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 $\mathbb{R}$, 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, $\mu (\alpha )\ge 2$. 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] |
$\mathrm{log}2$ | 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 $\zeta \mathit{}\mathrm{(}\mathrm{3}\mathrm{)}$’, 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 |
---|---|
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 |