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