Fuglede's conjecture

Template:Short description 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.

Spectral sets and translational tiles

Spectral sets in ${ℝ}^d$

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

Translational tiles of ${ℝ}^d$

A set $\mathrm{\Omega }\subset {ℝ}^d$ is said to tile ${ℝ}^d$ by translation (i.e. $\mathrm{\Omega }$ is a translational tile) if there exist a discrete set $\mathrm{T}$ such that $\bigcup _{t\in \mathrm{T}}(\mathrm{\Omega }+t)={ℝ}^d$ and the Lebesgue measure of $(\mathrm{\Omega }+t)\cap (\mathrm{\Omega }+t^{\prime })$ is zero for all $t\ne t^{\prime }$ in $\mathrm{T}$.

Partial results

• Fuglede proved in 1974 that the conjecture holds if $\mathrm{\Omega }$ is a fundamental domain of a lattice.
• In 2003, Alex Iosevich, Nets Katz and Terence Tao proved that the conjecture holds if $\mathrm{\Omega }$ is a convex planar domain.
• In 2004, Terence Tao showed that the conjecture is false on ${ℝ}^d$ for $d\ge 5$. 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 ${ℝ}^3$.
• In 2019, Nir Lev and Máté Matolcsi settled the conjecture for convex domains affirmatively in all dimensions.