  The ground states of interacting one-dimensional metals are generically
Luttinger liquids. Luttinger liquid theory is usually considered for
translation invariant systems. The Luttinger liquid description remains valid
for weak quasiperiodic modulations; however, as the quasiperiodic modulation
gets increasingly strong, it is increasingly renormalized and eventually fails,
as the system becomes localized. We explore how quasiperiodic modulation
renormalizes the Luttinger parameter characterizing this emergent Luttinger
liquid, using the renormalization of transmission coefficients across a barrier
as a proxy that remains valid for general quasiperiodic modulation. We find,
unexpectedly, that quasiperiodic modulation weakens the effects of short-range
interactions, but enhances those of long-range interactions. We support the
former finding with matrix-product numerics. We also discuss how interactions
affect the localization phase boundary.
