  Using a combination of analytic and numerical methods, we study the
polarizability of a (non-interacting) Anderson insulator in one, two, and three
dimensions and demonstrate that, in a wide range of parameters, it scales
proportionally to the square of the localization length, contrary to earlier
claims based on the effective-medium approximation. We further analyze the
effect of electron-electron interactions on the dielectric constant in
quasi-1D, quasi-2D and 3D materials with large localization length, including
both Coulomb repulsion and phonon-mediated attraction. The phonon-mediated
attraction (in the pseudogapped state on the insulating side of the
Superconductor-Insulator Transition) produces a correction to the dielectric
constant, which may be detected from a linear response of a dielectric constant
to an external magnetic field.
