$\varphi^+_b$ is the set $\mathbb{Z}_b\setminus (-\mathbb{N})$. Furthermore, $\varphi_b\circ \varphi^+_b$ is the identity map on $[0,1)$, and $\varphi^+_b\circ \varphi_b$ the identity on $\mathbb{N}_0 \subset \mathbb{Z}_b$. In general, $z\neq \varphi^+_b(\varphi_b(z))$, for $z\in \mathbb{Z}_b$. For example, if $z=-1$, then $\varphi^+_b(\varphi_b(-1))=\varphi^+_b(0)=0\neq -1$.

It has been shown in [@Hel11a] that the dual group $\hat{\mathbb{Z}}_b$ can be written in the form $\hat{\mathbb{Z}}_b= \{\chi_k: k\in \mathbb{N}_0 \},$ where $\chi_k: \mathbb{Z}_b\rightarrow \{ c\in \mathbb{C}: |c|=1 \}$, $\chi_k( \sum_{j\ge 0} z_j b^j ) =     e(\varphi_b(k)(z_0+z_1b+\cdots))$. We note that $\chi_k$ depends only on a finite number of digits of $z$ and, hence, this function is well defined.

As in [@Hel10a], we employ the function $\varphi_b^+$ to lift the characters $\chi_k$ to the torus.

For $k\in {\ensuremath{\mathbb{N}}}_0$, let $\gamma_k: [0,1) \rightarrow \{ c\in \mathbb{C}: |c|=1 \}$, $\gamma_k(x)=     \chi_k(\varphi^+_b(x))$, denote the $k$th $b$-adic function. We put $\Gamma_b=\{\gamma_k:  k\in \mathbb{N}_0  \}$ and call it the $b$-adic function system on $[0,1)$.

There is an obvious generalization of the preceding notions to the higher-dimensional case. Let $\mathbf{b}=(b_1, \ldots, b_s)$ be a vector of not necessarily distinct integers $b_i\ge 2$, let ${\mathbf{x}}= (x_1, \ldots, x_s)\in [0,1)^s$, let $\mathbf{z}=(z_1, \ldots, z_s)$ denote an element of the compact product group $\mathbb{Z}_\mathbf{b}= \mathbb{Z}_{b_1}\times \cdots \times \mathbb{Z}_{b_s}$ of $\mathbf{b}$-adic integers, 