Alexandru Popa (Bukarest): Pair correlation of hyperbolic lattice angles
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Abstract: The statistics of spacings, such as the pair correlation or the gap distribution, measure
the fine structure of sequences of real numbers, beyond the classical Weyl equidistribution. For
sequences of interest in number theory, these statistics often reveal an unexpected structure, as
evidenced by the well-known example of the gap distribution for the critical zeroes of the Riemann
zeta function.
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In this talk I will consider the sequence of angles formed by rays of the hyperbolic lattice, namely
rays between a fixed point w in the upper half plane and images of w by elements in the modular group
via fractional linear transformations. It is well known that the angles corresponding to lattice
points in a ball centered at w become equidistributed as the radius of the ball increases, and we are
interested in their pair correlation function. When w is one of the two elliptic points, we show that
the pair correlation function exists and we find an explicit formula for it. I will sketch the proof
of this formula, which uses estimates on the number of points in two dimensional regions based on
bounds for Kloosterman sums. This is joint work with Florin Boca and Alexandru Zaharescu.