For an irrational $q$ with continued fraction expansion $[q_0;q_1,q_2 \dots]$, say $q_i$ is the $i$-th patial quotient of $q$. I have the following question:
Are there any nontrivial examples of distinct irrationals $\alpha,\beta >0$ such that $\alpha, \beta,$ and $\alpha + \beta$ have bounded, aperiodic partial quotients?
My guess is that such pairs of irrationals do exist. But constructing an example seems beyond me. By nontrivial, I mean $\alpha,\beta$ are not rational multiples of each other, nor have integral difference. (Should 'integral' be replaced by 'rational'?) In fact, $\{ 1, \alpha, \beta\}$ should be $\mathbb{Q}$-linearly independent.