# irrationality measure

Let $\alpha\not\in\mathbb{Q}$. Let

 $M(\alpha)=\{\mu>0\mid\exists q_{0}=q_{0}(\alpha,\mu)>0\mbox{ such that }\left|% \alpha-\frac{p}{q}\right|>\frac{1}{q^{\mu}}\quad\forall p,q\in\mathbb{Z},q>q_{% 0}\}.$

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

 $\mu(\alpha)=\inf M(\alpha).$

If $M(\alpha)=\emptyset$, we set $\mu(\alpha)=\infty$.

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 $\mathbb{R}$, 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, $\mu(\alpha)\geq 2$. 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.

## References

• 1 Davis, C.S., Rational approximations to $e$, J. Austral. Math. Soc. Ser. A 25 (1978), 497–502.
• 2 Hata, M. ‘Legendre Type Polynomials and Irrationality Measures’, J. reine angew. Math. 407, 99–125, 1990.
• 3 Hata, M., ‘Rational approximations to $\pi$ and some other numbers’, Acta Arith. 63, 335–349 (1993).
• 4 Rhin, G. and Viola, C. ‘On a permutation group related to zeta(2)’, Acta Arith. 77 (1996), 23–56.
• 5 Rhin, G. and Viola, C. ‘The group structure for $\zeta(3)$, Acta Arith. 97 (2001), 269–293.
• 6 Roth, K.F., ‘Rational Approximations to Algebraic Numbers, Mathematika 2 (1955), 1–20.
• 7 Roth, K.F. ‘Corrigendum to ’Rational Approximations to Algebraic Numbers” Mathematika 2 (1955), 168.
• 8 Rukhadze, E.A. ‘A Lower Bound for the Rational Approximation of by Rational Numbers’ Vestnik Moskov Univ. Ser. I Math. Mekh., 6 (1987), 25-29 and 97.
Title irrationality measure IrrationalityMeasure 2013-03-22 14:12:22 2013-03-22 14:12:22 mathcam (2727) mathcam (2727) 8 mathcam (2727) Definition msc 11J82 DirichletsApproximationTheorem