Continued fractions of cubic irrationals

Orateur: Dzmitry BADZIAHIN
Localisation: ,
Type: Online Seminar in Diophantine Approximation and Related Topics
Site: N/A
Salle: Zoom
Date de début: 25/10/2022 - 14:00
Date de fin: 25/10/2022 - 15:00

It was discovered by Gauss that for any $r\in\mathbb{Q}$ the Laurent series of the function $(1+t)^r$ has an easy-to-describe continued fraction expansion. Later, A. Baker used the convergents of that fraction to produce the first effective upper bounds of the irrationality exponent of some algebraic numbers, including $\sqrt[3]{2}$, and thereby improved the classical result of Liouville for them. Later, his method was refined by many mathematicians, including Chudnovsky brothers, Rickert and Bennet. However, it only works for algebraic numbers of the form $\big(1+\frac{a}{N}\big)^r$. In this talk, I will show that there are many other cubic irrational Laurent series, apart from $(1+t)^r$, that enjoy a nice continued fraction expansion. We will see that there are many enough of them, so that their specializations cover all cubic irrational numbers and for at least some of them we can provide non-trivial upper bounds of their irrationality exponents.