Definition
Fuglede's conjecture is a statement in harmonic analysis and geometric measure theory concerning the relationship between two properties of a measurable set $ \Omega \subset \mathbb{R}^n $ of finite Lebesgue measure:
-
Spectral property – $ \Omega $ is called spectral if there exists a set $ \Lambda \subset \mathbb{R}^n $ such that the collection of exponential functions
$$ { e^{2\pi i \langle \lambda, x\rangle} : \lambda \in \Lambda } $$ forms an orthogonal basis for the Hilbert space $ L^{2}(\Omega) $. -
Tiling property – $ \Omega $ tiles $ \mathbb{R}^n $ by translations if there exists a set $ T \subset \mathbb{R}^n $ such that the translates $ { \Omega + t : t \in T } $ are pairwise disjoint (up to measure zero) and their union covers $ \mathbb{R}^n $ (again up to measure zero).
The conjecture, formulated by Bent Fuglede in 1974, posits that for any such set $ \Omega $ the spectral property is equivalent to the tiling property:
$$ \Omega \text{ is spectral } \iff \Omega \text{ tiles } \mathbb{R}^n \text{ by translations}. $$
Historical development
- 1974 – Bent Fuglede publishes the conjecture in Journal of Functional Analysis.
- 1990s – Partial positive results are obtained for special classes of sets (e.g., convex bodies, lattice tiles) and for low dimensions.
- 2004 – Terence Tao constructs a counterexample in dimension $ n \ge 5 $, disproving the conjecture in those dimensions.
- 2005–2009 – Subsequent work (by Kolountzakis, Matolcsi, and others) extends counterexamples to dimensions $ n = 3 $ and $ n = 4 $. Consequently, the conjecture is known to be false for all dimensions $ n \ge 3 $.
- Present – The conjecture remains open in dimensions $ n = 1 $ and $ n = 2 $. Various partial results are known, for instance:
- In one dimension, a set of finite measure is spectral if and only if it is a finite union of intervals of equal length; such sets also tile by translation, confirming the conjecture for this class.
- For certain convex planar domains (e.g., the disc, regular polygons), the spectral‑tiling equivalence has been verified, but a general proof or counterexample in $ \mathbb{R}^2 $ is lacking.
Mathematical significance
Fuglede's conjecture connects harmonic analysis (spectral sets) with geometric tiling theory, linking properties of Fourier bases to combinatorial tiling structures. Its resolution in various dimensions has driven the development of techniques in:
- Fourier analysis on finite and infinite groups,
- Additive combinatorics (e.g., cyclotomic integer sets, Hadamard matrices),
- Geometry of numbers and convex geometry.
The conjecture’s failure in higher dimensions illustrates the subtle interplay between algebraic and geometric constraints in Euclidean space.
Current status
| Dimension | Status |
|---|---|
| $ n \ge 3 $ | Disproved – explicit counterexamples exist. |
| $ n = 2 $ | Open – no known counterexample nor general proof. |
| $ n = 1 $ | Open in full generality; the conjecture holds for many natural subclasses (e.g., finite unions of equal‑length intervals). |
Research continues on refining the classification of spectral sets, identifying additional classes of tiling sets, and exploring analogues of the conjecture on other groups (e.g., finite abelian groups, locally compact abelian groups).