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Counting and equidistribution over primes in hyperbolic groups

Yiannis N. Petridis, Morten S. Risager

Abstract

We consider equidistribution of angles for certain hyperbolic lattice points in the upper half-plane. Extending work of Friedlander and Iwaniec we show that for the full modular group equidistribution persists for matrices with $a^2+b^2+c^2+d^2=p$ with $p$ prime; at least if we assume sufficiently good lower bounds in the hyperbolic prime number theorem by Friedlander and Iwaniec. We also investigate related questions for a specific arithmetic co-compact group and its double cosets by hyperbolic subgroups. The general equidistribution problem was studied by Good, and in this case, we show, that equidistribution holds unconditionally when restricting to primes.

Counting and equidistribution over primes in hyperbolic groups

Abstract

We consider equidistribution of angles for certain hyperbolic lattice points in the upper half-plane. Extending work of Friedlander and Iwaniec we show that for the full modular group equidistribution persists for matrices with with prime; at least if we assume sufficiently good lower bounds in the hyperbolic prime number theorem by Friedlander and Iwaniec. We also investigate related questions for a specific arithmetic co-compact group and its double cosets by hyperbolic subgroups. The general equidistribution problem was studied by Good, and in this case, we show, that equidistribution holds unconditionally when restricting to primes.
Paper Structure (22 sections, 30 theorems, 197 equations)

This paper contains 22 sections, 30 theorems, 197 equations.

Table of Contents

  1. Introduction
  2. Main results
  3. The parabolic case
  4. The elliptic case
  5. The hyperbolic case
  6. Bounding sums of multiplicative functions
  7. The elliptic case
  8. Cartan decomposition and angles between lattice points
  9. The Gaussian integers
  10. Analysis on Weyl sums in Z[i]
  11. Parametrisation of G in terms of Gaussian integers
  12. Counting group elements along primes in the elliptic case
  13. Equidistribution over primes in the elliptic case
  14. The hyperbolic case
  15. Hyperbolic decomposition in SL2(R) and Good's equidistribution
  16. ...and 7 more sections

Key Result

Theorem 1.3

Let $E$ be the sequence eq:E-sequence, i.e. the pair of normalized angles from the Cartan decomposition for ${\hbox{SL}_2( {\mathbb Z})}$. Assume $\pi_E(x)$ satisfies hyperbolic-counting-prime. Then $E$ is equidistributed on $({\mathbb R}\slash \mathbb{Z})^2$ over primes.

Theorems & Definitions (54)

  • Example 1.1: Angles of lattice points in $\mathbb{Z}^2$
  • Example 1.2: The sequence $\alpha n$ modulo 1 for $\alpha$ irrational
  • Theorem 1.3
  • Theorem 1.4
  • Theorem 1.5
  • Remark 1
  • Theorem 1.6
  • Remark 2
  • Theorem 1.7
  • Remark 3
  • ...and 44 more