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 $\mathbb{H}$. The angles of lattice points arising from the orbit of the modular group $PSL_{2}(\mathbb{Z})$, 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 $\mathbb{Z}^2$-lattice points (with certain parity conditions) lying on circles in $\mathbb{R}^2$, 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 $\frac{\pi}{2}$, unlike the Euclidean setting where all measures have this invariance property.