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.
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