Roth's theorem

For Roth's theorem on arithmetic progressions, see Roth's theorem on arithmetic progressions.

In mathematics, Roth's theorem or Thue–Siegel–Roth theorem is a fundamental result in diophantine approximation to algebraic numbers. It is of a qualitative type, stating that algebraic numbers cannot have many rational approximations that are 'very good'. Over half a century, the meaning of very good here was refined by a number of mathematicians, starting with Joseph Liouville in 1844 and continuing with work of Axel Thue (1909), Carl Ludwig Siegel (1921), Freeman Dyson (1947), and culminating with Klaus Roth (1955).

This machinery was originally developed to prove Thue's theorem in Diophantine geometry, that Thue equations (bivariate homogeneous of degree at least 3) possess only finitely many integer solutions.

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