Irrationality exponents of Mahler numbers

Speaker: 

Dmitry Badziahin

Affiliation: 

University of Sydney

Date: 

Wed, 08/08/2018 - 3:00pm

Venue: 

RC-4082, The Red Centre, UNSW

Abstract: 

We will consider the following question: what values irrationality exponents of Mahler numbers can take? For much smaller set of automatic numbers it was initially asked by Adamczewski and Rivoal and later this question was transformed into the conjecture that the set of irrationality exponents of such numbers coincides with the set of rational numbers not smaller than 2. In our talk we will consider the values $f(b)$ of the Mahler functions which satisfy the following equation:
$$f(x) = R(x) f(x^d),$$
where $R(x)$ is a rational function over $\mathbb{Q}$. We will see how to compute the irrationality exponent of $f(b)$, having the information about the continued fraction of $f(x)$ as a Laurent series. Finally, we will show that the irrationality exponent of $f(b)$ can only take rational values.

School Seminar Series: