Low-count positron emission tomography (PET) reconstruction is a challenging inverse problem due to severe degradations arising from Poisson noise, photon scarcity, and attenuation correction errors. Existing deep learning methods typically address these in the spatial domain with an undifferentiated optimization objective, making it difficult to disentangle overlapping artifacts and limiting correction effectiveness. In this work, we perform a Fourier-domain analysis and reveal that these degradations are spectrally separable: Poisson noise and photon scarcity cause high-frequency phase perturbations, while attenuation errors suppress low-frequency amplitude components. Leveraging this insight, we propose FourierPET, a Fourier-based unrolled reconstruction framework grounded in the Alternating Direction Method of Multipliers. It consists of three tailored modules: a spectral consistency module that enforces global frequency alignment to maintain data fidelity, an amplitude-phase correction module that decouples and compensates for high-frequency phase distortions and low-frequency amplitude suppression, and a dual adjustment module that accelerates convergence during iterative reconstruction. Extensive experiments demonstrate that FourierPET achieves state-of-the-art performance with significantly fewer parameters, while offering enhanced interpretability through frequency-aware correction.