When regarding a ternary quadratic form $Q(x,y,z)$, is a classic question to consider which integers $n$ can be represented by $Q$. It is also classic to wonder how "well distributed" are the lattice points $(x,y,z)\in\mathbb{Z}^3$ that satisfies $Q(x,y,z)=n$ when $n\rightarrow\infty$ along a certain sequence.


When $Q$ is just the modulus square, I have found a lot of literature but I am interested in more general $Q$. On chapter 11 of Topics in Classical
Automorphic Forms by Henryk Iwaniec, we find conditions to $n$ so that this condition holds. However, Iwaniec said that the result is due to Duke and that he asked a few less on $n$.


I have tried to read Duke papers:


Representation of integers
by positive ternary quadratic forms and
equidistribution of lattice points on ellipsoids


And


On ternary quadratic forms,


But I couldn't find the conditions he imposed over $n$ so that we have equidistribution. I am interested in asking the less possible to $n$ and knowing the difference between the hypothesis of Duke and those of Iwaniec.


Thank you to everyone.



Duke and Schulze-Pillot condition for equidistribution
I recommend that you study the paper by Duke and Schulze-Pillot, Representation of integers by positive ternary quadratic forms and equidistribution of lattice points on ellipsoids (Inventiones, 1990).


Theorem 1 shows that if the number of representations $r(n,Q)$ exceeds $n^{1/2-1/176}$, then the representations become equidistributed on the relevant ellipsoid as $n\to\infty$. So the question is how to guarantee that $r(n,Q)$ is large. By Theorem 3 (see also the subsequent Corollary), it suffices that $n$ is primitively represented by some form in the spinor genus of $Q$. By Theorem 2, if we exclude finitely many explicitly computable square classes for $n$, the condition simplifies to: $n$ is primitively represented by some form in the genus of $Q$. The latter is equivalent to: $n$ is primitively represented by $Q$ modulo $4\det(a\_{ij})$, where $(a\_{ij})\in\mathrm{M}\_3(\mathbb{Z})$ is the symmetric matrix such that
$$Q(x\_1,x\_2,x\_3)=\frac{1}{2}\sum\_{i,j}a\_{ij}x\_ix\_j.$$
It remains to determine the exceptional square classes. Regarding that, see the proof of Theorem 2 and the references given. In particular, if $n$ is coprime to $\det(a\_{ij})$, then the only exceptional square class is the set of squares (see Footnote 5 in [arXiv:1402.1332](https://arxiv.org/abs/1402.1332)).


