We give effective asymptotic counting results for pairs of Farey neighbours and for modular symbols in Q, in imaginary quadratic number fields and in definite quaternion algebras over Q, using the distribution of common perpendiculars between Margulis cusp neighbourhoods and divergent geodesics in hyperbolic manifolds. We describe the tangency properties of the canonical Margulis cusp neighbourhoods in Bianchi hyperbolic $3$-orbifolds.
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Parkkonen et al. (2026) studied this question.
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