## Liouville-Roth Constant

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

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

 (1)

has (at most) finitely many solutions for and Integers. Then the Liouville-Roth constant (or Irrationality Measure) is defined as the threshold at which Liouville's Rational Approximation Theorem kicks in and is no longer approximable by Rational Numbers,
 (2)

There are three regimes:
 (3)

The best known upper bounds for common constants are
 (4) (5) (6) (7) (8) (9)

where is Liouville's Constant, is Apéry's Constant, and the lower bounds are 2 for the inequalities.

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 and .'' Proc. Japan. Acad. Ser. A Math. Sci. 68, 283-286, 1992.

Hata, M. Rational Approximations to 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.