Angles in hyperbolic lattices: The pair correlation density

Risager, Morten; Södergren, Anders

doi:10.1090/tran/6770

Angles in hyperbolic lattices: The pair correlation density
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by Morten S. Risager and Anders Södergren PDF

Trans. Amer. Math. Soc. 369 (2017), 2807-2841 Request permission

Abstract:

It is well known that the angles in a lattice acting on hyperbolic $n$-space become equidistributed. In this paper we determine a formula for the pair correlation density for angles in such hyperbolic lattices. Using this formula we determine, among other things, the asymptotic behavior of the density function in both the small and large variable limits. This extends earlier results by Boca, Paşol, Popa and Zaharescu and Kelmer and Kontorovich in dimension 2 to general dimension $n$. Our proofs use the decay of matrix coefficients together with a number of careful estimates, and lead to effective results with explicit rates.

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