  The objective of this paper is to develop simple techniques to bound the
optimal uplink sum-rate of multi-user RIS-aided wireless systems. Specifically,
we develop a novel technique called \textit{channel separation} which provides
a new understanding as to how the RIS phases affect the sum-rate. Leveraging
channel separation, we derive upper and lower bounds on the optimal sum-rate.
In addition, we propose a low-complexity alternating optimization algorithm to
obtain near-optimal sum-rate results. Numerical results demonstrate the
tightness of the bounds and show that the alternating optimization approach
delivers sum-rate values similar to the results of a full numerical
optimization procedure. Furthermore, in practical scenarios where hardware
limitations cause the RIS phases to be quantized, our lower bound can still be
applied and shows that the sum-rate is robust to quantization, even with low
resolution.
