Continued fractions of some Mahler functions and applications to Diophantine approximation

Abstract: 

We consider a class of Mahler functions g(x)g(x) which can be written as an infinite product g(x)=1/x∗∏∞t=0P(x−dt)g(x)=1/x∗∏t=0∞P(x−dt), where dd is a positive integer and P(x)P(x) is a polynomial of degree less than dd. 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)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)g(b) for integer b>1b>1 and some functions g(x)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.