(a)), the correlations decay according to the well-known $r^{-\frac{1}{2}}$ power law behaviour of a free TG gas [@Cazalilla2011]. In the presence of a lattice for incommensurate particle density (such as $F=\frac{2}{5}$) an exponential decay is observed at small $r$ followed by a power-law decay. This demonstrates quasi-off-diagonal long range order (QODLRO) in the gas for the Hamiltonian and mimics the behavior of the hard-core BHM [@Rigol2005]. In the continuous case, however, the power-law decay has an additional, very intuitive super-structure: the auto-correlations are modulated with the lattice periodicity. This corresponds to the fact that the atoms are preferably located at the lattice minima and their probability to be at a lattice maximum is significantly decreased. In fact, if the auto-correlations at these distances are going to zero, the system will enter a regime in which the tight-binding approximation works well. Once the second energy-band is occupied a deeper lattice is required to create modulations in the auto-correlations (compare $F=\frac{7}{5}$ in (a) and (b)). For $F=1$ the pinned phase leads to an exponential decay of correlations, with no correlations surviving at distances longer than one lattice site for $V_0=20$. In the case of a microscopic number of defects ($F=\frac{48}{50}$) the initial 