irrationality measure

Let $\alpha \notin \mathbb{Q}$.Let

$$M(\alpha )=\{\mu >0\mid \exists q_0=q_0(\alpha ,\mu )>0\text{ such that }\left|\alpha -\frac{p}{q}\right|>\frac{1}{q^{\mu }}\mathit{\hspace{1em}}\forall p,q\in \mathbb{Z},q>q_0\}.$$

The irrationality measure of $\alpha$,denoted by $\mu (\alpha )$, is defined by

$$\mu (\alpha )=infM(\alpha ).$$

If $M(\alpha )=\mathrm{\varnothing }$, we set $\mu (\alpha )=\mathrm{\infty }$.

This definition is (loosely) a measure of the extent to which $\alpha$ can be approximated by rational numbers. Of course, by the fact that $\mathbb{Q}$ is dense in $ℝ$, we can make arbitrarily good approximations to real numbers by rationals. Thus this definition was made to represent a stronger statement: it is the ability of rational numbers to approximate $\alpha$ given a fixed growth bound on the denominators of those rational numbers.

By the Dirichlet’s Lemma, $\mu (\alpha )\ge 2$.Roth [6, 7]proved in 1955 that$\mu (\alpha )=2$ for every algebraic real number.It is well known alsothat $\mu (e)=2$. For almost all real numbersthe irrationality measure is 2.However, for special constants, only some upper boundsare known:

It is worth noting that the last column of the above table is simply a list of references, not a collection of discoverers. For example that fact that the irrationality measure of $e$ is 2 was known to Euler.