  We suggest that KAM theory could be extended for certain infinite-dimensional
systems with purely discrete linear spectrum. We provide empirical arguments
for the existence of square summable infinite-dimensional invariant tori in the
random discrete nonlinear Schr\"odinger equation, appearing with a finite
probability for a given initial condition with sufficiently small norm.
Numerical support for the existence of a fat Cantor set of initial conditions
generating almost-periodic oscillations is obtained by analyzing (i) sets of
recurrent trajectories over successively larger time scales, and (ii)
finite-time Lyapunov exponents. The norm region where such KAM-like tori may
exist shrinks to zero when the disorder strength goes to zero and the
localization length diverges.
