  Finite-size effects have been a major and justifiable source of concern for
studies of many-body localization, and several works have been dedicated to the
subject. In this paper, however, we discuss yet another crucial problem that
has received much less attention, that of the lack of self-averaging and the
consequent danger of reducing the number of random realizations as the system
size increases. By taking this into account and considering ensembles with a
large number of samples for all system sizes analyzed, we find that the
generalized dimensions of the eigenstates of the disordered Heisenberg spin-1/2
chain close to the transition point to localization are described remarkably
well by an exact analytical expression derived for the non-interacting
Fibonacci lattice, thus providing an additional tool for studies of many-body
localization.
