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.

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 $\mathrm {T}$.

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 $\mathbb {R} ^{d}$ for $d\geq 5$. 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 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.
• In 2017, Rachel Greenfeld and Nir Lev proved the conjecture for convex polytopes in $\mathbb {R} ^{3}$.
• In 2019, Nir Lev and Máté Matolcsi settled the conjecture for convex domains affirmatively in all dimensions.