Kelmer, Dubi; Kontorovich, Alex On the pair correlation density for hyperbolic angles. (English) Zbl 1318.30066 Duke Math. J. 164, No. 3, 473-509 (2015). Let \(\Lambda \subset \mathbb{R}^2\) be a Euclidean lattice. Then a classical result states that the angles of line segments connecting a point \(\alpha \in \mathbb{R}^2\) with the points of \(\Lambda\) contained in increasing balls around \(\alpha\) become equidistributed on the circle. A more detailed device for measuring the statistical properties of these angles is their pair correlation; under certain number-theoretic assumptions on \(\alpha\), the distribution of these pair correlations is so-called Poissonian, that is, in accordance with the distribution of the pair correlations in the random setting.A. Boca et al. [Int. J. Number Theory 10, No. 8, 1955–1989 (2014; Zbl 1304.11075)] studied the similar problem in the hyperbolic case, that is the problem concerning the pair correlations of the angles between geodesic rays of the lattice \(\Gamma \omega\) intersected with increasing balls centered at \(\omega\), where \(\Gamma < \text{PSL}_2(\mathbb{R})\) is a lattice and \(w \in \mathbb{H}\) is a point in the upper half-plane. Boca et al. [loc. cit.] formulated a conjecture for the formula of the limit distribution of the pair correlations in this setting, but could only give a proof in the case when \(\Gamma = \text{PSL}_2(\mathbb{Z})\) and \(\omega\) is an elliptic point. In the present paper, the full conjecture of Boca et al. [loc.cit.] is confirmed, by proving the existence of and giving a formula for the asymptotic density function for the distribution of the pair correlations for general \(\Gamma < \text{PSL}_2(\mathbb{R})\) and \(\omega \in \mathbb{H}\). The formula for this distribution is fully explicit, but too complicated to be reproduced here. Interestingly, the proof given for the main result in the present paper is totally different from the one given by Boca et al. [loc. cit.] for their result. Reviewer: Christoph Aistleitner (Kobe) Cited in 1 ReviewCited in 10 Documents MSC: 30F35 Fuchsian groups and automorphic functions (aspects of compact Riemann surfaces and uniformization) 22E40 Discrete subgroups of Lie groups 11K38 Irregularities of distribution, discrepancy Keywords:pair correlations; hyperbolic lattice angles Citations:Zbl 1304.11075 × Cite Format Result Cite Review PDF Full Text: DOI arXiv Euclid Link References: [1] F. P. Boca, Distribution of angles between geodesic rays associated with hyperbolic lattice points , Q. J. Math. 58 (2007), 281-295. · Zbl 1197.11134 · doi:10.1093/qmath/ham014 [2] F. P. Boca, C. Cobeli, and A. Zaharescu, Distribution of lattice points visible from the origin , Comm. Math. Phys. 213 (2000), 433-470. · Zbl 0989.11049 · doi:10.1007/s002200000250 [3] F. P. Boca, V. Paşol, A. A. Popa, and A. Zaharescu, Pair correlation of angles between reciprocal geodesics on the modular surface , Algebra Number Theory 8 (2014), 999-1035. · Zbl 1304.11073 · doi:10.2140/ant.2014.8.999 [4] F. P. Boca, A. A. Popa, and A. Zaharescu, Pair correlation of hyperbolic lattice angles , Int. J. Number Theory 10 (2014), 1955-1989. · Zbl 1304.11075 · doi:10.1142/S1793042114500651 [5] F. P. Boca and A. Zaharescu, The correlations of Farey fractions , J. Lond. Math. Soc. (2) 72 (2005), 25-39. · Zbl 1089.11037 · doi:10.1112/S0024610705006629 [6] F. P. Boca and A. Zaharescu, On the correlations of directions in the Euclidean plane , Trans. Amer. Math. Soc. 358 , no. 4 (2006), 1797-1825. · Zbl 1154.11022 · doi:10.1090/S0002-9947-05-03783-9 [7] M. Cowling, U. Haagerup, and R. Howe, Almost \(L^{2}\) matrix coefficients , J. Reine Angew. Math. 387 (1988), 97-110. [8] D. El-Baz, J. Marklof, and I. Vinogradov, The distribution of directions in an affine lattice: two-point correlations and mixed moments , preprint, [math.NT]. arXiv:1306.0028v2 · Zbl 1386.11101 [9] A. Good, Local Analysis of Selberg’s Trace Formula , Lecture Notes in Math. 1040 , Springer, Berlin, 1983. · Zbl 0525.10013 · doi:10.1007/BFb0073074 [10] H. H. Kim, Functoriality for the exterior square of \(\mathrm{GL}_{4}\) and the symmetric fourth of \(\mathrm{GL}_{2}\) , with an appendix by D. Ramakrishnan and an appendix by H. H. Kim and P. Sarnak, J. Amer. Math. Soc. 16 (2003), 139-183. · Zbl 1018.11024 · doi:10.1090/S0894-0347-02-00410-1 [11] J. Marklof and A. Strömbergsson, The distribution of free path lengths in the periodic Lorentz gas and related lattice point problems , Ann. of Math. (2) 172 (2010), 1949-2033. · Zbl 1211.82011 · doi:10.4007/annals.2010.172.1949 [12] P. Nicholls, A lattice point problem in hyperbolic space , Michigan Math. J. 30 (1983), 273-287. · Zbl 0537.30033 · doi:10.1307/mmj/1029002905 [13] M. S. Risager and J. L. Truelsen, Distribution of angles in hyperbolic lattices , Q. J. Math. 61 (2010), 117-133. · Zbl 1194.11081 · doi:10.1093/qmath/han033 [14] Y. Shalom, Bounded generation and Kazhdan’s property (T) , Publ. Math. Inst. Hautes Études Sci. 90 (1999), 145-168. · Zbl 0980.22017 · doi:10.1007/BF02698832 [15] A. Venkatesh, Sparse equidistribution problems, period bounds and subconvexity , Ann. of Math. (2) 172 (2010), 989-1094. · Zbl 1214.11051 · doi:10.4007/annals.2010.172.989 [16] G. Warner, Harmonic Analysis on Semi-simple Lie Groups, I , Grundlehren Math. Wiss. 188 , Springer, New York, 1972. · Zbl 0265.22020 This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. 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