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 expansion of an algebraic irrational number satisfies , where is the number of distinct blocks of digits of length occurring in the expansion.
Theorem. Let be a positive real number, irrational and not quadratic. If the continued fraction expansion of is a sequence of coefficients taking finitely many values and beginning in arbitrarily long prefixes for some , then is transcendental.
Other results include simultaneous rational approximations to a real number and its square, continued fractions and palindromes, etc.