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.
- Publication:
-
arXiv e-prints
- Pub Date:
- September 2020
- DOI:
- 10.48550/arXiv.2009.10546
- arXiv:
- arXiv:2009.10546
- Bibcode:
- 2020arXiv200910546C
- Keywords:
-
- Mathematics - Number Theory;
- 11E25;
- 11H06;
- 11H56;
- 11N35;
- 11N36;
- 11N37;
- 11N13
- E-Print:
- 22 pages, 3 figures