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

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

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

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

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

• Fuglede proved in 1974 that the conjecture holds if ${\displaystyle \Omega }$ is a fundamental domain of a lattice.
• In 2003, Alex Iosevich, Nets Katz and Terence Tao proved that the conjecture holds if ${\displaystyle \Omega }$ is a convex planar domain.^[3]
• In 2004, Terence Tao showed that the conjecture is false on ${\displaystyle \mathbb {R} ^{d}}$ for ${\displaystyle d\geq 5}$.^[4] It was later shown by Bálint Farkas, Mihail N. Kolounzakis, Máté Matolcsi and Péter Móra that the conjecture is also false for ${\displaystyle d=3}$ and ${\displaystyle 4}$.^[5][6][7][8] However, the conjecture remains unknown for ${\displaystyle d=1,2}$.
• In 2015, Alex Iosevich, Azita Mayeli and Jonathan Pakianathan showed that an extension of the conjecture holds in ${\displaystyle \mathbb {Z} _{p}\times \mathbb {Z} _{p}}$, where ${\displaystyle \mathbb {Z} _{p}}$ is the cyclic group of order p.^[9]
• In 2017, Rachel Greenfeld and Nir Lev proved the conjecture for convex polytopes in ${\displaystyle \mathbb {R} ^{3}}$.^[10]
• In 2019, Nir Lev and Máté Matolcsi settled the conjecture for convex domains affirmatively in all dimensions.^[11]