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arXiv · 0706.0431

Abstract numeration systems on bounded languages and multiplication by a constant

Emilie Charlier·Michel Rigo·Wolfgang Steiner
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Abstract

A set of integers is $S$-recognizable in an abstract numeration system $S$ 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 $λ\ge2$ does not preserve $S$-recognizability, meaning that there always exists a $S$-recognizable set $X$ such that $λX$ is not $S$-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.

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Emilie Charlier, Michel Rigo, Wolfgang Steiner. 2008-09-16. Abstract numeration systems on bounded languages and multiplication by a constant. https://arxiv.org/abs/0706.0431

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