  We consider interacting spinless fermions in one dimension embedded in
self-similar quasiperiodic potentials. We examine generalizations of the
Fibonacci potential known as precious mean potentials. Using a bosonization
technique and a renormalization group analysis, we study the low-energy physics
of the system. We show that it undergoes a metal-insulator transition for any
filling factor, with a critical interaction that strongly depends on the
position of the Fermi level in the Fourier spectrum of the potential. For some
positions of the Fermi level the metal-insulator transition occurs at the non
interacting point. The repulsive side is an insulator with a gapped spectrum
whereas in the attractive side the spectrum is gapless and the properties of
the system are described by a Luttinger liquid. We compute the transport
properties and give the characteristic exponents associated to the frequency
and temperature dependence of the conductivity.
