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Sur les ensembles d'entiers reconnaissables

Fabien Durand — 1998

Journal de théorie des nombres de Bordeaux

Soient U et V deux systèmes de numération de Bertrand, α et β deux β -nombres multiplicativement indépendants tels que L ( U ) = L ( α ) et L ( V ) = L ( β ) , et E un sous-ensemble de . Si E est U -reconnaissable et V -reconnaissable alors E est une réunion finie de progressions arithmétiques.

Cobham's theorem for substitutions

Fabien Durand — 2011

Journal of the European Mathematical Society

The seminal theorem of Cobham has given rise during the last 40 years to a lot of work about non-standard numeration systems and has been extended to many contexts. In this paper, as a result of fifteen years of improvements, we obtain a complete and general version for the so-called substitutive sequences. Let α and β be two multiplicatively independent Perron numbers. Then a sequence x A , where A is a finite alphabet, is both α -substitutive and β -substitutive if and only if x is ultimately periodic....

Ergodic averages with deterministic weights

Fabien DurandDominique Schneider — 2002

Annales de l’institut Fourier

We study the convergence of the ergodic averages 1 N k = 0 N - 1 θ ( k ) f T u k where ( θ ( k ) ) k is a bounded sequence and ( u k ) k a strictly increasing sequence of integers such that Sup α | k = 0 N - 1 θ ( k ) exp ( 2 i π α u k ) | = O ( N δ ) for some δ < 1 . Moreover we give explicit such sequences θ and u and we investigate in particular the case where θ is a q -multiplicative sequence.

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