Let $(L,q)$ be a non-degenerate quadratic lattice of signature $(n_{+},n_{-})$ and dimension $d\geq 5$. Consider the quadric $C_{k}=\{x\in L_{\mathbb{R}},\, q(x)=k \}$ for some non-zero integer $k$.

Several results are known about the distribution of $L$-points on caps on $C_{k}$, that is, intersection of $C_{k}$ with a ball centered on a point of $C_{k}$. For instance, if the lattice is definite, $C_{k}$ is an ellipsoid $(k>0)$ and the question has been adressed here.

I would like to know if there is such asymatotics in the case where $(L,q)$ is an indefinite lattice. Namley, what is the behaviour of the following function, with respect to $|k|$ and $\epsilon$:
$$F(\lambda_{0},k,\epsilon)=|\{x\in L,\, q(x)=k, \, |x-\lambda_{0}|<\epsilon\}|$$
where $\lambda_{0}$ is some point on $C_{k}$.

An account of known results would be also very helpful.