We present an elementary three-pass algorithm for computing addition in Ostrowski numeration systems. When is quadratic, addition in the Ostrowski numeration system based on is recognizable by a finite automaton. We deduce that a subset of is definable in , where is the function that maps a natural number to the smallest denominator of a convergent of that appears in the Ostrowski representation based on of with a nonzero coefficient if and only if the set of Ostrowski representations of elements of is recognizable by a finite automaton. The decidability of the theory of follows.
"Ostrowski Numeration Systems, Addition, and Finite Automata." Notre Dame J. Formal Logic 59 (2) 215 - 232, 2018. https://doi.org/10.1215/00294527-2017-0027