Transcendence of numbers with a low complexity expansion.(English)Zbl 0895.11029

The (block-) complexity of a sequence on a finite alphabet is the function $$n\to p(n)$$, where $$p(n)$$ counts the number of (different) blocks of length $$n$$ occurring in the sequence. A small step towards the conjecture that any algebraic irrational number is normal in base $$k$$ (hence contains all possible blocks in its base $$k$$ expansion) would be to prove that, if too many blocks are missing (i.e., $$p(n)$$ is small), then the number is either rational or transcendental. Let us call $$C$$ the set of sequences on a given alphabet $$\{0,1,\dots, k-1\}$$ having low complexity, and such that the corresponding real numbers in base $$k$$ are either rational or transcendental. For automatic sequences, we have $$p(n)= O(n)$$ and results of Loxton and van der Poorten seem to indicate that these sequences are in $$C$$.
In the paper under review the authors translate in combinatorial terms Ridout’s theorem; they obtain a combinatorial criterion of transcendence that implies: Sturmian sequences (i.e., sequences of complexity $$n+k-1$$ on the alphabet $$\{0,1,\dots, k-1\}$$) are in $$C$$ (this was known only for subclasses); Arnoux-Rauzy sequences are in $$C$$ (these sequences have complexity $$2n+1)$$; fixed points with overlaps of primitive morphisms are in $$C$$. This last result has been extended in the binary case by Luca Q. Zamboni (misspelled in the paper under review) and the reviewer to any fixed point of a binary morphism primitive or of constant length and non-trivial [J.-P. Allouche and L. Q. Zamboni, J. Number Theory 69, No. 1, 119–124 (1998; Zbl 0918.11016)].

MSC:

 11J81 Transcendence (general theory) 11B85 Automata sequences

Zbl 0918.11016
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References:

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