Liouville's Approximation Theorem

For any algebraic number  of degree , a rational approximation  to  must satisfy

for sufficiently large . Writing  leads to the definition of the irrationality measure of a given number. Apostol (1997) states the theorem in the slightly modified but equivalent form that there exists a positive constant  depending only on  such that for all integers  and  with ,

REFERENCES:

Apostol, T. M. "Liouville's Approximation Theorem." §7.3 in Modular Functions and Dirichlet Series in Number Theory, 2nd ed. New York: Springer-Verlag, pp. 146-148, 1997.

Courant, R. and Robbins, H. "Liouville's Theorem and the Construction of Transcendental Numbers." §2.6.2 in What Is Mathematics?: An Elementary Approach to Ideas and Methods, 2nd ed. Oxford, England: Oxford University Press, pp. 104-107, 1996.