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Transcendence and Diophantine approximation. (English) Zbl 1271.11073
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+, 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 j v j w for some wge2, then x is transcendental.

Other results include simultaneous rational approximations to a real number and its square, continued fractions and palindromes, etc.

MSC:
11J81Transcendence (general theory)
11-02Research monographs (number theory)
11J04Homogeneous approximation to one number
11J13Simultaneous homogeneous approximation, linear forms
11J70Continued fractions and generalizations
11J87Schmidt Subspace Theorem and applications
11B85Automata sequences
68R15Combinatorics on words