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 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 group Z53, and this observation quickly led to the failure of the Spectral Set Conjecture in R [16], and subsequently in R [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. In a computational search for counterexamples in lower dimension (one and two) one needs, at the very least, to be able to decide efficiently if a set is a tile (in, say, a cyclic group) and if it is spectral. Such efficient procedures are lacking however and we make a few comments for the computational complexity of some related problems. 2000 Mathematics Subject Classification. Primary 52C22, Secondary 20K01, 42B99.