2025/08/11 by Chu, Michelle, Murillo, Plinio G. P., Romero, Otto +1 · 1 citation
#Differential Geometry (math.DG) #FOS: Mathematics #Geometric Topology (math.GT) #Group Theory (math.GR) #Number Theory (math.NT)
paper · doi:10.48550/arxiv.2508.08003
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 ℚ×/ℚ× 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.