In the context of global function fields, the strong approximation theorem can be stated as follows: Let $F$ be a global function field, and $P_1,P_2,\dots,P_r$ be a finite set of places of $F$ (with respective valuations $\nu_{P_i}$). Then given integers $n_1,n_2,\dots,n_r$ and a proper subset $S$ of places (which contains $P_1,\dots,P_r$), there exists an element $z \in F$ such that $$\nu_{P_i}(z)=n_i$$ for all $1 \leq i \leq r$, and for any place $P$ not in $S$, $$\nu_{P}(z) = 0$$

Recently, I came to know of another strong approximation theorem for groups. The version I read of it is as follows: For a finitely generated Zariski dense subgroup $\Gamma$ of $SL_n(\mathbb{Z})$, the congruence modulo $p$, $\Gamma_p$ is the whole of $SL_n(\mathbb{Z}/p\mathbb{Z})$ for all large enough primes $p$. I also read that the above statement is a generalization of the fact that the quotient map $SL_2(\mathbb{Z}) \mapsto SL_2(\mathbb{Z}/p\mathbb{Z})$ is surjective, and also that it is essentially the Chinese remainder theorem on steroids.

Could someone explain to me how the above two strong approximation theorems are related, if they are. As in, what exactly are we "approximating"? In general, what is a "strong approximation"?


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