Zaremba's conjecture

For positive integers $a_1,\ldots,a_\ell\geq1$, define the continued fraction

$$[a_1,a_2,\ldots,a_\ell]=\cfrac{1}{a_1+\cfrac{1}{a_2+\cdots+\cfrac{1}{a_\ell}}}.$$

For a positive integer $A$, let $Q_A$ be the set of denominators of reduced fractions in $(0,1)$ admitting such an expansion with all partial quotients at most $A$, together with $1$. Zaremba's conjecture. For $A=5$, the set $Q_A$ consists of all positive integers. This is described as a longstanding open problem in continued-fraction theory and is used in the paper's proofs, but the assertion itself remains open.