Liouville-Roth Constant

N.B. A detailed on-line essay by S. Finch was the starting point for this entry.

Let $x$ be a Real Number, and let $R$ be the Set of Positive Real Numbers for which

$$\left\vert{x-{p\over q}}\right\vert < {1\over q^r}$$ (1)

has (at most) finitely many solutions $p/q$ for $p$ and $q$ Integers. Then the Liouville-Roth constant (or Irrationality Measure) is defined as the threshold at which Liouville's Rational Approximation Theorem kicks in and $x$ is no longer approximable by Rational Numbers,

$$r(x)\equiv \inf_{r\in R} r.$$ (2)

There are three regimes:

$$\cases{ r(x)=1 & $x$\ is rational\cr r(x)=2 & $x$\ is algebraic irrational\cr r(x)\geq 2 & $x$\ is transcendental.\cr}$$ (3)

The best known upper bounds for common constants are

$$r(L) = \infty$$ (4)

$$r(e) = 2$$ (5)

$$r(\pi) < 8.0161$$ (6)

$$r(\ln 2) < 4.13$$ (7)

$$r(\pi^2) < 6.3489$$ (8)

$$r(\zeta(3)) < 13.42,$$ (9)

where $L$ is Liouville's Constant, $\zeta(3)$ is Apéry's Constant, and the lower bounds are 2 for the inequalities.

See also Liouville's Rational Approximation Theorem, Roth's Theorem, Thue-Siegel-Roth Theorem

References

Borwein, J. M. and Borwein, P. B. Pi & the AGM: A Study in Analytic Number Theory and Computational Complexity. New York: Wiley, 1987.

Finch, S. ``Favorite Mathematical Constants.'' http://www.mathsoft.com/asolve/constant/lvlrth/lvlrth.html

Hardy, G. H. and Wright, E. M. An Introduction to the Theory of Numbers, 5th ed. Oxford: Clarendon Press, 1979.

Hata, M. ``Improvement in the Irrationality Measures of $\pi$ and $\pi^2$.'' Proc. Japan. Acad. Ser. A Math. Sci. 68, 283-286, 1992.

Hata, M. ``Rational Approximations to $\pi$ and Some Other Numbers.'' Acta Arith. 63 335-349, 1993.

Hata, M. ``A Note on Beuker's Integral.'' J. Austral. Math. Soc. 58, 143-153, 1995.

Stark, H. M. An Introduction to Number Theory. Cambridge, MA: MIT Press, 1978.