linear map. For each $G$ that is not a symmetry of the torus, a zero-one law for the $3$-adic analytic dynamics of the mapping class group on the Markoff surface shows that the solutions $v$ have density zero.