On effective irrationality exponents of cubic irrationals

Wednesday, 1 March 2023

Abstract

Thanks to the remarkable work of Roth, it is known that the irrationality exponents of all irrational algebraic numbers equal two, i.e. they are the smallest possible. However, that result is ineffective and hence it only gives that for algebraic irrational $x$ and $\lambda>2$ the inequality $|x - p/q| < q^{-\lambda}$ has finitely many solutions but does not give a method for finding all of them. This ineffectiveness does not allow to use Roth's theorem in many applications, in particular for Thue equations of degree at least 3. 

After Roth, the problem of estimating the effective irrationality exponents of algebraic irrationals attracted many mathematicians. In this talk we will show how recently discovered continued fractions of cubic irrationals can be applied to obtain non-trivial upper bounds of the effective irrationality exponents for a large class of cubic irrationals. In particular, for $x = (1+a)^{1/3}$, they give the same bounds as the classical hypergeometric method of Baker. On the other hand, they also provide non-trivial bounds for solutions of the equation $x^3 - ax^2 - t = 0$ for $|t|>19.71 a^{4/3}$. That is currently the best known result of this kind.