Translational tilings of the integers with long periods.
Filling a box with translates of two bricks.
Reconstruction of functions from their triple correlations.
On the additive complements of the primes and sets of similar growth
Tiling and spectral properties of near-cubic domains
We prove that if a measurable domain tiles ℝ or ℝ² by translations, and if it is "close enough" to a line segment or a square respectively, then it admits a lattice tiling. We also prove a similar result for spectral sets in dimension 1, and give an example showing that there is no analogue of the tiling result in dimensions 3 and higher.
Complex Hadamard matrices and the spectral set conjecture.
By analyzing the connection between complex Hadamard matrices and spectral sets, we prove the direction "spectral ⇒ tile" of the Spectral Set Conjecture, for all sets A of size |A| ≤ 5, in any finite Abelian group. This result is then extended to the infinite grid Z for any dimension d, and finally to R.
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