Energy Separation in Oscillatory Hamiltonian Systems without any Non-resonance Condition

We consider multiscale Hamiltonian systems in which harmonic oscillators with several high frequencies are coupled to a slow system. It is shown that the oscillatory energy is nearly preserved over long times $${\varepsilon^{-N}}$$ for arbitrary N > 1, where $${\varepsilon^{-1}}$$ is the size of the smallest high frequency. The result is uniform in the frequencies and does not require non-resonance conditions.