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Tiling and spectral properties of near-cubic domains

Mihail N. Kolountzakis, Izabella Łaba (2004)

Studia Mathematica

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.

Tilings associated with non-Pisot matrices

Maki Furukado, Shunji Ito, E. Arthur Robinson (2006)

Annales de l’institut Fourier

Suppose A G l d ( ) has a 2-dimensional expanding subspace E u , satisfies a regularity condition, called “good star”, and has A * 0 , where A * is an oriented compound of A . A morphism θ of the free group on { 1 , 2 , , d } is called a non-abelianization of A if it has structure matrix A . We show that there is a tiling substitution Θ whose “boundary substitution” θ = Θ is a non-abelianization of A . Such a tiling substitution Θ leads to a self-affine tiling of E u 2 with A u : = A | E u G L 2 ( ) as its expansion. In the last section we find conditions on A so...

Tilings of convex polygons

Richard Kenyon (1997)

Annales de l'institut Fourier

Call a polygon rational if every pair of side lengths has rational ratio. We show that a convex polygon can be tiled with rational polygons if and only if it is itself rational. Furthermore we give a necessary condition for an arbitrary polygon to be tileable with rational polygons: we associate to any polygon P a quadratic form q ( P ) , which must be positive semidefinite if P is tileable with rational polygons.The above results also hold replacing the rationality condition with the following: a polygon...

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