Complex Hadamard matrices and the Spectral Set Conjecture
| dc.creator | Kolountzakis, Mihail N. | |
| dc.creator | Matolcsi, Mate | |
| dc.date | 2004-11-23 | |
| dc.date.accessioned | 2026-07-07T05:14:37Z | |
| dc.date.available | 2026-07-07T05:14:37Z | |
| dc.description | 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 at most 5 in any finite Abelian group. This result is then extended to the infinite grid $\Z^d$ for any dimension d, and finally to Euclidean space. It was pointed out recently by Tao that the corresponding statement fails for |A|=6 in the group $\Z_3^5$, and this observation quickly led to the failure of the Spectral Set Conjecture in $\R^5$ (Tao), and subsequently in $\R^4$ (Matolcsi). 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. | |
| dc.identifier | https://arxiv.org/abs/math/0411512 | |
| dc.identifier | http://arxiv.org/abs/math/0411512 | |
| dc.identifier.uri | http://salesiana.dossiersoluciones.com/handle/123456789/73341 | |
| dc.subject | Classical Analysis and ODEs | |
| dc.subject | Combinatorics | |
| dc.subject | 52C22; 68Q17 | |
| dc.title | Complex Hadamard matrices and the Spectral Set Conjecture | |
| dc.type | text |