Salem numbers and the spectrum of hyperbolic surfaces

TL;DR

This paper links Salem's conjecture to the spectral gap of arithmetic hyperbolic surfaces, offering a new perspective on the distribution of Salem numbers near one.

Contribution

It reformulates Salem's conjecture in terms of spectral gaps of hyperbolic surfaces, connecting number theory and geometric analysis.

Findings

Reformulation of Salem's conjecture via spectral gaps

Establishment of a connection between Salem numbers and hyperbolic surface spectra

Potential implications for understanding Salem numbers near one

Abstract

We give a reformulation of Salem's conjecture about the absence of Salem numbers near one in terms of a uniform spectral gap for certain arithmetic hyperbolic surfaces.