If $G$ is a $\mathbb Q$-defined subgroup of $\operatorname{GL}\_n(\mathbb C)$, $\Lambda$ is a subgroup of $G(\mathbb Z)$, and $U$ is a unipotent subgroup of $G(\mathbb C)$ such that $\Lambda \cap U$ is Zariski-dense in $U$, why must $\Lambda$ contain $U(k\mathbb Z)$ (the principal congruence subgroup of $U(\mathbb Z)$ of level $k$) for some $k \ge 1$?


This is a bit of generalization of a line I did not understand in the article [Meiri - Generating pairs for finite index subgroups of $\operatorname{SL}(n, \mathbb Z)$](https://arxiv.org/abs/1511.07798) (*page 4, second paragraph*), that I'm trying to expand to the Chevalley groups.



If $\Lambda \cap U$ is Zariski-dense in $U$, then $\Lambda$ contains $U(k\mathbb Z)$ for some $k ≥ 1$?
In this answer i will follow Yves' comment and add references. If $U = \mathbf{U}(R)$ with $\mathbf U$ an algebraic unipotent $\mathbb Q$-group then the two following facts hold : 


* If $\Lambda \le U$ is Zariski-dense subgroup then $U/\Lambda$ is compact (Theorem 2.1 in Raghunathan's book *Discrete subgroups of Lie groups*).
* If $\Lambda\_1, \Lambda\_2 \le \mathbf U(\mathbb Q)$ are both cocompact then $\Lambda\_1 \cap \Lambda\_2$ has finite index in both (this is true for abelian $U$ and any subset generating the abelianisation generates a cocompact subgroup).


Together these imply that for any $\mathbf U, \Lambda$ as in your question the subgroup $\mathbf U(\mathbb Z) \cap \Lambda$ has finite index in $\mathbf U(\mathbb Z)$. Then the (positive) answer to your question follows from the congruence subgroup property for unipotent groups : any finite-index subgroup of $\mathbf U(\mathbb Z)$ contains a subgroup of the form $\mathbf U(k\mathbb Z)$. 


That the latter holds can be proven by elementary means (let me give a sketch : this is true if $U$ is abelian, and if $\Lambda/[\Lambda, \Lambda]$ containes the image of $\mathbf U(k\mathbb Z)$ in the abelianisation then $\Lambda$ should contain $\mathbf U(k^r\mathbb Z)$ where $r$ is the step of $U$). Or you can probably adapt the proof in the case of upper triangular matrices given in B. Sury's book (*The congruence subgroup problem--an elementary approach aimed at applications*, Theorem 2-5.2) to the general unipotent case. 


