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.{{Cite journal|last=Fuglede|first=Bent|date=1974|title=Commuting self-adjoint partial differential operators and a group theoretic problem|journal=Journal of Functional Analalysis|volume=16|pages=101–121|doi=10.1016/0022-1236(74)90072-X|doi-access=}}

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 $\Tau$ such that $\bigcup_{t\in\Tau}(\Omega + t)=\mathbb{R}^d$ and the Lebesgue measure of $(\Omega + t) \cap (\Omega + t')$ is zero for all $t\neq t'$ in $\Tau$ .{{Cite journal |arxiv = 1301.0814|doi = 10.1017/S0305004113000558|title = Some reductions of the spectral set conjecture to integers|journal = Mathematical Proceedings of the Cambridge Philosophical Society|volume = 156|issue = 1|pages = 123–135|year = 2014|last1 = Dutkay|first1 = Dorin Ervin|last2 = Lai|first2 = Chun–Kit|bibcode = 2014MPCPS.156..123D|s2cid = 119153862}}

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.{{Cite journal|last1=Iosevich|first1=Alex|last2=Katz|first2=Nets|last3=Terence|first3=Tao|date=2003|title=The Fuglede spectral conjecture hold for convex planar domains|journal=Mathematical Research Letters|volume=10|issue= 5–6|pages=556–569|doi=10.4310/MRL.2003.v10.n5.a1|doi-access=free}}
In 2004, Terence Tao showed that the conjecture is false on $\mathbb{R}^{d}$ for $d\geq5$ .{{Cite journal|last=Tao|first=Terence|date=2004|title=Fuglede's conjecture is false on 5 or higher dimensions|journal=Mathematical Research Letters|volume=11|issue= 2–3|pages=251–258|doi=10.4310/MRL.2004.v11.n2.a8|arxiv=math/0306134|s2cid=8267263}} 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 $4$ .{{Cite journal|last1=Farkas|first1=Bálint|last2=Matolcsi|first2=Máté|last3=Móra|first3=Péter|date=2006|title=On Fuglede's conjecture and the existence of universal spectra|journal=Journal of Fourier Analysis and Applications|volume=12 |issue=5|pages=483–494|doi=10.1007/s00041-005-5069-7|bibcode=2006JFAA...12..483F|arxiv=math/0612016|s2cid=15553212}}{{Cite journal|last1=Kolounzakis|first1=Mihail N.|last2=Matolcsi|first2=Máté|date=2006|title=Tiles with no spectra|journal=Forum Mathematicum|volume=18 |issue=3|pages=519–528|bibcode=2004math......6127K|arxiv=math/0406127}}{{Cite journal|last=Matolcsi|first=Máté|date=2005|title=Fuglede's conjecture fails in dimension 4|journal=Proceedings of the American Mathematical Society|volume=133 |issue=10|pages=3021–3026|doi=10.1090/S0002-9939-05-07874-3|doi-access=free}}{{Cite journal|last1=Kolounzakis|first1=Mihail N.|last2=Matolcsi|first2=Máté|date=2006|title=Complex Hadamard Matrices and the spectral set conjecture|journal=Collectanea Mathematica|volume= Extra|pages=281–291|bibcode=2004math.....11512K|arxiv=math/0411512}} 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.{{Cite journal|last1=Iosevich|first1=Alex|last2=Mayeli|first2=Azita|last3=Pakianathan|first3=Jonathan|title=The Fuglede Conjecture holds in Zp×Zp|journal=Analysis & PDE|volume=10|issue=4|pages=757–764|arxiv=1505.00883|doi=10.2140/apde.2017.10.757|date=2017}}
In 2017, Rachel Greenfeld and Nir Lev proved the conjecture for convex polytopes in $\mathbb{R}^3$ .{{Cite journal|last1=Greenfeld|first1=Rachel|last2=Lev|first2=Nir|title=Fuglede's spectral set conjecture for convex polytopes|journal=Analysis & PDE|volume=10|issue=6|pages=1497–1538|arxiv=1602.08854|doi=10.2140/apde.2017.10.1497|year=2017|s2cid=55748258}}
In 2019, Nir Lev and Máté Matolcsi settled the conjecture for convex domains affirmatively in all dimensions.{{Cite journal|last1=Lev|first1=Nir|last2=Matolcsi|first2=Máté|title=The Fuglede conjecture for convex domains is true in all dimensions

|journal=Acta Mathematica |arxiv=1904.12262|year=2022|volume=228 |issue=2 |pages=385–420 |doi=10.4310/ACTA.2022.v228.n2.a3 |s2cid=139105387 }}

References

Category:Conjectures