  We study the single file diffusion problem on a one-dimensional lattice with
a self-similar distribution of hopping rates. We find that the time dependence
of the mean-square displacement of both a tagged particle and the center of
mass of the system present anomalous power laws modulated by logarithmic
periodic oscillations. The anomalous exponent of a tagged particle is one half
of the exponent of the center of mass, and always smaller than 1/4. Using
heuristic arguments, the exponents and the periods of oscillation are
analytically obtained and confirmed by Monte Carlo simulations.
