High-density spectral sets in finite cyclic groups

Let A be a spectral set in ℤ/Nℤ, with |A| > N/6. We prove that N/|A| is an integer at most five. Given a complete spectrum of A, we construct a single tiling complement for all sets with that spectrum. The proof uses prime-power digit packings and the observation that the uncovered part of a packing by a spectral set still carries the orthogonal characters of its spectrum.