  In this paper we study the differential capacitance of a single-level quantum
dot attached to quantum wires in the Luttinger liquid phase, or to fractional
quantum Hall edges, by both tunneling and interactions. We show that
logarithmic or even power-law divergence of the capacitance at low temperatures
may result in a substantial region of the parameter space (including the
$\nu=1/3$ filling). This behavior is a manifestation of generalized two channel
Kondo physics, and can be used to extract Luttinger liquid parameters from
thermodynamic measurements.
