11,078 research outputs found

    Generalization of automatic sequences for numeration systems on a regular language

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    Let L be an infinite regular language on a totally ordered alphabet (A,<). Feeding a finite deterministic automaton (with output) with the words of L enumerated lexicographically with respect to < leads to an infinite sequence over the output alphabet of the automaton. This process generalizes the concept of k-automatic sequence for abstract numeration systems on a regular language (instead of systems in base k). Here, I study the first properties of these sequences and their relations with numeration systems.Comment: 10 pages, 3 figure

    Dynamical Directions in Numeration

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    International audienceWe survey definitions and properties of numeration from a dynamical point of view. That is we focuse on numeration systems, their associated compactifications, and the dynamical systems that can be naturally defined on them. The exposition is unified by the notion of fibred numeration system. A lot of examples are discussed. Various numerations on natural, integral, real or complex numbers are presented with a special attention payed to beta-numeration and its generalisations, abstract numeration systems and shift radix systems. A section of applications ends the paper

    Abstract numeration systems on bounded languages and multiplication by a constant

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    A set of integers is SS-recognizable in an abstract numeration system SS if the language made up of the representations of its elements is accepted by a finite automaton. For abstract numeration systems built over bounded languages with at least three letters, we show that multiplication by an integer λ≥2\lambda\ge2 does not preserve SS-recognizability, meaning that there always exists a SS-recognizable set XX such that λX\lambda X is not SS-recognizable. The main tool is a bijection between the representation of an integer over a bounded language and its decomposition as a sum of binomial coefficients with certain properties, the so-called combinatorial numeration system

    Syndeticity and independent substitutions

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    We associate in a canonical way a substitution to any abstract numeration system built on a regular language. In relationship with the growth order of the letters, we define the notion of two independent substitutions. Our main result is the following. If a sequence xx is generated by two independent substitutions, at least one being of exponential growth, then the factors of xx appearing infinitely often in xx appear with bounded gaps. As an application, we derive an analogue of Cobham's theorem for two independent substitutions (or abstract numeration systems) one with polynomial growth, the other being exponential

    Sturmian numeration systems and decompositions to palindromes

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    We extend the classical Ostrowski numeration systems, closely related to Sturmian words, by allowing a wider range of coefficients, so that possible representations of a number nn better reflect the structure of the associated Sturmian word. In particular, this extended numeration system helps to catch occurrences of palindromes in a characteristic Sturmian word and thus to prove for Sturmian words the following conjecture stated in 2013 by Puzynina, Zamboni and the author: If a word is not periodic, then for every Q>0Q>0 it has a prefix which cannot be decomposed to a concatenation of at most QQ palindromes.Comment: Submitted to European Journal of Combinatoric
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