Comptes Rendus
no. 1
Sous-ensembles homogènes de ℤ2 et pavages du plan
Maurice Nivat1
1 LIAFA CNRS UMR 7089, Université Paris 7, case 7014, 2, place Jussieu, 75251 Paris cedex 05, France
Comptes Rendus. Mathématique, Volume 335 (2002) no. 1, pp. 83-86.

Nous appelons sous-ensemble homogène de degré k pour F du plan discret ℤ2 tout sous-ensemble tel qu'à travers toutes les positions possibles d'une fenêtre finie que l'on translate apparait toujours le même nombre k de points de A. Nous montrons deux propriétés, il existe un sous-ensemble homogène de degré 1 pour F si et seulement si F pave le plan par translation. Si la fenêtre est rectangulaire tout sous-ensemble homogène de degré k pour F est l'union disjointe de k sous-ensembles homogènes de degré 1 pour F.

We say that the subset A of the discrete plane ℤ2 is k-homogeneous for F if and only if whichever is the position of a finite window F which we translate over ℤ2 the same number k of points of A appears in the window. And we prove two properties. There exists a 1-homogeneous subset for F if and only if F tiles the plane by translation. If the window is a rectangle every k-homogeneous subset is the disjoint union of k 1-homogeneous subset.

Reçu le :
Accepté le :
Publié le :
DOI : 10.1016/S1631-073X(02)02377-4

Maurice Nivat 1

1 LIAFA CNRS UMR 7089, Université Paris 7, case 7014, 2, place Jussieu, 75251 Paris cedex 05, France 
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Maurice Nivat. Sous-ensembles homogènes de $ \mathbb{Z}^{2}$ et pavages du plan. Comptes Rendus. Mathématique, Volume 335 (2002) no. 1, pp. 83-86. doi : 10.1016/S1631-073X(02)02377-4. https://comptes-rendus.academie-sciences.fr/mathematique/articles/10.1016/S1631-073X(02)02377-4/

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[2] R. Tijdeman Decomposition of the integers as a direct sum of two subsets (S. David, ed.), Number Theory, Number Theory Seminar, Paris, 1992–1993, Cambridge University Press, 1995, pp. 261-276

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