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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.

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This is Theorem 1.10 of Sardari (2017). Let $d\geq 5$, and for simplicity let $L=\mathbb{Z}^d$.

Define the ball $B_0=\{x:|x-\lambda_0|<\epsilon\}$ and the normalised ball $B = |k|^{-1/2}B$. Provided that (1) the normalised ball $B$ is contained in some fixed compact set $\Omega$, and (2) the ball $B_0$ has a radius $\epsilon\gg |k|^{\frac{1}{4}+\delta}$ for some fixed $\delta>0$, he proves $$ F(\lambda_0,k,\epsilon) = \mathfrak{S}_\infty(\lambda_0) \Big(\prod_{p\text{ prime}}\mathfrak{S}_p(N)\Big)\epsilon^{d-1}|k|^{-\frac{1}{2}}\big(1+O_{q,\Omega,\delta}(|k|^{-\delta'})\big) $$ for some $\delta'>0$. Provided $q(x)=k$ is soluble everywhere locally, $\Big(\prod_{p\text{ prime}}\mathfrak{S}_p(N)\Big)$ is bounded above and below by constants depending only on $q$. Also $ \mathfrak{S}_\infty(\lambda_0)$ is bounded above and below by constants depending only on $q$ and $\Omega$. (See Remark 1.4 in the same paper)

The exponent in condition (2) is optimal, see the second part of Theorem 1.10. It is possible that condition (1) is included to keep things manageable, rather than because it is essential to the proof. If so, then the same proof might show that as soon as $\epsilon\gg |\lambda_0|^{\frac{1}{2}+\delta}$ for some fixed $\delta>0$ we have $$ F(\lambda_0,k,\epsilon) =\overline{\mathfrak{S}}_\infty(\lambda_0) \Big(\prod_{p\text{ prime}}\mathfrak{S}_p(N)\Big)\epsilon^{d-1}|\lambda_0|^{-1}\big(1+O_{q,\Omega,\delta}(|\lambda_0|^{-\delta'})\big) $$ where the modified constant $\overline{\mathfrak{S}}_\infty(\lambda_0)$ is still bounded above and below by constants depending only on $q$. I would need to read through the proof in more detail to be confident of this last part.

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