If $q = [q\_0;q\_1 \dots]$, say $q\_i$ is the $i$-th partial quotient of $q$. My question is the following:



> 
> Can one construct an explicit example of irrational $r,s > 0$ such that
> 
> 
> 1. $\{ 1,r,s\}$ is $\mathbb{Q}$-linearly independent,
> 2. $r,s$ have bounded, aperiodic partial quotients, and
> 3. $r+s$ has bounded, aperiodic partial quotients?
> 
> 
> 


That is, assuming such a pair exists. I would guess one does--in fact I would guess every irrational with bounded, aperiodic partial quotients has such a decomposition. But this general statement is not what I'm interested in. I am interested in seeing even one example.


As to known results which apply, all I have found is [this](https://arxiv.org/abs/1904.09392) article posted on ArXiv, pointed out to me by user3733558. The titular theorem of the paper would immediately answer my question. However, I've no idea as to its validity.


[Here](https://math.stackexchange.com/q/4083370/592365) is the MSE post corresponding to this post.



Bounded, aperiodic irrationals with bounded, aperiodic sum
Theorem 3.1 page 972 in [1] ensures that every number in a certain interval I is a sum of two real numbers with partial quotients bounded by 4. If you consider numbers in I with partial quotients bounded by 5 (Call this set $I \cap F(5)$), they could not all be obtained via periodic summands or linearly dependent summands over the rationals, since $F(5)$ has Hausdorff dimension strictly larger than $F(4)$. Note there are many later refinements of Hall's theorem- his paper has 200 citations.


[1] Hall, Marshall. "On the sum and product of continued fractions." Annals of Mathematics (1947): 966-993.


