When does a Bohr set have the right size?

Question

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.

Answer 1

To some extent one can obtain upper bounds on $B(\Gamma,\epsilon)$ when $\Gamma$ is quite special. For instance, when $\Gamma$ is dissociated (which is to say, the sums $\sum_{s\in S}s$ are distinct for each $S\subset \Gamma$) then one can say something. Indeed, any dissociated set of size $d$ cannot be contained in an interval of size $\epsilon p$ if $2^d\geq d\epsilon p$ which shows that if $\epsilon< 2^d/dp$, then $B(\Gamma,\epsilon)$ consists only of the zero element. From this one can obtain bounds for larger $\epsilon$ using $|B(\Gamma,2\epsilon)|\leq 4^d|B(\Gamma,\epsilon)|$. One could also obtain bounds when $\Gamma$ is dissociated by appealing to Rudin's inequality.