  We introduce a complete axiomatization for the structure $Z_{\alpha}=(Z, +,
0, 1, f)$ where $f : x \to \lfloor{\alpha} x \rfloor $ is a unary function with
${\alpha}$ a fixed transcendental number. This result fits into the more
general theme of adding traces of multiplication to integers without losing
decidability. When ${\alpha}$ is either a quadratic number or an irrational
number less than one, similar decidability results have been already obtained
by applying heavy techniques from the theory of autamota. Nevertheless, our
approach is based on a clear axiomatization and involves only elementary
techniques from model theory.

Logic