On the distribution of lattice points on hyperbolic circles

Chatzakos, Dimitrios; Kurlberg, Pär; Lester, Stephen; Wigman, Igor

doi:10.2140/ant.2021.15.2357

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On the distribution of lattice points on hyperbolic circles

Dimitrios Chatzakos, Pär Kurlberg, Stephen Lester and Igor Wigman

Vol. 15 (2021), No. 9, 2357–2380

DOI: 10.2140/ant.2021.15.2357

Abstract

We study the fine distribution of lattice points lying on expanding circles in the hyperbolic plane $ℍ$ . The angles of lattice points arising from the orbit of the modular group ${PSL}_{2} (ℤ)$ , and lying on hyperbolic circles, are shown to be equidistributed for generic radii. However, the angles fail to equidistribute on a thin set of exceptional radii, even in the presence of growing multiplicity. Surprisingly, the distribution of angles on hyperbolic circles turns out to be related to the angular distribution of $ℤ^{2}$ -lattice points (with certain parity conditions) lying on circles in $ℝ^{2}$ , along a thin subsequence of radii. A notable difference is that measures in the hyperbolic setting can break symmetry; on very thin subsequences they are not invariant under rotation by $π ∕ 2$ , unlike in the Euclidean setting where all measures have this invariance property.

Keywords

hyperbolic plane, hyperbolic circle, lattice points, equidistribution

Mathematical Subject Classification

Primary: 11E25, 11H06, 11H56, 11N36, 11N37

Secondary: 11N13, 11N35

Milestones

Received: 14 October 2020

Revised: 28 January 2021

Accepted: 28 February 2021

Published: 23 December 2021

Authors

Dimitrios Chatzakos
Institut de Mathématiques de Bordeaux Université de Bordeaux France

Pär Kurlberg
Department of Mathematics KTH Stockholm Sweden

Stephen Lester
Department of Mathematics King’s College London United Kingdom

Igor Wigman
Department of Mathematics King’s College London United Kingdom

Vol. 15, No. 9, 2021