Vol. 15, No. 9, 2021

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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
Abstract

We study the fine distribution of lattice points lying on expanding circles in the hyperbolic plane . The angles of lattice points arising from the orbit of the modular group PSL2(), 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 2-lattice points (with certain parity conditions) lying on circles in 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 π2, 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