## Angles in hyperbolic lattices: The pair correlation density

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- by Morten S. Risager and Anders SĂ¶dergren PDF
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**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, PaĹźol, 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.## References

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## Additional Information

**Morten S. Risager**- Affiliation: Department of Mathematical Sciences, University of Copenhagen, Universitet- sparkenÂ 5, 2100 Copenhagen, Denmark
- MR Author ID: 740566
- Email: risager@math.ku.dk
**Anders SĂ¶dergren**- Affiliation: Department of Mathematical Sciences, University of Copenhagen, Universitet- sparkenÂ 5, 2100 Copenhagen, Denmark
- MR Author ID: 931224
- ORCID: 0000-0001-6061-0319
- Email: sodergren@math.ku.dk
- 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).
- © Copyright 2016 American Mathematical Society
- Journal: Trans. Amer. Math. Soc.
**369**(2017), 2807-2841 - MSC (2010): Primary 11N45; Secondary 11P21, 20H10
- DOI: https://doi.org/10.1090/tran/6770
- MathSciNet review: 3592529