Suppose that we have a variety $X \subset \mathbb{P}^{n}$ defined over $\mathbb{Q}$. Suppose we have $S$ many rational points on $X$ inside the box defined by $|x_i| \leq B_i$ for $i = 0, \cdots, n$, and $B_i \in \mathbb{R}_{\geq 0}$ for $i = 0, \cdots, n$. We denote this set by $$\displaystyle X(\mathbb{Q}; B_0, \cdots, B_n).$$ For a prime $p$, let $X_p$ denote the reduction of $X$ modulo $p$.
Now suppose that we fix a point $P$ on $X_p$, and consider the pre-image of $P$ in $X(\mathbb{Q}; B_0, \cdots, B_n)$ which we denote by $U_P$. Now consider a prime $q < p$. If $q, p, S$ are all sufficiently large, so that we don't have problems with biases towards small primes, can we say that the points in $U_P$ are well-distributed among the congruence classes modulo $q$?