By analyzing the connection between complex Hadamard matrices and spectral sets we prove the direction “spectral ⇒ tile” of the Sectral Set Conjecture for all sets A of size |A| ≤ 5 in any finite Abelian group. This result is then extended to the infinite grid Z for any dimension d, and finally to R. It was pointed out recently in [16] that the corresponding statement fails for |A| = 6 in the g...