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arXiv · 2007.08147

Ultimate periodicity problem for linear numeration systems

E. Charlier·A. Massuir·M. Rigo·E. Rowland
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Abstract

We address the following decision problem. Given a numeration system $U$ and a $U$-recognizable set $X\subseteq\mathbb{N}$, i.e. the set of its greedy $U$-representations is recognized by a finite automaton, decide whether or not $X$ is ultimately periodic. We prove that this problem is decidable for a large class of numeration systems built on linearly recurrent sequences. Based on arithmetical considerations about the recurrence equation and on $p$-adic methods, the DFA given as input provides a bound on the admissible periods to test.

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BibTeXRIS

E. Charlier, A. Massuir, M. Rigo, E. Rowland. 2020-07-16. Ultimate periodicity problem for linear numeration systems. https://doi.org/10.1142/s0218196722500254

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