Number Theory Seminar

Dmitry Badziahin, Durham University

"On a generalised Thue-Morse generating function and its continued fraction expansion"

In the talk we consider the following function F_n(x) = \prod_{i=1}^\infty (1 - x^{-n^i}). In particular for n=2 it is the generating function of the classical Thue-Morse sequence. We will show that the continued fraction expansion of some very closely related function G_n(x) = x^{-n+1}F_n(x) admits several remarkable properties. This allows us to construct the precise continued fraction expansion of G_n(x) for n=2 and n=3. Finally we will discuss how to use this construction to show that the Thue-Morse constant is not badly approximable.

MC 5479