Fuglede's conjecture
Fuglede's conjecture is an open problem in mathematics proposed by Bent Fuglede in 1974. It states that every domain of [math]\displaystyle{ \mathbb{R}^{d} }[/math] (i.e. subset of [math]\displaystyle{ \mathbb{R}^{d} }[/math] with positive finite Lebesgue measure) is a spectral set if and only if it tiles [math]\displaystyle{ \mathbb{R}^{d} }[/math] by translation.[1]
Spectral sets and translational tiles
Spectral sets in [math]\displaystyle{ \mathbb{R}^d }[/math]
A set [math]\displaystyle{ \Omega }[/math] [math]\displaystyle{ \subset }[/math] [math]\displaystyle{ \mathbb{R}^{d} }[/math] with positive finite Lebesgue measure is said to be a spectral set if there exists a [math]\displaystyle{ \Lambda }[/math] [math]\displaystyle{ \subset }[/math] [math]\displaystyle{ \mathbb{R}^d }[/math] such that [math]\displaystyle{ \left \{ e^{2\pi i\left \langle \lambda, \cdot \right \rangle} \right \}_{\lambda\in\Lambda} }[/math]is an orthogonal basis of [math]\displaystyle{ L^2(\Omega) }[/math]. The set [math]\displaystyle{ \Lambda }[/math] is then said to be a spectrum of [math]\displaystyle{ \Omega }[/math] and [math]\displaystyle{ (\Omega, \Lambda) }[/math] is called a spectral pair.
Translational tiles of [math]\displaystyle{ \mathbb{R}^d }[/math]
A set [math]\displaystyle{ \Omega\subset\mathbb{R}^d }[/math] is said to tile [math]\displaystyle{ \mathbb{R}^d }[/math] by translation (i.e. [math]\displaystyle{ \Omega }[/math] is a translational tile) if there exist a discrete set [math]\displaystyle{ \Tau }[/math] such that [math]\displaystyle{ \bigcup_{t\in\Tau}(\Omega + t)=\mathbb{R}^d }[/math] and the Lebesgue measure of [math]\displaystyle{ (\Omega + t) \cap (\Omega + t') }[/math] is zero for all [math]\displaystyle{ t\neq t' }[/math]in [math]\displaystyle{ \Tau }[/math].[2]
Partial results
- Fuglede proved in 1974 that the conjecture holds if [math]\displaystyle{ \Omega }[/math] is a fundamental domain of a lattice.
- In 2003, Alex Iosevich, Nets Katz and Terence Tao proved that the conjecture holds if [math]\displaystyle{ \Omega }[/math] is a convex planar domain.[3]
- In 2004, Terence Tao showed that the conjecture is false on [math]\displaystyle{ \mathbb{R}^{d} }[/math] for [math]\displaystyle{ d\geq5 }[/math].[4] It was later shown by Bálint Farkas, Mihail N. Kolounzakis, Máté Matolcsi and Péter Móra that the conjecture is also false for [math]\displaystyle{ d=3 }[/math] and [math]\displaystyle{ 4 }[/math].[5][6][7][8] However, the conjecture remains unknown for [math]\displaystyle{ d=1,2 }[/math].
- In 2015, Alex Iosevich, Azita Mayeli and Jonathan Pakianathan showed that an extension of the conjecture holds in [math]\displaystyle{ \mathbb{Z}_{p}\times\mathbb{Z}_{p} }[/math], where [math]\displaystyle{ \mathbb{Z}_{p} }[/math] is the cyclic group of order p.[9]
- In 2017, Rachel Greenfeld and Nir Lev proved the conjecture for convex polytopes in [math]\displaystyle{ \mathbb{R}^3 }[/math].[10]
- In 2019, Nir Lev and Máté Matolcsi settled the conjecture for convex domains affirmatively in all dimensions.[11]
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. doi:10.1017/S0305004113000558. Bibcode: 2014MPCPS.156..123D.
- ↑ 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. 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. doi:10.1007/s00041-005-5069-7. Bibcode: 2006math.....12016F.
- ↑ Kolounzakis, Mihail N.; Matolcsi, Máté (2006). "Tiles with no spectra". Forum Math. 18 (3): 519–528. 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. Bibcode: 2004math.....11512K.
- ↑ Iosevich, Alex; Mayeli, Azita; Pakianathan, Jonathan (2015). The Fuglede Conjecture holds in Zp×Zp. doi:10.2140/apde.2017.10.757.
- ↑ Greenfeld, Rachel; Lev, Nir (2017). "Fuglede's spectral set conjecture for convex polytopes". Analysis & PDE 10 (6): 1497–1538. doi:10.2140/apde.2017.10.1497.
- ↑ Lev, Nir; Matolcsi, Máté (2022). "The Fuglede conjecture for convex domains is true in all dimensions". Acta Mathematica 228 (2): 385–420. doi:10.4310/ACTA.2022.v228.n2.a3.
 |  |
