Balian–Low Type Theorems on Homogeneous Groups

We prove strict necessary density conditions for coherent frames and Riesz sequences on homogeneous groups. Let N be a connected, simply connected nilpotent Lie group with a dilation structure (a homogeneous group) and let (π,H_π) be an irreducible, square-integrable representation modulo the center Z(N) of N on a Hilbert space H_π of formal dimension d_π. If g ∈ H_π is an integrable vector and the set {π(λ)g : λ ∈ Λ} for a discrete subset Λ ⊆ N/Z(N) forms a frame for H_π, then its density satisfies the strict inequality D − (Λ) > d_π, where D − (Λ) is the lower Beurling density. An analogous density condition D⁺(Λ) < d_π holds for a Riesz sequence in H_π contained in the orbit of (π,H_π). The proof is based on a deformation theorem for coherent systems, a universality result for p-frames and p-Riesz sequences, some results from Banach space theory, and tools from the analysis on homogeneous groups.