We provide a non-trivial measure of irrationality for a class of Mahler numbers defined by infinite products. This class includes the Thue–Morse constant. Among other things, our results imply a generalisation to [12].

In this paper we extend and generalize, up to a natural bound of the method, our previous work Badziahin and Zorin [‘Thue–Morse constant is not badly approximable’, Int. Math. Res. Not. IMRN 19 (2015), 9618–9637] where we proved, among other things, that the Thue–Morse constant is not badly approximable. Here we consider Laurent series defined with infinite products $f_{d}(x)=\prod _{n=0}^{\infty }(1-x^{-d^{n}})$ , $d\in \mathbb{N}$ , $d\geq 2$ , which generalize the generating function $f_{2}(x)$ of the Thue–Morse number, and study their continued fraction expansion. In particular, we show that the convergents of $x^{-d+1}f_{d}(x)$ have a regular structure. We also address the question of whether the corresponding Mahler numbers $f_{d}(a)\in \mathbb{R}$ , $a,d\in \mathbb{N}$ , $a,d\geq 2$ , are badly approximable.

We prove a new general multiplicity estimate applicable to sets of functions without any assumption on algebraic independence. The multiplicity estimates are commonly used in determining measures of algebraic independence of values of functions, for instance within the context of Mahler's method. For this reason, our result provides an important tool for the proofs of algebraic independence of complex numbers. At the same time, these estimates can be considered as a measure of algebraic independence of functions themselves. Hence our result provides, under some conditions, the measure of algebraic independence of elements in ${\bf F}_q[[T]]$, where ${\bf F}_q$ denotes a finite field.

In this paper we develop the convergence theory of simultaneous, inhomogeneous Diophantine approximation on manifolds. A consequence of our main result is that if the manifold $M \subset \mathbb{R}^n$ is of dimension strictly greater than $(n+1)/2$ and satisfies a natural non-degeneracy condition, then $M$ is of Khintchine type for convergence. The key lies in obtaining essentially the best possible upper bound regarding the distribution of rational points near manifolds.