TOPICS
Search

Fuglede's Conjecture


Fuglede (1974) conjectured that a domain Omega admits an operator spectrum iff it is possible to tile R^d by a family of translates of Omega. Fuglede proved the conjecture in the special case that the tiling set or the spectrum are lattice subsets of R^d and Iosevich et al. (1999) proved that no smooth symmetric convex body Omega with at least one point of nonvanishing Gaussian curvature can admit an orthogonal basis of exponentials.

Using complex Hadamard matrices of orders 6 and 12, Tao (2003) constructed counterexamples to the conjecture in some small Abelian groups, and lifted these to counterexamples in R^5 or R^(11).

However, the conjecture has been proved in a great number of special cases (e.g., all convex planar bodies) and remains an open problem in small dimensions. For example, it has been shown in dimension 1 that a nice algebraic characterization of finite sets tiling Z indeed implies one side of Fuglede's conjecture (Coven and Meyerowitz 1999). Furthermore, it is sufficient to prove these conditions when the tiling gives a factorization of a non-Hajós cyclic group (Amiot 2005).


See also

Operator Spectrum

Portions of this entry contributed by Emmanuel Amiot

Explore with Wolfram|Alpha

References

Amiot, E. "Rhythmic Canons and Galois Theory." Grazer Math. Ber. 347, 1-21, 2005.Coven, E. M. and Meyerowitz, A. "Tiling the Integers with Translates of One Finite Set." J. Algebra 212, 161-174, 1999.Fuglede, B. "Commuting Self-Adjoint Partial Differential Operators and a Group Theoretic Problem." J. Func. Anal. 16, 101-121, 1974.Iosevich, A.; Katz, N. H.; and Tao, T. "Convex Bodies with a Point of Curvature Do Not Have Fourier Bases." 23 Nov 1999. http://arxiv.org/abs/math.CA/9911167.Iosevich, A.; Katz, N. H.; and Tao, T. "The Fuglede Spectral Conjecture Holds For Convex Planar Domains." Math. Res. Lett. 10, 559-569, 2003. http://www.mrlonline.org/mrl/2003-010-005/2003-010-005-001.pdf.Jorgensen, P. E. T. and Pedersen, S. "Orthogonal Harmonic Analysis of Fractal Measures." Elec. Res. Announc. Amer. Math. Soc. 4, 35-42, 1998.Kolountzakis, M. N. and Matolcsi, M. "Complex Hadamard Matrices and the Spectral Set Conjecture." In Proceedings of the 7th International Conference on Harmonic Analysis and Partial Differential Equations (El Escorial, 2004). http://fourier.math.uoc.gr/~mk/ps/hadamard.pdf.Laba, I. "Tiling Problems and Spectral Sets." http://www.math.ubc.ca/~ilaba/tiling.html.Lagarias, J. and Wang, Y. "Spectral Sets and Factorizations of Finite Abelian Groups." J. Func. Anal. 145, 73-98, 1997.Tao, T. "Fuglede's Conjecture Is False in 5 and Higher Dimensions." 9 Jun 2003. http://arxiv.org/abs/math.CO/0306134.

Cite this as:

Amiot, Emmanuel and Weisstein, Eric W. "Fuglede's Conjecture." From MathWorld--A Wolfram Web Resource. https://mathworld.wolfram.com/FugledesConjecture.html

Subject classifications