  Given a Chevalley group ${\mathbf G}(q)$ and a parabolic subgroup $P\subset
{\mathbf G}(q)$, we prove that for any set $A$ there is a certain growth of $A$
relatively to $P$, namely, either $AP$ or $PA$ is much larger than $A$. Also,
we study a question about intersection of $A^n$ with parabolic subgroups $P$
for large $n$. We apply our method to obtain some results on a modular form of
Zaremba's conjecture from the theory of continued fractions and make the first
step towards Hensley's conjecture about some Cantor sets with Hausdorff
dimension greater than $1/2$.
