McShane-Rivin norm balls and simple-length multiplicities

Abstract

We use normal-turn estimates for McShane--Rivin norm balls to prove that, for every complete finite-area hyperbolic once-punctured torus $X$, the number of simple closed geodesics of length exactly $L\geq 2$ is at most $C_X(\log L)^2$. For the modular torus, this gives $$ \#λ_M^{-1}(m)\leq C(\log\log(3m))^2 $$ for every Markoff number $m$, improving the previous logarithmic Markoff-fiber bounds. These estimates also give new quantitative information on the local geometry of McShane--Rivin norm balls, including obstructions to infinite-order flatness at certain irrational directions.