  For trigonometric and modified trigonometric integrators applied to
oscillatory Hamiltonian differential equations with one or several constant
high frequencies, near-conservation of the total and oscillatory energies are
shown over time scales that cover arbitrary negative powers of the step size.
This requires non-resonance conditions between the step size and the
frequencies, but in contrast to previous results the results do not require any
non-resonance conditions among the frequencies. The proof uses modulated
Fourier expansions with appropriately modified frequencies.
