Degree-One Mahler Functions: Asymptotics, Applications and Speculations

Abstract

We present a complete characterisation of the radial asymptotics of degree-one Mahler functions as $z$ approaches roots of unity of degree $k^{n}$ , where $k$ is the base of the Mahler function, as well as some applications concerning transcendence and algebraic independence. For example, we show that the generating function of the Thue–Morse sequence and any Mahler function (to the same base) which has a nonzero Mahler eigenvalue are algebraically independent over $\mathbb{C}(z)$ . Finally, we discuss asymptotic bounds towards generic points on the unit circle.

Radial behavior of Mahler functions

“…Remark 1. • In particular, the function C(z) (z ∈ [0, 1]) in the Bell-Coons asymptotic expansion (1.2) (see also [9] in this "order 1" case), is given in our situation by…”