$x$.

Let $\mathbb{Z}_b$ denote the compact group of the $b$-adic integers. We refer the reader to [@Hewitt79a; @Mahler81a] for details. An element $z$ of $\mathbb{Z}_b$ will be written as a formal sum $z=\sum_{j\ge 0} z_j\, b^j,$ with digits $z_j\in \{0,1,\ldots, b-1  \}$. The set $\mathbb{Z}$ of integers is embedded in $\mathbb{Z}_b$. If $z\in \mathbb{N}_0$, then at most finitely many digits $z_j$ are different from 0. If $z\in \mathbb{Z}$, $z< 0$, then at most finitely many digits $z_j$ are different from $b-1$. In particular, $-1 = \sum_{j\ge 0} (b-1)\, b^j.$

We recall the following concepts from [@Hel10a; @Hel10c; @Hel11a].

The map $\varphi_b: \mathbb{Z}_b \rightarrow [0,1)$, given by $\varphi_b(\sum_{j\ge 0} z_j\, b^j) = \sum_{j\ge 0} z_j\, b^{-j-1}\pmod{1}$, will be called the [*$b$-adic Monna map*]{}.

The restriction of $\varphi_b$ to $\mathbb{N}_0$ is often called the [*radical-inverse function*]{} in base $b$. The Monna map is surjective, but not injective. It may be inverted in the following sense.

We define the [*pseudoinverse*]{} $\varphi^+_b$ of the $b$-adic Monna map $\varphi_b$ by $$\varphi^+_b: [0,1) \rightarrow \mathbb{Z}_b, \quad
\varphi^+_b(\sum_{j\ge 0} x_j\, b^{-j-1}) = \sum_{j\ge 0} x_j\, b^{j}\;,$$ where $\sum_{j\ge 0} x_j\, b^{-j-1}$ stands for the regular $b$-adic representation of the element $x\in [0,1)$.

The image of $[0,1)$ under 