  We establish a density one version of Braak's conjecture on the fine
structure of the spectrum of the quantum Rabi model, as well as a recent
conjecture of Braak, Nguyen, Reyes-Bustos and Wakayama on the nearest neighbor
spacings of the spectrum. The proof uses a three-term asymptotic expansion for
large eigenvalues due to Boutet de Monvel and Zielinski, and a number theoretic
argument from uniform distribution theory.
