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Let $Q(x_1, \dots, x_n)$ be a non-degenerate indefinite quadratic form with integer coefficients. Let $N(Q,T)$ be the set of vectors $x=(x_1, \dots, x_n) \in {\mathbb Z}^n$ such that $|x|<T$ and $Q(x)=0$, where $|x|$ denotes the (say) Euclidean norm of $x$. Let $N(Q,T;x_0)$ denote the subset of $N(Q,T)$ consisting of those vectors $x$ with $x \equiv x_0 \pmod{m}$ for some $m \ge 1$ and $x_0 \in {\mathbb Z}^n$. Is it true that if there are no local obstructions, then a positive proportion of the elements of $N(Q,T)$ are in $N(Q,T;x_0)$, i.e., $$ \liminf_{T \to \infty} \frac{|N(Q,T;x_0)|}{|N(Q,T)|}>0. $$

Of course, I am implicitly assuming that $Q(x)=0$ has non-trivial solutions.

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  • $\begingroup$ I think you had better include some examples with, say, indefinite binary forms. There are some results, heavily dependent on dimension, but I'm not sure what you want to do with the $\pmod m$ or why you want to do so. $\endgroup$
    – Will Jagy
    Commented Nov 19, 2015 at 17:23
  • $\begingroup$ Will Jagy: Here is an example: Take (say) the form $Q(x,y,z)=x^2+y^2-5z^2$. Suppose I want to have solutions $(x,y,z)$ to $Q(x,y,z)=0$ such that $x$ is not divisible by a certain prime number $p$. Is there a guarantee I can get $cT log T$ of them with $|x|,|y|,|z|<T$? $\endgroup$
    – Keivan Karai
    Commented Nov 19, 2015 at 20:02
  • $\begingroup$ Alright. I have an appointment. Dimension 3 is rather special, in that we can parametrize all solutions with a finite number of recipes, similar to the formula for primitive Pythagorean triples; in your example, this follows from simple stereographic projection, in general, it comes from a fact in Fricke and Klein (1897). Meanwhile, dimension 4 has this problem: arxiv.org/abs/1205.4416 $\endgroup$
    – Will Jagy
    Commented Nov 19, 2015 at 20:16
  • $\begingroup$ Thanks for the reference. I mentioned that particular form as an example, but I am indeed interested in dimensions 3 and 4. I will look up Fricke and Klein. $\endgroup$
    – Keivan Karai
    Commented Nov 19, 2015 at 20:23
  • $\begingroup$ It's pages 507-508 in FK. The result has been repeated in later summaries. In brief, an indefinite ternary form, integer coefficients, that really is isotropic over $\mathbb Q$ and therefore $\mathbb Q,$ is integrally equivalent to an integer multiple of $y^2 - zx.$ The value of this is that we can write down the primitive null vectors of this latter form, $(u^2, uv, v^2)$ up to $\pm.$ $\endgroup$
    – Will Jagy
    Commented Nov 19, 2015 at 23:05

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