  We study the transport properties and the spectral statistics of a
one-dimensional closed quantum system of interacting spinless fermions in a
quasiperiodic potential which produces a single particle mobility edge in the
absence of interaction. For such systems, it has been shown that the many body
eigenstates can be of three different kinds: extended and ETH (eigenstate
thermalization hypothesis) obeying (thermal), localized and ETH violating (many
body localized) and extended and ETH violating (non-ergodic extended). Here we
investigate the non-ergodic extended phase from the point of view of level
spacing statistics and charge transport. We calculate the dc conductivity and
the low frequency conductivity $\sigma(\omega)$ and show that both are
consistent with sub-diffusive transport. This is contrasted with diffusive
transport in the thermal phase and blocked transport in the MBL phase.
