The reconfigurability of fluid antenna systems (FASs) and reconfigurable intelligent surfaces (RISs) provides significant flexibility in optimizing channel conditions by jointly adjusting the positions of fluid antennas and the phase shifts of RISs. However, it is challenging to acquire the instantaneous channel state information (CSI) for both fluid antennas and RISs, while frequent adjustment of antenna positions and phase shifts will significantly increase the system complexity. To tackle this issue, this paper investigates the two-timescale design for FAS-RIS multi-user systems with linear precoding, where only the linear precoder design requires instantaneous CSI of the end-to-end channel, while the FAS and RIS optimization relies on statistical CSI. The main challenge comes from the complex structure of channel and inverse operations in linear precoding, such as regularized zero-forcing (RZF) and zero-forcing (ZF). Leveraging on random matrix theory (RMT), we first investigate the fundamental limits of FAS-RIS systems with RZF/ZF precoding by deriving the ergodic sum rate (ESR). This result is utilized to determine the minimum number of activated antennas to achieve a given ESR. Based on the evaluation result, we propose an algorithm to jointly optimize the antenna selection, regularization factor of RZF, and phase shifts at the RIS. Numerical results validate the accuracy of performance evaluation and demonstrate that the performance gain brought by joint FAS and RIS design is more pronounced with a larger number of users.

翻译：流体天线系统（FAS）与可重构智能表面（RIS）的可重构性，通过联合调整流体天线位置与RIS相位偏移，为优化信道条件提供了显著的灵活性。然而，同时获取流体天线与RIS的瞬时信道状态信息（CSI）具有挑战性，且频繁调整天线位置与相位偏移将显著增加系统复杂度。为解决这一问题，本文研究了采用线性预编码的FAS-RIS多用户系统的双时间尺度设计，其中仅线性预编码器设计需要端到端信道的瞬时CSI，而FAS与RIS的优化则依赖于统计CSI。主要挑战源于信道的复杂结构以及线性预编码（如正则化迫零（RZF）与迫零（ZF））中的逆运算。借助随机矩阵理论（RMT），我们首先通过推导遍历和速率（ESR）来研究采用RZF/ZF预编码的FAS-RIS系统的基本性能极限。该结果用于确定实现给定ESR所需激活天线的最小数量。基于评估结果，我们提出一种算法，用于联合优化天线选择、RZF的正则化因子以及RIS的相位偏移。数值结果验证了性能评估的准确性，并表明联合FAS与RIS设计带来的性能增益在用户数量较大时更为显著。