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Angles in hyperbolic lattices: The pair correlation density


Authors: Morten S. Risager and Anders Södergren
Journal: Trans. Amer. Math. Soc. 369 (2017), 2807-2841
MSC (2010): Primary 11N45; Secondary 11P21, 20H10
DOI: https://doi.org/10.1090/tran/6770
Published electronically: December 7, 2016
MathSciNet review: 3592529
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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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Additional Information

Morten S. Risager
Affiliation: Department of Mathematical Sciences, University of Copenhagen, Universitet- sparken 5, 2100 Copenhagen, Denmark
Email: risager@math.ku.dk

Anders Södergren
Affiliation: Department of Mathematical Sciences, University of Copenhagen, Universitet- sparken 5, 2100 Copenhagen, Denmark
Email: sodergren@math.ku.dk

DOI: https://doi.org/10.1090/tran/6770
Keywords: Pair correlation, hyperbolic $n$-space, hyperbolic lattice points
Received by editor(s): December 5, 2014
Received by editor(s) in revised form: December 11, 2014, and June 19, 2015
Published electronically: December 7, 2016
Additional Notes: The first author was supported by a Sapere Aude grant from The Danish Council for Independent Research (grant id:0602-02161B). The second author was supported by a grant from The Danish Council for Independent Research and FP7 Marie Curie Actions-COFUND (grant id: DFF-1325-00058).
Article copyright: © Copyright 2016 American Mathematical Society