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Algebraic irrational binary numbers cannot be fixed points of non-trivial constant length or primitive morphisms. (English) Zbl 0918.11016

Heuristics say that an irrational real number whose sequence of digits in its base-2 expansion has some kind of regularity, then it is transcendental. In particular, J. Loxton and A. J. van der Poorten state in [J. Reine Angew. Math. 392, 57-69 (1988; Zbl 0656.10033)] that the digits in the \(k\)-ary expansion of an algebraic irrational number cannot be generated by a finite \(k\)-automaton (note that this result should be considered as a conjecture since there is still a gap in their proof).
The paper under review proves that a positive real number whose binary expansion is a fixed point of a non-trivial morphism (over \(\{0,1\}\)) which is either of constant length or primitive, must be rational or transcendental. This paper is a continuation of [S. Ferenczi and C. Mauduit, J. Number Theory 67, 146-161 (1997; Zbl 0895.11029)] and uses a combinatorial translation of a transcendence criterium due to Ridout, given in this last paper: if the expansion in some basis of an irrational number contains infinitely may \(2+\varepsilon\)-powers of blocks at distances from the origin which are not too much larger than the lengths of the considered blocks, then it is transcendental. In the case of the fixed point of a primitive or of a non-trivial constant length morphism, this criterium reduces to the existence of overlaps. P. Séébold [Discrete Appl. Math. 11, 255-264 (1985; Zbl 0583.20047)] gave a description of the overlap-free fixed points of non-trivial morphisms (these are the two fixed points of the Thue-Morse substitution). Since the Thue-Morse number is known to be transcendental, this completes the proof of the paper under review.

MSC:

11B85 Automata sequences
11J81 Transcendence (general theory)
11A67 Other number representations
11K16 Normal numbers, radix expansions, Pisot numbers, Salem numbers, good lattice points, etc.
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References:

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