Fuglede's conjecture

Fuglede's conjecture is an open problem in mathematics proposed by Bent Fuglede in 1974. It states that every domain of ${\displaystyle \mathbb {R} ^{d}}$ (i.e. subset of ${\displaystyle \mathbb {R} ^{d}}$ with positive finite Lebesgue measure) is a spectral set if and only if it tiles ${\displaystyle \mathbb {R} ^{d}}$ by translation. ^[1]

Spectral sets and translational tiles

Spectral sets in${\displaystyle \mathbb {R} ^{d}}$

A set ${\displaystyle \Omega }$${\displaystyle \subset }$${\displaystyle \mathbb {R} ^{d}}$ with positive finite Lebesgue measure is said to be a spectral set if there exists a ${\displaystyle \Lambda }$${\displaystyle \subset }$${\displaystyle \mathbb {R} ^{d}}$ such that ${\displaystyle \left\{e^{2\pi i\left\langle \lambda ,\cdot \right\rangle }\right\}_{\lambda \in \Lambda }}$is an orthogonal basis of ${\displaystyle L^{2}(\Omega )}$. The set ${\displaystyle \Lambda }$ is then said to be a spectrum of ${\displaystyle \Omega }$ and ${\displaystyle (\Omega ,\Lambda )}$ is called a spectral pair.

Translational tiles of${\displaystyle \mathbb {R} ^{d}}$

A set ${\displaystyle \Omega \subset \mathbb {R} ^{d}}$ is said to tile ${\displaystyle \mathbb {R} ^{d}}$ by translation (i.e. ${\displaystyle \Omega }$ is a translational tile) if there exist a discrete set ${\displaystyle \mathrm {T} }$ such that ${\displaystyle \bigcup _{t\in \mathrm {T} }(\Omega +t)=\mathbb {R} ^{d}}$ and the Lebesgue measure of ${\displaystyle (\Omega +t)\cap (\Omega +t')}$ is zero for all ${\displaystyle t\neq t'}$in ${\displaystyle \mathrm {T} }$. ^[2]

Partial results

• Fuglede proved in 1974 that the conjecture holds if ${\displaystyle \Omega }$ is a fundamental domain of a lattice.
• In 2003, Alex Iosevich, Nets Katz and Terence Tao proved that the conjecture holds if ${\displaystyle \Omega }$ is a convex planar domain. ^[3]
• In 2004, Terence Tao showed that the conjecture is false on ${\displaystyle \mathbb {R} ^{d}}$ for ${\displaystyle d\geq 5}$. ^[4] It was later shown by Bálint Farkas, Mihail N. Kolounzakis, Máté Matolcsi and Péter Móra that the conjecture is also false for ${\displaystyle d=3}$ and ${\displaystyle 4}$. ^{[5] [6] [7] [8]} However, the conjecture remains unknown for ${\displaystyle d=1,2}$.
• In 2015, Alex Iosevich, Azita Mayeli and Jonathan Pakianathan showed that the conjecture holds in ${\displaystyle \mathbb {Z} _{p}\times \mathbb {Z} _{p}}$, where ${\displaystyle \mathbb {Z} _{p}}$ is the cyclic group of order p. ^[9]
• In 2017, Rachel Greenfeld and Nir Lev proved the conjecture for convex polytopes in ${\displaystyle \mathbb {R} ^{3}}$. ^[10]
• In 2019, Nir Lev and Máté Matolcsi settled the conjecture for convex domains affirmatively in all dimensions. ^[11]

References

1. 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.
2. 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. S2CID   119153862.
3. 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 .
4. 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. S2CID   8267263.
5. 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. S2CID   15553212.
6. Kolounzakis, Mihail N.; Matolcsi, Máté (2006). "Tiles with no spectra". Forum Math. 18 (3): 519–528. arXiv: math/0406127 . Bibcode:2004math......6127K.
7. 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 .
8. 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.
9. 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}}: Cite journal requires |journal= (help)
10. 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. S2CID   55748258.
11. Lev, Nir; Matolcsi, Máté (2022). "The Fuglede conjecture for convex domains is true in all dimensions". Acta Mathematica. 228 (2): 385–420. arXiv: 1904.12262 . doi:10.4310/ACTA.2022.v228.n2.a3. S2CID   139105387.