Fuglede's conjecture
Fuglede's conjecture is an open problem in mathematics proposed by Bent Fuglede in 1974. It states that every domain of (i.e. subset of with positive finite Lebesgue measure) is a spectral set if and only if it tiles by translation.[1]
Spectral sets and translational tiles
Spectral sets in
A set with positive finite Lebesgue measure is said to be a spectral set if there exists a such that is an orthogonal basis of . The set is then said to be a spectrum of and is called a spectral pair.
Translational tiles of
A set is said to tile by translation (i.e. is a translational tile) if there exist a discrete set such that and the Lebesgue measure of is zero for all in .[2]
Partial results
- Fuglede proved in 1974, that the conjecture holds if is a fundamental domain of a lattice.
- In 2003, Alex Iosevich, Nets Katz and Terence Tao proved that the conjecture holds if is a convex planar domain.[3]
- In 2004, Terence Tao showed that the conjecture is false on for .[4] It was later shown by Bálint Farkas, Mihail N. Kolounzakis, Máté Matolcsi and Péter Móra that the conjecture is also false for and .[5][6][7][8] However, the conjecture remains unknown for .
- Alex Iosevich, Azita Mayeli and Jonathan Pakianathan showed that the conjecture holds in , where is the finite group of order p.[9]
- In 2017, Rachel Greenfeld and Nir Lev proved the conjecture for convex polytopes in .[10]
- In 2019, Nir Lev and Máté Matolcsi settled the conjecture for convex domains affirmatively in all dimensions.[11]
gollark: Implode enes.
gollark: It's been too long. I no longer know mods.
gollark: It's possible.
gollark: I am aware of this game, yes.
gollark: Observe, HOTBAR.
References
- Fuglede, Bent (1974). "Commuting self-adjoint partial differential operators and a group theoretic problem". J. Funct. Anal. 16: 101–121. doi:10.1016/0022-1236(74)90072-X.
- Dutkay, Dorin Ervin; Lai, Chun–KIT (2014). "Some reductions of the spectral set conjecture to integers". Mathematical Proceedings of the Cambridge Philosophical Society. 156 (1): 123–135. arXiv:1301.0814. Bibcode:2014MPCPS.156..123D. doi:10.1017/S0305004113000558.
- Iosevich, Alex; Katz, Nets; Terence, Tao (2003). "The Fuglede spectral conjecture hold for convex planar domains". Math. Res. Lett. 10 (5–6): 556–569. doi:10.4310/MRL.2003.v10.n5.a1.
- Tao, Terence (2004). "Fuglede's conjecture is false on 5 or higher dimensions". Math. Res. Lett. 11 (2–3): 251–258. arXiv:math/0306134. doi:10.4310/MRL.2004.v11.n2.a8.
- Farkas, Bálint; Matolcsi, Máté; Móra, Péter (2006). "On Fuglede's conjecture and the existence of universal spectra". J. Fourier Anal. Appl. 12 (5): 483–494. arXiv:math/0612016. Bibcode:2006math.....12016F. doi:10.1007/s00041-005-5069-7.
- Kolounzakis, Mihail N.; Matolcsi, Máté (2006). "Tiles with no spectra". Forum Math. 18 (3): 519–528. arXiv:math/0406127. Bibcode:2004math......6127K.
- Matolcsi, Máté (2005). "Fuglede's conjecture fails in dimension 4". Proc. Amer. Math. Soc. 133 (10): 3021–3026. doi:10.1090/S0002-9939-05-07874-3.
- Kolounzakis, Mihail N.; Matolcsi, Máté (2006). "Complex Hadamard Matrices and the spectral set conjecture". Collect. Math. Extra: 281–291. arXiv:math/0411512. Bibcode:2004math.....11512K.
- Iosevich, Alex; Mayeli, Azita; Pakianathan, Jonathan (2015). "The Fuglede Conjecture holds in Zp×Zp". arXiv:1505.00883. doi:10.2140/apde.2017.10.757. Cite journal requires
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(help) - Greenfeld, Rachel; Lev, Nir (2017). "Fuglede's spectral set conjecture for convex polytopes". Analysis & PDE. 10 (6): 1497–1538. arXiv:1602.08854. doi:10.2140/apde.2017.10.1497.
- Lev, Nir; Matolcsi, Máté (2019). "The Fuglede conjecture for convex domains is true in all dimensions". arXiv:1904.12262 [math.CA].
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