Some applications of growth in $\mathrm{SL}_2(\pmb{\mathbb{F}}_p)$ to the proof of modular versions of Zaremba's conjecture

Abstract

Using growth in $\mathrm{SL}_2(\mathbb{F}_p)$ we prove that for every prime number $p$ and any positive integer $u$ there are positive integers $q=O(p^{2+\varepsilon})$, $\varepsilon > 0$, $q \equiv u \pmod{p}$, and $a < p$, $(a, p)=1$, such that the partial quotients of the continued fraction of $a/q$ are bounded by an absolute constant. Bibliography: 21 titles.