Fuglede's Conjecture

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

Spectral sets and translational tiles

Spectral sets in $\mathbb^d$

A set $\Omega$ $\subset$ $\mathbb^$ with positive finite Lebesgue measure is said to be a spectral set if there exists a $\Lambda$ $\subset$ $\mathbb^d$ such that $\left \_$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 $\mathbb^d$

A set $\Omega\subset\mathbb^d$ is said to tile $\mathbb^d$ by translation (i.e. $\Omega$ is a translational tile) if there exist a discrete set $\Tau$ such that $\bigcup_(\Omega + t)=\mathbb^d$ and the Lebesgue measure of $(\Omega + t) \cap (\Omega + t')$ is zero for all $t\neq t'$in $\Tau$.

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.
* In 2004, Terence Tao showed that the conjecture is false on $\mathbb^$ for $d\geq5$. 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 $4$. However, the conjecture remains unknown for $d=1,2$.
* In 2015, Alex Iosevich, Azita Mayeli and Jonathan Pakianathan showed that the conjecture holds in $\mathbb_\times\mathbb_$, where $\mathbb_$ is the cyclic group of order p.
* In 2017, Rachel Greenfeld and Nir Lev proved the conjecture for convex polytopes in $\mathbb^3$.
*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

Conjectures