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Comptes Rendus Mathématique
Volume 335, n° 1
pages 83-86 (2002)
Doi : S1631-073X(02)02377-4
Received : 11 February 2002 ;  accepted : 7 Mars 2002
Sous-ensembles homogènes de Z2 et pavages du plan
Homogeneous subsets of Z2 and plane tilings
 

Maurice Nivat
LIAFA CNRS UMR 7089, Université Paris 7, case 7014, 2, place Jussieu, 75251 Paris cedex 05, France 

Note présentée par Maurice Nivat

Résumé

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 . Pour citer cet article : M. Nivat, C. R. Acad. Sci. Paris, Ser. I 335 (2002) 83-86.

The full text of this article is available in PDF format.
Abstract

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. To cite this article: M. Nivat, C. R. Acad. Sci. Paris, Ser. I 335 (2002) 83-86.

The full text of this article is available in PDF format.


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