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 also that $\mu (e)=2$. For almost all real numbers the irrationality measure is 2. However, for special constants, only some upper bounds are known:

Constant	Upper bound	Reference
$\pi$	8.0161	Hata (1993) [3]
$\pi /\sqrt{3}$	4.6016	Hata (1993) [3]
$e$	2	Davis (1978) [1]
${\pi }^2$	5.4413	Rhin and Viola (1996) [4]
$\log 2$	3.8914	Rukhadze (1987) [8], Hata (1990) [2]
$\zeta (3)$	5.5139	Rhin and Viola (2001) [5]

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.

Title	irrationality measure
Canonical name	IrrationalityMeasure
Date of creation	2013-03-22 14:12:22
Last modified on	2013-03-22 14:12:22
Owner	mathcam (2727)
Last modified by	mathcam (2727)
Numerical id	8
Author	mathcam (2727)
Entry type	Definition
Classification	msc 11J82
Related topic	DirichletsApproximationTheorem