  The average charge Q on a quantum wire, modeled as a single-channel Luttinger
liquid, connected to metallic leads and coupled to a gate is studied
theoretically. We find that the behavior of the charge as the gate voltage V_G
varies depends strongly on experimentally adjustable parameters (length,
contact transmission, temperature,...). When the intrinsic backscattering at
the contacts is weak (i.e. the conductance is close to 2e^2/h at high
temperature), we predict that this behavior should be described by a universal
function. For short such wires, the charge increases roughly linearly with V_G,
with small oscillations due to quantum interference between electrons scattered
at the contacts. For longer wires at low temperature, Coulomb blockade behavior
sets in, and the charge increases in steps. In both limits $\partial Q/\partial
V_G$, which should characterize the linear response conductance, exhibits
periodic peaks in V_G. We show that due to Coulomb interactions the period in
the former limit is twice that of the latter, and describe the evolution of the
peaks through this crossover. The study can be generalized to multi-channel
Luttinger liquids, and may explain qualitatively the recent observation by
Liang et al (Phys. Rev. Lett. 88, 126801) of a four-electron periodicity for
electron addition in single-walled carbon nanotubes.
