Thue–Siegel–Roth theorem

In mathematics, Roth's theorem is a fundamental result in diophantine approximation to algebraic numbers. It is of a qualitative type, stating that a given algebraic number ${\displaystyle \alpha }$ may not have too many rational number 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 Klaus Roth (1955).

Roth's theorem states that every irrational algebraic number ${\displaystyle \alpha }$ has approximation exponent equal to 2, i.e., for given ${\displaystyle \varepsilon >0}$, the inequality

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