Consider the polynomial ring $\mathbb{Z}\left[x\right]$ in the indeterminate x. Fix an integer $b\geq2$. Applying the evaluation map $x\mapsto \frac{1}{b}$ then gives us the “$b$-adic” rationals; apologies for the bad name—I do not mean, $\mathbb{Q}_{b}$, but rather, the set of all rational numbers whose denominators are non-negative integer powers of $b$. For instance, $\mathbb{Z}\left[\frac{1}{2}\right]$ is the set of dyadic rationals.
The wikipedia article on the dyadic rationals gives a helpful characterization of the Pontryagin dual of $\mathbb{Z}\left[\frac{1}{2}\right]$ (denoted $\widehat{\mathbb{Z}\left[\frac{1}{2}\right]}$ ) as the group of all infinite sequences $\mathbf{s}=\left\{ s_{n}\right\} _{n\geq0}\subseteq\partial\mathbb{D}$ of unimodular complex numbers satisfying $s_{n}^{2}=s_{n-1}$ for all $n\geq1$. This dual is called the dyadic solenoid. The group operation on the solenoid is given by $$\mathbf{s}\mathbf{s}^{\prime}=\left\{ s_{n}s_{n}^{\prime}\right\} _{n\geq0}$$ Now, my questions.
(1) Am I correct in assuming that $\widehat{\mathbb{Z}\left[\frac{1}{b}\right]}$ is the set of all $\mathbf{s}$ whose terms satisfy $s_{n}^{b}=s_{n-1}$?
(2) Since $\mathbb{Z}\left[\frac{1}{b}\right]$ is a discrete group, its Haar measure is simply the counting measure. What is the “standard” (since there is no canonical normalization) Haar measure of $\widehat{\mathbb{Z}\left[\frac{1}{b}\right]}$? Is there an elementary family of borel sets in $\widehat{\mathbb{Z}\left[\frac{1}{b}\right]}$ whose Haar measure can be explicitly computed (compare to the $p$-adic integers (with $\mu\left(\mathbb{Z}_{p}\right)=1$), in which case $\mu\left(p^{k}\mathbb{Z}_{p}\right)=p^{-k}$)?
(3) More generally, is there any theorem or other useful result that describes how the Haar measures of the additive group-structure of a given ring $R$ (as well as the Haar measure of the dual of said additive group) change when one moves from $R$ to a ring extension of $R$ (finite or infinite)? That is, If you know the Haar measures $\mu$ and $\hat{\mu}$ of the groups $\left(R,+\right)$ and $\widehat{\left(R,+\right)}$ and if $S$ is a ring extension of $R$, do the characterizations of $\mu$ and $\hat{\mu}$ then similarly “extend” to characterizations of the Haar measures $\nu$ and $\hat{\nu}$ of the groups $\left(S,+\right)$ and $\widehat{\left(S,+\right)}$. An explanation or a reference would be most appreciated.
(4) My current work deals with concrete instances of abstract harmonic analysis—that is, doing harmonic analysis on specific locally compact abelian groups—and as such, requires doing explicit calculations of fourier transforms and what-not. Are there any good references which contain (preferably rather extensive) lists/tables with explicit descriptions of their character groups, their Pontryagin duality brackets, their Haar measures, and the like? Doing it by hand every time I come across a new group to work with is extraordinarily tedious.
