For the stratum H(2) of translation surfaces, long horocycle orbits have discrete transverse dimension arbitrarily close to 1, with effective exponential error rates.
Almost l^ 2 matrix coefficients
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Effective Exponential Drifts on Strata of Abelian Differentials
For the stratum H(2) of translation surfaces, long horocycle orbits have discrete transverse dimension arbitrarily close to 1, with effective exponential error rates.