I've just started to read about additive combinatorics and I'd like to know how I can use Bohr sets to make a statement about arithmetic progressions in a given subset $A$ of an Abelian group $Z$ (the ambient group).
For example: Green-Tao states that if $A$ is the set of all primes then we can always find an arithmetic progression of length $k$ in $A$ for all $k \in \mathbb N$.
Can I do the following: If $B(S, \rho)$ is a Bohr set with certain properties then it contains an arithmetic progression of length say, $k=10$?
If yes: how exactly does it work?
If no: why not?
A question directly related to this is: why were Bohr sets introduced? I've read that they replace the orthogonal complement $S^\bot$ of a subset $S$ of $Z$ when $S^\bot$ is not a subgroup of $Z$ because for example $Z = Z_p$ for $p$ prime.
And one more question is: if $F$ is a linear space then every linear subspace is a Bohr set (Tao/Vu, p. 166). So for some reason we want to study linear subspaces of $Z$ or subgroups if possible. How do we use this structure or in particular the closedness with respect to addition of subsets? I'm slightly confused about whether we're interested in subsets or subspaces of $Z$.
Edit
Let $S \subset \widehat{Z}$ be a set of characters of $Z$ and let $\rho > 0$ be a real number. Then a Bohr set is defined as $$ \operatorname{Bohr}(S, \rho) := \{ z \in Z \mid \sup_{\chi \in S} \left | \chi(z) - 1 \right | < \rho \}$$
Thanks for your help.