  We investigate a dilute Bose gas confined in a tight one-dimensional (1D)
optical lattice plus a superimposed random potential at zero temperature.
Accordingly, the ground state energy, quantum depletion and superfluid density
are calculated. The presence of the lattice introduces a crossover to the
quasi-2D regime, where we analyze asymptotically the 2D behavior of the system,
particularly the effects of disorder. We thereby offer an analytical expression
for the ground state energy of a purely 2D Bose gas in a random potential. The
obtained disorder-induced normal fluid density $n_n$ and quantum depletion
$n_d$ both exhibit a characteristic $1/\ln\left(1/n_{2D}a_{2D}^{2}\right)$
dependence. Their ratio $n_n/n_d$ increases to $2$ compared to the familiar
$4/3$ in lattice-free 3D geometry, signifying a more pronounced contrast
between superfluidity and Bose-Einstein condensation in low dimensions.
Conditions for possible experimental realization of our scenario are also
proposed.
