Roth’s theorem

The following theorem is due to Klaus Roth and it is a generalizationPlanetmathPlanetmath of a previous theorem of Liouville (see Liouville approximation theoremMathworldPlanetmath). 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 numberPlanetmathPlanetmathPlanetmath t in reduced form, the denominator of t is denoted by d(t).

Theorem 1.

For any algebraic numberMathworldPlanetmath α and for any ϵ>0 there are only finitely many rational numbers t with:


In other words, the equation:


has only finitely many solutions with pZ and qZ+.

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