We study the transport properties of quantum dots and quantum point contacts in the Coulomb blockade regime and in the limit where the quantum point contact has nearly fully transmitting channels. We argue that the properties of the charge fluctuations through the contact can be dealt with using a one-dimensional representation and study the scaling behavior of the model. We find a cross-over between a low-energy regime with Coulomb blockade to a high-energy regime where quantum charge fluctuations are dominant. The cross-over energy defines an effective charging energy given by U^* ∼ U[1 - T] ^{N 2}, where U is the bare charging energy, T is the transmission coefficient, and N is the number of channels given by twice the number of quantum point contacts connected to the dot.