Fix a set $ \Gamma\subset \mathbb F_p$, the field with $p$ elements and a parameter $\epsilon>0$. The Bohr set $B(\Gamma,\epsilon)$ consists of those $x$ for which $x\cdot \Gamma\subseteq[-\epsilon p,\epsilon p]$ - I am identifying the interval with its image mod $p$. The standard lower bound for $B(\Gamma,\epsilon)$ is then $\epsilon^{|\Gamma|}p$ (which is given by the pigeon-hole principle) and this is in fact what one expects (more or less since one expects the events $x\gamma\in[-\epsilon p,\epsilon p]$ with $\gamma\in\Gamma$ to be independent trials). Since $\Gamma\subset B(B(\Gamma,\epsilon),\epsilon)$, this lower bound can miss the mark. What I want to know is if there is something about $\Gamma$ that will tell me it should have the expected size.

EDIT: I have figured out some estimates in the case that $\Gamma$ is dissociated, see below. I am wondering if one can do better in this case, or if one can be more general.