On the distribution of lattice points on hyperbolic circles

TL;DR

This paper investigates how lattice points on expanding hyperbolic circles are distributed in angle, revealing equidistribution for most radii but exceptions, and connecting hyperbolic and Euclidean lattice point distributions.

Contribution

It establishes the equidistribution of angles for generic radii in hyperbolic circles and uncovers a surprising link to Euclidean lattice point distributions on certain subsequences.

Findings

Angles are equidistributed for most radii in hyperbolic circles.

On thin sets of radii, angles fail to be equidistributed.

Hyperbolic measures can break rotational symmetry, unlike Euclidean measures.

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.