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Let $x$ be a real number. Then the irrationality measure $\mu(x)$ can be defined as the smallest positive real number $\mu(x)$ such that the inequality $ \left|x-\dfrac{p}{q}\right|>\dfrac{1}{q^{\mu+\epsilon}}$ holds for any $\epsilon>0$ and all integers $p,q$ with $q$ sufficiently large. If no such $\mu(x)$ exists, then $\mu(x)=\infty$ and $x$ is said to be a Liouville number

Per the Mathworld page, it is known that $\mu(x)=1$ if $x$ is rational, $\mu(x)=2$ if $x$ is algebraic of degree $>1$, and $\mu(x)\geq 2$ if $x$ is transcendental. (Hence all Liouville numbers are transcendent, but not vice versa). It goes on to cite various upper bounds on $\mu(x)$ for various common constants.

Are there effective algorithms for computing such upper bounds numerically?

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  • $\begingroup$ How do you suppose the input to such an algorithm would be given? A floating-point approximation to the real number in question won't do, because such an approximation just indicates an interval on the real number line, and any such interval contains numbers of all possible irrationality measures. $\endgroup$ – Henning Makholm Sep 29 '16 at 15:48
  • $\begingroup$ @HenningMakholm I suppose I had in mind something involving the continued fraction representation, which certainly is computable. But one can't actually draw rigorous conclusions about the limiting behavior of this representation; at most, such an observation would be heuristic. So it's entirely possible that the answer to my question is simply "no, there isn't." $\endgroup$ – Semiclassical Sep 29 '16 at 15:54

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