Continued fractions of some Mahler functions and applications to Diophantine approximation

Speaker: 

Dzmitry Badziahin

Affiliation: 

University of Sydney

Date: 

Thu, 06/04/2017 - 2:00pm

Venue: 

RC-4082, The Red Centre, UNSW

Abstract: 

We consider a class of Mahler functions $g(x)$ which can be written as an infinite product $g(x) = 1/x * \prod_{t=0}^\infty P(x^{-d^t})$, where $d$ is a positive integer and $P(x)$ is a polynomial of degree less than $d$. These functions attracted the attention of van der Poorten (along many others). Starting from 1991 he together with the Allouche and Mendes France wrote a series of papers on the continued fraction expansion of the most classical example of Mahler function, the Thue-Morse function. In this talk we substantially extend the results from there. In particular, we show that the continued fraction of $g(x)$, written as a Laurent series, can be computed by a recurrent formula. Then we will use this fact to establish several approximational properties of Mahler numbers $g(b)$ for integer $b>1$ and some functions $g(x)$. In particular we will compute their irrationality exponent in some cases and make non-trivial estimates on it in the other cases. Also, if time permits, we will show that the Thue-Morse number is not badly approximable.

School Seminar Series: