  We consider a many body fermionic system with an incommensurate external
potential and a short range interaction in one dimension. We prove that, for
certain densities and weak interactions, the zero temperature thermodynamical
correlations are exponentially decaying for large distances, a property
indicating persistence of localization in the interacting ground state. The
analysis is based on Renormalization Group, and convergence of the renormalized
expansion is achieved using fermionic cancellations and controlling the small
divisor problem assuming a Diophantine condition for the frequency.
