Berthé, Valérie (ed.) et al., Combinatorics, automata, and number theory. Cambridge: Cambridge University Press (ISBN 978-0-521-51597-9/hbk). Encyclopedia of Mathematics and its Applications 135, 410-451 (2010).
This paper is a chapter of a book, devoted to transcendence and Diophantine approximation in relation with combinatorics on words. Several nice results are presented and some of the proofs are given. Let us cite two theorems revisited in that chapter:
Theorem. The base $b$ expansion of an algebraic irrational number satisfies $p(n)/n\to +\infty$, where $p(n)$ is the number of distinct blocks of digits of length $n$ occurring in the expansion.
Theorem. Let $x$ be a positive real number, irrational and not quadratic. If the continued fraction expansion of $x$ is a sequence of coefficients taking finitely many values and beginning in arbitrarily long prefixes $u_jv^w_j$ for some $w ge 2$, then $x$ is transcendental.
Other results include simultaneous rational approximations to a real number and its square, continued fractions and palindromes, etc. For the entire collection see [Zbl 1197.68006
|11J81||Transcendence (general theory)|
|11-02||Research monographs (number theory)|
|11J04||Homogeneous approximation to one number|
|11J13||Simultaneous homogeneous approximation, linear forms|
|11J70||Continued fractions and generalizations|
|11J87||Schmidt Subspace Theorem and applications|
|68R15||Combinatorics on words|