Localization and delocalization in the quantum kicked prime number rotator

The quantum kicked prime number rotator (QKPR) is defined as the rotator whose energy levels are prime numbers. The long time behavior is decided by the kick period $\tau$ and kick strength $k$. When $\frac{\tau}{2\pi}$ is irrational, QKPR is localized because of the equidistribution theorem. When $\frac{\tau}{2\pi}$ is rational, QKPR is localized for small $k$, because the system seems like a generalized kicked dimer model. We argue for rational $\frac{\tau}{2\pi}$ QKPR delocalizes for large k.