​We study the achievable rate of Cognitive Radio (CR) spectrum sharing systems at low power regime for general ​fading channels then for Nakagami fading. We formally define the low power regime and present the corresponding closed form expressions of the achievable rate lower-bound under various types of interference and/or power constraints, depending on the available channel state information (CSI) of the cross link (CL) between the secondary user transmitter and the primary user receiver. We explicitly characterize two regimes where either the interference constraint or the power constraint dictates the optimal power profile. Our framework also highlights the effects of different fading parameters on the secondary link ergodic achievable rate. We, also, study more realistic scenarios when there is either a 1-bit quantized channel feedback from the CL alone or 2-bit feedback from both the CL and the secondary link (SL), and propose simple power control schemes and show that these schemes achieve the previously achieved rate at low power regime. Interestingly, we show that the low power regime analysis provides a specific insight on the maximum achievable rate behavior of CR that has not been reported by previous studies.​