On the distribution of lattice points on hyperbolic circles

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 ${\text{PSL}}_2\left(\mathbb{Z}\right)$, 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 ${\mathbb{Z}}^2$-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 $\pi ∕2$, unlike in the Euclidean setting where all measures have this invariance property.