Submission date: 25 July 2014.

Abstract:

Addition in the Ostrowski numeration system based on a quadratic number a is recognizable by a finite automaton. We deduce that a subset of X\subseteq N^n is definable in (N,+,V_a), where V_a is the function that maps a natural number x to the smallest denominator of a convergent of a that appears in the Ostrowski representation based on a of x with a non-zero coefficient, if and only if the set of Ostrowski representations of elements of X is recognizable by a finite automaton. The decidability of the theory of (N,+,V_a) follows.

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Full text arXiv 1407.7000: pdf, ps.