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 $L\geq 2$ 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 bounds for Markoff fibers. Our second result shows that the boundary $\partial B_X$ 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 $β$ the graph-flatness order admits an explicit formula in terms of the exponential rate at which rational directions approach $β$ and the $\ell^\infty$-radius of $B_X$ in the projective direction $β$. 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.