Corpus ID: 221836650

On the distribution of lattice points on hyperbolic circles

@article{Chatzakos2020OnTD,
  title={On the distribution of lattice points on hyperbolic circles},
  author={Dimitrios Chatzakos and Par Kurlberg and Stephen Lester and Igor Wigman},
  journal={arXiv: Number Theory},
  year={2020}
}
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… Expand

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References

SHOWING 1-10 OF 44 REFERENCES
Hyperbolic prime number theorem
AbstractWe count the number S(x) of quadruples $ {\left( {x_{1} ,x_{2} ,x_{3} ,x_{4} } \right)} \in \mathbb{Z}^{4} $ for which $$ p = x^{2}_{1} + x^{2}_{2} + x^{2}_{3} + x^{2}_{4} \leqslant x $$is aExpand
Spectral methods of automorphic forms
Introduction Harmonic analysis on the Euclidean plane Harmonic analysis on the hyperbolic plane Fuchsian groups Automorphic forms The spectral theorem. Discrete part The automorphic Green functionExpand
On the angular distribution of Gaussian integers with fixed norm
TLDR
It is able to show that when n is representable then it is almost surely representable with min(a, b) small, with an explicit bound. Expand
Bombieri's theorem in short intervals
AbstractThe well-known Bombieri-A. I. Vinogradov theorem states that(1) $$\sum\limits_{q \leqslant x^{\tfrac{1}{2}} (\log x)^{ - s} } {\mathop {\max }\limits_{(a,q) = 1} \mathop {\max }\limits_{yExpand
A lattice point problem in hyperbolic space.
Superscars for Arithmetic Point Scatterers II
We consider momentum push-forwards of measures arising as quantum limits (semi-classical measures) of eigenfunctions of a point scatterer on the standard flat torus $\mathbb T^2 = \mathbb R^2/\mathbbExpand
On the fractal structure of attainable probability measures
Intervals between numbers that are sums of two squares
In this paper, we improve the moment estimates for the gaps between numbers that can be represented as a sum of two squares of integers. We consider certain sum of Bessel functions and prove theExpand
On probability measures arising from lattice points on circles
TLDR
The set of attainable measures is investigated and it is shown that it contains all extreme points, in the sense of convex geometry, of the set of all probability measures that are invariant under some natural symmetries. Expand
Angles in hyperbolic lattices : The pair correlation density
It is well known that the angles in a lattice acting on hyperbolic $n$-space become equidistributed. In this paper we determine a formula for the pair correlation density for angles in suchExpand
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