i\le s$. $\mathbb{N}$ represents the positive integers, and we put $\mathbb{N}_0=\mathbb{N}\cup \{0\}.$

The underlying space is the $s$-dimensional torus $\mathbb{R}^s/\mathbb{Z}^s$, which will be identified with the half-open interval $[0,1)^s$. Haar measure on the $s$-torus $[0,1)^s$ will be denoted by $\lambda_s$. We put $e(y)={\ensuremath{\mathrm{e}}}^{2\pi \mathrm{i} y}$ for $y\in\mathbb{R}$, where $\mathrm{i}$ is the imaginary unit.

We will use the standard convention that empty sums have the value 0 and empty products value 1.

For a nonnegative integer $k$, let $k =\sum_{j\ge0} k_j\, b^j, k_j \in \{0,1,\ldots ,b-1\},$ be the unique $b$-adic representation of $k$ in base $b$. With the exception of at most finitely many indices $j$, the digits $k_j$ are equal to 0.

Every real number $x\in [0,1)$ has a representation in base $b$ of the form $x = \sum_{j\ge 0} x_j \,b^{-j-1},$ with digits $x_j\in \{0,1, \ldots, b-1\}.$ If $x$ is a [*$b$-adic rational*]{}, which means that $x=ab^{-g}$, $a$ and $g$ integers, $0\le a< b^g$, $g\in \mathbb{N}$, and if $x\ne 0$, then there exist two such representations. The $b$-adic representation of $x$ is uniquely determined under the condition that $x_j \ne b-1$ for infinitely many $j$. In the following, we will call this particular representation the [*regular*]{} ($b$-adic) representation of 