2026/05/14 by Nhat Minh Doan, Xiaobin Li, Van Nguyen
#math.GT #math.MG #math.NT
We use normal-turn estimates to study the global and local geometry of the boundaries of McShane--Rivin norm balls BX 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≥ 2 is at most CX(log L)2. For the modular torus, this gives #λM-1(m)≤ C(loglog(3m))2 for every Markoff number m, improving the previous logarithmic bounds for Markoff fibers. Our second result shows that the boundary ∂ BX 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 ℓ^∞-radius of BX 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.