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Counting Salem numbers arising from arithmetic hyperbolic orbifolds

Michelle Chu, Plinio G. P. Murillo, Otto Romero, Lola Thompson

Abstract

The relationship between Salem numbers and short geodesics has been fruitful in quantitative studies of arithmetic hyperbolic orbifolds, particularly in dimensions 2 and 3. In this article, we push these connections even further. The primary goals are: (1) to bound the proportion of Salem numbers of degree up to $n+1$ in the commensurability class of classical arithmetic lattices in any odd dimension $n$; (2) to improve lower bounds for the strong exponential growth of averages of multiplicities in the geodesic length spectrum of non-compact arithmetic orbifolds. In order to accomplish these goals, we bound, for a fixed square-free integer $D$, the count of Salem numbers with minimal polynomial $f$ satisfying $f(1)f(-1)=-D$ in $\mathbb{Q}^{\times}/\mathbb{Q}^{\times 2}$. To do this, we make use of results on the distribution of Salem numbers, as well as classical methods for counting Pythagorean triples and Gauss' lattice-counting argument. To this end, we give a generalization of the count of Pythagorean triples and provide an elementary proof which may be of independent interest.

Counting Salem numbers arising from arithmetic hyperbolic orbifolds

Abstract

The relationship between Salem numbers and short geodesics has been fruitful in quantitative studies of arithmetic hyperbolic orbifolds, particularly in dimensions 2 and 3. In this article, we push these connections even further. The primary goals are: (1) to bound the proportion of Salem numbers of degree up to in the commensurability class of classical arithmetic lattices in any odd dimension ; (2) to improve lower bounds for the strong exponential growth of averages of multiplicities in the geodesic length spectrum of non-compact arithmetic orbifolds. In order to accomplish these goals, we bound, for a fixed square-free integer , the count of Salem numbers with minimal polynomial satisfying in . To do this, we make use of results on the distribution of Salem numbers, as well as classical methods for counting Pythagorean triples and Gauss' lattice-counting argument. To this end, we give a generalization of the count of Pythagorean triples and provide an elementary proof which may be of independent interest.

Paper Structure

This paper contains 10 sections, 16 theorems, 81 equations.

Table of Contents

  1. Introduction
  2. Background
  3. Quadratic forms
  4. Hyperbolic space
  5. Arithmetic lattices
  6. Salem numbers and lengths of geodesics
  7. Counting Diophantine solutions
  8. Counting Salem numbers
  9. Counting lengths of closed geodesics in commensurability classes
  10. Bounds on multiplicity of the length spectrum

Key Result

Theorem 1

Let $m$ be a positive integer and $D$ a square-free positive integer. The number of Salem polynomials $f$ of degree $2m$ for which $f(1)f(-1)\equiv -D$ in $\mathbb{Q}^{\times}/\mathbb{Q}^{\times 2}$ with Mahler measure $\leq Q$ is bounded above by where $\kappa(m,D)$ is a constant depending only on $D$ and $m$.

Theorems & Definitions (31)

  • Theorem 1
  • Theorem 1.1
  • Theorem 2
  • Corollary 3
  • Corollary 4
  • Theorem 5
  • Conjecture 2.1: Lehmer's conjecture
  • Definition 2.2
  • Theorem 2.3
  • Conjecture 2.4: Short Geodesic Conjecture
  • ...and 21 more