Fuglede's conjecture

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

Spectral sets and translational tiles

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

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

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

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

Partial results

Fuglede proved in 1974 that the conjecture holds if $\Omega$ is a fundamental domain of a lattice.
In 2003, Alex Iosevich, Nets Katz and Terence Tao proved that the conjecture holds if $\Omega$ is a convex planar domain.^[3]
In 2004, Terence Tao showed that the conjecture is false on $\mathbb {R} ^{d}$ for $d\geq 5$ .^[4] It was later shown by Bálint Farkas, Mihail N. Kolounzakis, Máté Matolcsi and Péter Móra that the conjecture is also false for $d=3$ and $4$ .^[5]^[6]^[7]^[8] However, the conjecture remains unknown for $d=1,2$ .
In 2015, Alex Iosevich, Azita Mayeli and Jonathan Pakianathan showed that an extension of the conjecture holds in $\mathbb {Z} _{p}\times \mathbb {Z} _{p}$ , where $\mathbb {Z} _{p}$ is the cyclic group of order p.^[9]
In 2017, Rachel Greenfeld and Nir Lev proved the conjecture for convex polytopes in $\mathbb {R} ^{3}$ .^[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. arXiv:1301.0814. Bibcode:2014MPCPS.156..123D. doi:10.1017/S0305004113000558. S2CID 119153862.
^ 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. S2CID 8267263.
^ 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.
^ 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}}: Cite journal requires |journal= (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. S2CID 55748258.
^ 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.

[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.

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