Growth in Chevalley groups relatively to parabolic subgroups and some applications

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