Let $Q_0, Q_1, Q_2 \in \mathbb{Z}[x_0, x_1, x_2]$ be three non-singular ternary quadratic forms with integer coefficients. Let $T$ be a large real number, and let $p, q$ be two distinct primes having size $T^\alpha/\log T < p, q < T^\alpha$ for some $\alpha \geq 1$. Let $I(T) = [1, T^{1/2}) \cap \mathbb{Z}$, and let $I_p(T), I_q(T)$ respectively denote the reduction of $I(T)$ mod $p,q$.
Consider the polynomial map $\mathcal{F}(\mathbf{x}) = (Q_0(\mathbf{x}), Q_1(\mathbf{x}), Q_2(\mathbf{x}))$. Define the set
$\displaystyle W_{\mathcal{F},p,q}(T) = \{\mathbf{x} \in [1,T]^3 \cap \mathbb{Z}^3 : \mathcal{F}(\mathbf{x}) \pmod{p} \in I_p(T)^3, \mathcal{F}(\mathbf{x}) \pmod{q} \in I_q(T)^3 \}.$
In other words, $W_{\mathcal{F},p,q}(T)$ is the set of integer points in the box $[1,T]^3$ such that the triple $(Q_0(\mathbf{x}), Q_1(\mathbf{x}), Q_2(\mathbf{x}))$ satisfies $Q_i(\mathbf{x}) \pmod{p} \in I_p(T)$ and $Q_i(\mathbf{x}) \pmod{q} \in I_q(T)$ for $i = 0,1,2$.
Is there a way to control the size of $W_{\mathcal{F}, p, q,}(T)$, especially for $\alpha > 1$?
The idea is that typically one would expect at least one of $Q_i(\mathbf{x})$ to be "large" mod $p,q$ (i.e., does not land in $I_p(T), I_q(T)$) when $\mathbf{x} \in [1,T]^3$, and for all three quadratic forms to be "small" for both moduli seems a very difficult condition to satisfy. Of course, this notion of "large/small" is virtually meaningless in the finite field setting.
