Roth’s theorem

The following theorem is due to Klaus Roth and it is a generalization of a previous theorem of Liouville (see Liouville approximation theorem). Roth was awarded the Fields Medal for his work on the geometry of numbers. W. M. Schmidt generalized the result even further. The result is widely used to prove that a certain number is transcendental. Here, for a rational number $t$ in reduced form, the denominator of $t$ is denoted by $d(t)$ .

Theorem 1.

For any algebraic number $\alpha$ and for any $\epsilon>0$ there are only finitely many rational numbers $t$ with:

|\alpha-t|<\frac{1}{d(t)^{2+\epsilon}}.

In other words, the equation:

\left|\alpha-\frac{p}{q}\right|<\frac{1}{q^{2+\epsilon}}

has only finitely many solutions with $p\in\mathbb{Z}$ and $q\in\mathbb{Z}^{+}$ .

Title	Roth’s theorem
Canonical name	RothsTheorem
Date of creation	2013-03-22 15:02:23
Last modified on	2013-03-22 15:02:23
Owner	alozano (2414)
Last modified by	alozano (2414)
Numerical id	7
Author	alozano (2414)
Entry type	Theorem
Classification	msc 11J81
Classification	msc 11J68
Related topic	ExampleOfTranscendentalNumber