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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 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 is the MSE post corresponding to this post.

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    $\begingroup$ The abstract of that arXiv paper says, "the sequence of partial quotients of continued fractions expansion of any real algebraic number of degree 3 is bounded. In fact, we can conjecture that the sequence of partial quotients of continued fractions expansion of any real algebraic number is bounded." This would be remarkable, as it has been widely believed that for every real algebraic number of degree three or more, the partial quotients are unbounded. $\endgroup$
    – Gerry Myerson
    Commented Apr 3, 2021 at 0:49

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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.

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  • $\begingroup$ This doesn't give an explicit example of $r,s$ satisfying the conditions in the question, does it? $\endgroup$
    – Gerry Myerson
    Commented Apr 4, 2021 at 2:42
  • $\begingroup$ @GerryMyerson I do not think so. I have not yet looked at the cited paper, though. Although I would suspect it would not help toward constructing such $r,s$. But I get the feeling this is the best I can get toward this problem on MO/MSE, so I accept it as an answer. $\endgroup$
    – Descartes Before the Horse
    Commented Apr 4, 2021 at 3:59

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