The paper under review deals with a certain representation, called zeta-expansion, of complex numbers in a base \(\zeta \in \{z\in \mathbb{C} \mid z\notin \mathbb{R}\) and \(\left\vert z\right\vert >1\}.\) This representation can be considered as a complex version of the well-known beta-expansion, and is defined on the fundamental domain \(D:=\{a-b\overline{ \zeta }\mid (a,b)\in \lbrack -\varepsilon ,1-\varepsilon )^{2}\},\) where \( \varepsilon \) is fixed in \([0,1),\) for the lattice \(L:=\{k+l\overline{\zeta } \mid (k,l)\in \mathbb{Z}^{2}\},\) by the digit sequence \[ \mathbf{d}(z):=(\zeta f^{(n-1)}(z)-f^{(n)}(z))_{n\in \mathbb{N}}, \] where \(f^{(0)}(z)=z\in D,\) \(f\) \(^{(1)}(z)=\zeta z(\mathrm{mod}L)\in D\) and \(f\) \(^{(n)}(z)=f(f\) \(^{(n-1)}(z)).\) Then, \[ z=\sum_{_{n\in \mathbb{N}}}\frac{(\zeta f^{(n-1)}(z)-f^{(n)}(z))}{\zeta ^{n}} , \] and the exact shape of the digit set \(\mathcal{N}:=\cup _{z\in D}\mathbf{d} (z),\) obtained in the present manuscript, implies that \(\mathcal{N}\) is a finite subset of \(\mathbb{Z},\) and allows the author to describe the related admissible sequences. In particular, for \(\zeta \) being a multiple of certain roots of unity, these admissible sequences are described by inequalities with respect to the lexicographical order or the alternating order.
Also, the author proves that the condition \(\zeta ^{-1}D\subset D\) is sufficient to represent every complex number, in a unique way. More precisely, he shows that for each \(z\in \mathbb{C}^{\ast }\) there are an integer \(m\) and an admissible sequence \((d_{n})_{n\in \mathbb{N}},\) both uniquely determined, such that \(d_{1}\neq 0\) and \(z=\sum_{_{n\in \mathbb{N} }}d_{n}\) \(\zeta ^{-n+m}.\) Moreover, he establishes the relation with shift radix systems and discusses finiteness and periodicity properties, when \( \zeta \) is an algebraic integer.