Almost all simple lengths are unique up to isometry
For every fixed hyperbolic torus with one cusp or one geodesic boundary, we prove that, among simple curves of length at most $L$, the number of isometry orbits and the number of distinct lengths differ by $o(L^2)$. Thus almost every complete length level is a single isometry orbit of at most six curves, and multiplicity converges in mean to the generic orbit size. The modular specialization is exactly Chen's density theorem for Markoff uniqueness. We also prove the full Schmutz--Schaller bound for every punctured torus having a simple trace that is not an algebraic integer, including any…
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