Algebraic Irrational Binary Numbers Cannot Be Fixed Points of Non-trivial Constant Length or Primitive Morphisms

On the complexity of algebraic numbers I. Expansions in integer bases

“…As far as we are aware, no better result is known, although it has been proved [3], [6], [34] that some special real numbers with linear complexity are transcendental. Our first result is a considerable improvement of (1).…”

“…In particular, the b-ary expansion of every real number normal in base b satisfies p(n) = b n for any positive integer n. Using a clever reformulation of a theorem of Ridout [33], Ferenczi and Mauduit [20] established the transcendence of the real numbers whose b-ary expansion is a non eventually periodic sequence of minimal complexity, that is, which satisfies p(n) = n + 1 for every n ≥ 1 (such a sequence is called a Sturmian sequence, see the seminal papers by Morse and Hedlund [28], [29]). The combinatorial criterion given in [20] has been used subsequently to exhibit further examples of transcendental numbers with low complexity [3], [6], [4], [34]. It also implies that the complexity of the b-ary expansion of every irrational algebraic number satisfies lim inf n→∞ (p(n) − n) = +∞.…”

Combinatorics on Words – A Tutorial

“…For instance, the Fibonacci morphism is primitive. The result is the following, see [AZ98] or already [FM97] for primitive morphisms:…”

Approximations diophantiennes des nombres sturmiens

“…De nombreux travaux, certains récents, s'intéressent à la transcendance d'irrationnels définis par leur développe-ment, en base entière ou en fraction continue, lorsque celui-ci est une suite de faible complexité : rappelons que la fonction complexzté p compte, pour chaque n, le nombre de facteurs distincts de longueur n dans la suite. Une liste non exhaustive de ces résultats apparaît dans la bibliographie ~1~, [2], [3], [6], [8], [9], [10], [11], [13], [18], [24], [27]. D'autres développements peuvent être envisagés : développement de Engel [17], développement en base de Pisot par exemple.…”