HAL : hal-00626725, version 2

arXiv : 1109.5801

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v1 (27-09-2011)

v2 (03-08-2012)

Multidimensional extension of the Morse--Hedlund theorem

Fabien Durand 1, Michel Rigo 2

(03/08/2012)

A celebrated result of Morse and Hedlund, stated in 1938, asserts that a sequence $x$ over a finite alphabet is ultimately periodic if and only if, for some $n$, the number of different factors of length $n$ appearing in $x$ is less than $n+1$. Attempts to extend this fundamental result, for example, to higher dimensions, have been considered during the last fifteen years. Let $d\ge 2$. A legitimate extension to a multidimensional setting of the notion of periodicity is to consider sets of $\ZZ^d$ definable by a first order formula in the Presburger arithmetic $\langle\ZZ;<,+\rangle$. With this latter notion and using a powerful criterion due to Muchnik, we exhibit a complete extension of the Morse--Hedlund theorem to an arbitrary dimension $d$ and characterize sets of $\ZZ^d$ definable in $\langle\ZZ;<,+\rangle$ in terms of some functions counting recurrent blocks, that is, blocks occurring infinitely often.

1 :	Laboratoire Amiénois de Mathématique Fondamentale et Appliquée (LAMFA)
	CNRS : UMR6140 – Université de Picardie Jules Verne
2 :	Département de Mathématique
	Université de Liège

Domaine

:

Mathématiques/Combinatoire Mathématiques/Logique Informatique/Mathématique discrète

Mots Clés : tilings – block complexity – periodicity – Presburger arithmetic – Muchnik's criterion – Nivat's conjecture – definable set

hal-00626725, version 2

http://hal.archives-ouvertes.fr/hal-00626725

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Contributeur : Fabien Durand <>

Soumis le : Vendredi 3 Août 2012, 11:56:23

Dernière modification le : Vendredi 3 Août 2012, 14:10:01