Syntactic Complexity of Ultimately Periodic Sets of Integers and Application to a Decision Procedure

We compute the cardinality of the syntactic monoid of the language 0a repb(m$\mathbb{N}$) made of base b expansions of the multiples of the integer m. We also give lower bounds for the syntactic complexity of any (ultimately) periodic set of integers written in base b. We apply our results to a well studied problem: decide whether or not a b-recognizable set of integers is ultimately periodic.

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