A family $H$ is $(r,cr,p_1,p_2)$-sensitive if for all $x,y \in \mathbb{R}^d$ we have:
$\lVert x-y\rVert <r\quad \Rightarrow\quad \Pr[h(x)=h(y)] \geq p_1$, and
$\lVert x-y\rVert > cr \quad \Rightarrow\quad \Pr[h(x)=h(y)] \leq p_2 $
We can amplify the gap between $p_1,p_2$ by using concatenation: $g(x) = h_1(x)h_2(x)\cdots h_k(x)$ and $L$ tabels, so we have $g_1,\dots,g_L$
For each $\delta > 0$ we can choose $k,L$ s.t. for a query point $q$, if $\lVert x-q\rVert <r$ then $\Pr[\exists i \: \: s.t. \: g_i(x)=g_i(q)]>1-\delta$
My question is: what guarantees do I have for the "bad" points? Given a query point $q$, what is the ratio between good points (i.e., $r$-close or $cr$-close points) and the bad points (far points) in the bucket $g_i(q)$. Can I bound it for some (carefully chosen) $k,L$ ?