Fuglede's conjecture (Fuglede 1974) asserts that a measurable set 
with finite positive Lebesgue measure admits
a spectrum iff it is possible
to tile 
by a family of translates of 
. Fuglede proved the conjecture in the special case that
the tiling set or the spectrum are lattice subsets of 
and Iosevich et al. (2001) proved that no smooth
symmetric convex body 
with at least one point of nonvanishing Gaussian
curvature can admit an orthogonal basis of exponentials.
Using complex Hadamard matrices of orders 6 and 12, Tao (2003) constructed counterexamples to the conjecture in some small Abelian
groups, and lifted these to counterexamples in 
or 
.
However, the conjecture has been proved in a great number of special cases (e.g., all convex planar bodies) and remains an open problem in small dimensions. For example,
it has been shown in dimension 1 that a nice algebraic characterization of finite
sets tiling 
indeed implies one side of Fuglede's conjecture (Coven and Meyerowitz 1999). Furthermore,
it is sufficient to prove these conditions when the tiling gives a factorization
of a non-Hajós cyclic group (Amiot 2005).
The finite-group version uses restrictions of group characters in place of complex exponentials. Kadowaki (2026) proved that a nonempty
subset 
of 
admits an orthogonal basis
of characters iff it tiles 
by translations. Kadowaki
(2026) also proved that every 16-element subset of a binary
vector space admitting such a basis tiles that vector space by translations.
Here a subset is spectral when it admits the orthogonal basis of characters just specified. Ferguson and Iosevich (2026) extended the
finite-field results to 
and 
: a nonempty subset is spectral
iff it tiles the ambient vector
space by translations. In 
this is also equivalent to having a tiling complement
that is a subspace. In 
they proved the tiling-to-spectral implication, but the
converse remains unresolved for spectral sets of size 64.
Ferguson and Iosevich (2026) credit ChatGPT with assistance writing the computational
verification programs.
for 
and 
." 22 Oct 2026. 
." 16 Oct 2026.