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Let $\alpha\in\mathbb R_{>2}\setminus\mathbb Q$ be an irrational number and let $\beta$ be such that $$\alpha+\beta+\frac{1}{\beta}=0.$$ Is there a known relationship between the irrationality measure of $\beta$ and that of $\alpha$?

A maybe easier question: assume $\alpha$ is a Bruno number (very well approximated by rationals, see https://en.wikipedia.org/wiki/Brjuno_number); then is $\beta$ Bruno? It more or less boils down to knowing whether the set $B$ of Bruno numbers is stable by taking quadratic extensions.

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    $\begingroup$ You can solve $\beta=\frac{1}{2}(\pm\sqrt{a^2-4}-a)$ (WA). The question therefore seems to essentially come down to questions about relating irrationality measure of a number and its square root. $\endgroup$
    – Wojowu
    Commented Jan 8, 2021 at 15:29
  • $\begingroup$ Yes, that's basically what I meant in the question. $\endgroup$
    – Loïc Teyssier
    Commented Jan 8, 2021 at 15:52

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