Complex Hadamard matrices and the spectral set conjecture

By analyzing the connection between complex Hadamard matrices and spectral sets, we prove the direction ?spectral ) tile? of the Spectral Set Conjecture, for all sets A of size |A|  5, in any finite Abelian group. This result is then extended to the infinite grid Zd for any dimension d, and finally to Rd. It was pointed out recently in [16] that the corresponding statement fails for |A| = 6 in the group Z5 3, and this observation quickly led to the failure of the Spectral Set Conjecture in R5 [16], and subsequently in R4 [13]. In the second part of this note we reduce this dimension further, showing that the direction ?spectral ) tile? of the Spectral Set Conjecture is false already in dimension 3.