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Substitution in two symbols and transcendence. (English) Zbl 0940.11015

The authors study substitutions in two symbols in connection with the dyadic expansion of real algebraic irrationals. Let \(A=\{a,b\}\) be a set of two symbols, \(A^\ast\) be the sets of all finite words over \(A\) and \(\lambda\) be the empty word. A substitution over \(A\) is a map \(\sigma:A\to A^\ast\backslash\{\lambda\}\). Let \(\Phi(X)=X^2-(t_{aa}+t_{bb})X +(t_{aa}t_{bb}-t_{ab}t_{ba})\), where \(t_{\alpha\beta}=|\sigma(\alpha)|_{\beta}\), the number of occurrences of the symbol \(\beta\) in the word \(\sigma(\alpha)\). Let \(w\) be any fixed point of a substitution \(\sigma\) in two symbols and let \(f_a(z)\) and \(f_b(z)\) be the generating functions of \(w\) for \(a\) and for \(b\), respectively. If \(t_{ab}t_{ba}\Phi(0)\Phi(-1)\neq 0\), then the numbers \(f_a(l^{-1})\) and \(f_b(l^{-1})\) are transcendental for any integer \(l\geq 2\). In the case of substitutions in two symbols of constant length, the condition \(t_{ab}t_{ba}\Phi(0)\Phi(-1)\neq 0\) can be removed. But in the case of nonconstant length this condition still remains.
Reviewer’s remark: S. Ferenczi and C. Mauduit [J. Number Theory 67, 146-161 (1997; Zbl 0895.11029)] and J.-P. Allouche and L. Q. Zamboni [J. Number Theory 69, 119-124 (1998; Zbl 0918.11016)] proved the same theorems by completely different methods.

MSC:

11B85 Automata sequences
11J81 Transcendence (general theory)
11A67 Other number representations
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