Concentration of measure for uniform distribution on sphere

Question

I am looking for a (concentration-type) inequality for the following probability: if $\theta$ is distributed according to the uniform measure on the sphere $S^{d-1}$, then does there exist a bound for $$P\left(|\theta \cdot x| \leq t\|x\|, \hspace{2mm} \forall x \in B \right),$$ where $B$ is some set in $R^d$. I know that for a fixed point, the above probability is bounded like $\leq t\sqrt{d}$, but what could I do for the general case? In my particular case, I have control on the length of the interval $|\theta \cdot x|$ lies in i.e. w.h.p, $\forall x\in B$, $|\theta \cdot x| \in [-\delta, \delta]$.