  In a recent work [3], the authors established new results about general
linear Mahler systems in several variables from the perspective of
transcendental number theory, such as a multivariate extension of Nishioka's
theorem. Working with functions of several variables and with different Mahler
transformations leads to a number of complications, including the need to prove
a general vanishing theorem and to use tools from ergodic Ramsey theory and
Diophantine approximation (e.g., a variant of the $p$-adic Schmidt subspace
theorem). These complications make the proof of the main results proved in [3]
rather intricate. In this article, we describe our new approach in the special
case of linear Mahler systems in one variable. This leads to a new, elementary,
and self-contained proof of Nishioka's theorem, as well as of the lifting
theorem more recently obtained by Philippon [22] and the authors [1]. Though
the general strategy remains the same as in [3], the proof turns out to be
greatly simplified. Beyond its own interest, we hope that reading this article
will facilitate the understanding of the proof of the main results obtained in
[3].
