  In this work we investigate the stability of an algebraically localized phase
subject to periodic driving. First, we focus on a non-interacting model
exhibiting algebraically localized single-particle modes. For this model we
find numerically that the algebraically localized phase is stable under
driving, meaning that the system remains localized at arbitrary frequencies. We
support this result with analytical considerations using simple renormalization
group arguments. Second, we inspect the case in which short-range interactions
are added. By studying both, the eigenstates properties of the Floquet
Hamiltonian and the out-of-equilibrium dynamics in the interacting model, we
provide evidence that ergodicity is restored at any driving frequencies. In
particular, we observe that for the accessible system sizes localization sets
in at driving frequency that are comparable with the many-body bandwidth and
thus it might be only transient, suggesting that the system might thermalize in
the thermodynamic limit.
