McShane-Rivin norm balls and simple-length multiplicities

We use normal-turn estimates to study the global and local geometry of the boundaries of McShane–Rivin norm balls $B_X$ for complete finite-area hyperbolic once-punctured tori $X$. This yields a logarithmic-square bound for the number of integer points on the boundary of each dilated norm ball. Consequently, the number of simple closed geodesics of length exactly $L\ge 2$ is at most $C_X(\log L{)}^2$. For the modular torus, this gives $$\mathrm{\#}{\lambda }_M^{-1}(m)\le C(\log \log (3m){)}^2$$ for every Markoff number $m$, improving the previous logarithmic bounds for Markoff fibers. Our second result shows that the boundary $\mathrm{\partial }{B}_X$ is a convex-geometric detector of exponential Diophantine approximation: a rational direction gives genuine corner with exponentially small exterior angle in the hyperbolic length of the corresponding simple closed geodesic, while at an irrational direction $\beta$ the graph-flatness order admits an explicit formula in terms of the exponential rate at which rational directions approach $\beta$ and the ${\mathrm{\ell }}^{\mathrm{\infty }}$-radius of $B_X$ in the projective direction $\beta$. Thus, irrational directions are not uniformly flat to infinite order, correcting the McShane–Rivin local picture. We also determine all possible irrational flatness orders and the size of the corresponding level sets; in particular, every intermediate finite-flatness level determines the marked torus.