Model-completeness and decidability of the additive structure of integers expanded with a function for a Beatty sequence

Abstract

We introduce a model-complete theory which completely axiomatizes the structure Zα=〈Z,+,0,1,f〉 where f:x↦⌊αx⌋ is a unary function with α a fixed transcendental number. When α is computable, our theory is recursively enumerable, and hence decidable as a result of completeness. Therefore, this result fits into the more general theme of adding traces of multiplication to integers without losing decidability.