On the distribution of lattice points on hyperbolic circles
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 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 the standard lattice points (with certain parity conditions) lying on circles on the plane, 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 \pi/2, unlike the Euclidean setting where all measures have this invariance property.
This talk is based on a joint work with Dimitris Chatzakos, Steve Lester and Par Kurlberg.