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.
Translational tilings of the integers with long periods.
Filling a box with translates of two bricks.
Reconstruction of functions from their triple correlations.
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