The wireless channel is undergoing continuous changes, and the block-fading assumption, despite its popularity in theoretical contexts, never holds true in practical scenarios. This discrepancy is particularly critical for user activity detection in grant-free random access, where joint processing across multiple resource blocks is usually undesirable. In this paper, we propose employing a low-dimensional approximation of the channel to capture variations over time and frequency and robustify activity detection algorithms. This approximation entails projecting channel fading vectors onto their principal directions to minimize the approximation order. Through numerical examples, we demonstrate a substantial performance improvement achieved by the resulting activity detection algorithm.