  We consider the problem of a single level quantum dot coupled to the edge of
a one-dimensional Luttinger liquid wire by both a hopping term and
electron-electron interactions. Using bosonization and Coulomb gas mapping of
the Anderson-Yuval type we show that thermodynamic properties of the level, in
particular, its occupation, depend on the various interactions in the system
only through a single quantity --- the corresponding Fermi edge singularity
exponent. However, dynamical properties, such as the level density of states,
depend in a different way on each type of interaction. Hence, we can construct
different models, with and without interactions in the wire, with equal Fermi
edge singularity exponents, which have identical population curves, although
they originate from very different level densities of states. The latter may
either be regular or show a power-law suppression or enhancement at the Fermi
energy. These predictions are verified to a high degree of accuracy using the
density matrix renormalization group algorithm to calculate the dot occupation,
and classical Monte Carlo simulations on the corresponding Coulomb gas model to
extract the level density of states.
