Complex Hadamard matrices and the spectral set conjecture

Autores: Mihail N. Kolounzakis, Máté Matolcsi
Localización: Collectanea mathematica, ISSN 0010-0757, Vol. 57, Fasc. Extra 1, 2006 (Ejemplar dedicado a: Proceedings of the 7th International Conference on Harmonic Analysis and Partial Differential Equations), págs. 281-291
Idioma: español
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Resumen
- 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.