Fuglede's conjecture explained

Fuglede's conjecture is an open problem in mathematics proposed by Bent Fuglede in 1974. It states that every domain of

R ^{d} (i.e. subset of R ^{d} with positive finite Lebesgue measure) is a spectral set if and only if it tiles R ^{d} by translation.^[1]

Spectral sets and translational tiles

Spectral sets in

R ^{d}

A set

\Omega \subset R ^{d} with positive finite Lebesgue measure is said to be a spectral set if there exists a Λ \subset R ^{d} such that \left \{ e ^{2\pi} \right \} _{λ\inΛ} is an orthogonal basis of L ^{2(\Omega)} . The set Λ is then said to be a spectrum of \Omega and (\Omega, Λ ) is called a spectral pair.

Translational tiles of

R ^{d}

A set

\Omega\subset R ^{d} is said to tile R ^{d} by translation (i.e. \Omega is a translational tile) if there exist a discrete set \Tau such that c up _{t\in\Tau} (\Omega + t) = R ^{d} and the Lebesgue measure of (\Omega + t) \cap (\Omega + t') is zero for all t ≠ t' in \Tau .^[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

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

Z _{p} x Z _{p} , where Z _{p} is the cyclic group of order p.^[9]

• In 2017, Rachel Greenfeld and Nir Lev proved the conjecture for convex polytopes in

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. 10.1016/0022-1236(74)90072-X.
2. 1301.0814. 10.1017/S0305004113000558. Some reductions of the spectral set conjecture to integers. Mathematical Proceedings of the Cambridge Philosophical Society. 156. 1. 123–135. 2014. Dutkay. Dorin Ervin. Lai. Chun–KIT. 2014MPCPS.156..123D. 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. 10.4310/MRL.2003.v10.n5.a1. free.
4. Tao. Terence. 2004. Fuglede's conjecture is false on 5 or higher dimensions. Math. Res. Lett.. 11. 2–3. 251–258. 10.4310/MRL.2004.v11.n2.a8. math/0306134. 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. 10.1007/s00041-005-5069-7. 2006math.....12016F. math/0612016. 15553212.
6. Kolounzakis. Mihail N.. Matolcsi. Máté. 2006. Tiles with no spectra. Forum Math.. 18 . 3. 519–528. 2004math......6127K. math/0406127.
7. Matolcsi. Máté. 2005. Fuglede's conjecture fails in dimension 4. Proc. Amer. Math. Soc.. 133 . 10. 3021–3026. 10.1090/S0002-9939-05-07874-3. free.
8. Kolounzakis. Mihail N.. Matolcsi. Máté. 2006. Complex Hadamard Matrices and the spectral set conjecture. Collect. Math.. Extra. 281–291. 2004math.....11512K. math/0411512.
9. Iosevich. Alex. Mayeli. Azita. Pakianathan. Jonathan. The Fuglede Conjecture holds in Zp×Zp. 1505.00883. 10.2140/apde.2017.10.757. 2015.
10. Greenfeld. Rachel. Lev. Nir. Fuglede's spectral set conjecture for convex polytopes. Analysis & PDE. 10. 6. 1497–1538. 1602.08854. 10.2140/apde.2017.10.1497. 2017. 55748258.
11. Lev. Nir. Matolcsi. Máté. The Fuglede conjecture for convex domains is true in all dimensions. Acta Mathematica . 1904.12262. 2022. 228 . 2 . 385–420 . 10.4310/ACTA.2022.v228.n2.a3 . 139105387 .