Growth in Chevalley groups relatively to parabolic subgroups and some applications

Given a Chevalley group $\mathbf{G}(q)$ and a parabolic subgroup $P\subset {\mathbf G}(q)$,weprovethatforanyset$A$thereisacertaingrowthof$A$relativelyto$P$,namely,either$AP$or$PA$ismuchlargerthan$A$. 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 $1\mathrm{/}2$.