Holomorphic Functions and Moduli I

Volume 10 of the series Mathematical Sciences Research Institute Publications pp 215-228

Lower bounds for the number of saddle connections and closed trajectories of a quadratic differential

  • Howard MasurAffiliated withDepartment of Mathematics, Statistics, and Computer Science, University of Illinois at Chicago

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Suppose X is the torus the complex plane C divided by the group of translations generated by zz + 1 and 2 → z + i and q = dz 2 is the unique up to scalar multiple quadratic differential on X. The trajectories of q are straight lines and a trajectory is closed if and only if its slope is a rational p/q. Its length is (p 2 + q 2)1/2. Parallel closed trajectories fill up X. The number N(T) of parallel families of length ≤ T is then the number of lattice points (p, q) with p, q relatively prime inside a circle of radius T. It is classical that
$$ \mathop{{T \to \infty }}\limits^{{\lim }} \frac{{N\left( T \right)}}{{{T^{2}}}} = \frac{6}{{{\pi ^{2}}}}. $$