Fuglede's conjecture

Fuglede's conjecture asserts that a measurable set in Euclidean space tiles by translations if and only if its indicator function admits an orthogonal basis of exponential functions.

Connect

Spectral set: A measurable set whose square-integrable functions have an orthogonal basis of exponentials. This names the analytic property paired with tiling in the conjecture.
Fourier transform: A transform that represents a function through its frequency components. It converts inner products of exponentials on a set into values of the set's Fourier transform.
Børge Jessen: A Danish mathematician whose work included the geometry of numbers and harmonic analysis. Fuglede credited Jessen with inspiring the question linking tilings and spectra.
Unit interval: The interval [0,1] in the real line. Its integer translates tile the line, and the integers form an exponential spectrum.
Spectral implies tile: The implication that every spectral set tiles Euclidean space by translations. This direction fails in sufficiently high dimensions, independently of the reverse implication.
Translation tiling: A covering of space by translated copies of a set, with overlaps only on sets of measure zero. This is the geometric property the conjecture equates with spectrality.
Fourier transform of an indicator function: The Fourier transform of a function equal to one on a set and zero outside it. Its zero set encodes orthogonality between distinct frequencies on the set.
Bent Fuglede: A Danish mathematician known for work in potential theory and harmonic analysis. He formulated the conjecture in 1974.
Cube (geometry): A three-dimensional solid with six congruent square faces; higher-dimensional analogues have equal coordinate side lengths. Cubes provide standard examples that both tile by translations and admit exponential bases.
Tile implies spectral: The implication that every measurable translational tile has an orthogonal exponential basis. This direction also fails in sufficiently high dimensions.