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A number $x\in\mathbb R$ is said to be $\tau$-approximable if there are $c,C>0$ such that for infinitely many couples of integers $(p,q)$, $$|x-\frac pq|<Cq^{-\tau}$$ and for all couples $(p,q)$ with $q\neq 0$, $$|x-\frac pq|>cq^{-\tau}$$

It is known since the fourties that there are uncountably many such numbers for $\tau>2$, does anyone know an explicit construction of such a number that works for every $\tau$?

Comment: I know from a paper by Bugeaud that $$\sum_{n=1}^\infty 3^{-[\tau^n]}$$ has irrationality measure $\tau$, so it would be perfect, unfortunately the inequalities are satisfied with $q^{-\tau}$ multiplied by some logarithmic term

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    $\begingroup$ You misstated the definition. It should be $q^{-\tau}$, not the power of $x$. $\endgroup$
    – Oleg Eroshkin
    Commented Oct 27, 2020 at 13:26
  • $\begingroup$ Indeed, thanks! $\endgroup$
    – kaleidoscop
    Commented Oct 27, 2020 at 14:00

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