Approximate Control of the Marked Length Spectrum by Short Geodesics

The paper’s Hypothesis 1.1 and Theorem 1.2 assume multiplicative closeness of the marked length spectra only at a single sufficiently large scale L ≥ L0 and then deduce a global bound 1 ± (ε + C L^{-α}) for all conjugacy classes, with constants depending only on geometric/topological data . The proof uses a covering argument for the unit tangent bundle and Hölder regularity of an orbit equivalence to approximate long geodesic lengths by sums of short ones, leading to the L^{-α} tail (absorbing a factor from δ ∼ L^{-1/2n}) . By contrast, the model’s solution assumes the hypothesis at every scale L ≥ L0 (a strictly stronger assumption than the paper’s) and then obtains the uniform 1 ± ε bound via power invariance. This solves a different, easier problem and does not recover the paper’s main conclusion from finite data; moreover, it incorrectly treats the decay term C L^{-α} as arbitrary “for any C > 0 and 0 < α < 1,” whereas in the paper C and α are derived, not chosen freely, from controlled dynamical/geometric estimates .