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Let $\alpha\not\in\mathbb Q$ . Let $$M(\alpha)=\{\mu>0 \mid \exists q_0=q_0(\alpha, \mu)>0 \mbox{ such that } \left|\alpha-\frac pq\right|>\frac1{q^\mu} \quad \forall p,q\in\mathbb Z, q>q_0\}.$$ 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, $\mu(\alpha)\ge 2$ . 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] |
| $\pi/\sqrt3$ |
4.6016 |
Hata (1993) [3] |
| $e$ |
2 |
Davis (1978) [1] |
| $\pi^2$ |
5.4413 |
Rhin and Viola (1996) [4] |
| $\log2$ |
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.
- 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.
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