Counting Salem numbers arising from arithmetic hyperbolic orbifolds

TL;DR

This paper explores the connection between Salem numbers and the geometry of arithmetic hyperbolic orbifolds, providing bounds on Salem numbers and improving growth estimates of geodesic multiplicities.

Contribution

It introduces new bounds on Salem numbers in arithmetic orbifolds and generalizes Pythagorean triple counts using classical lattice methods.

Findings

Bound the proportion of Salem numbers in certain classes.

Improve lower bounds for geodesic multiplicity growth.

Generalize Pythagorean triple counting methods.

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.