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Zbl 1195.11094
Adamczewski, Boris; Bugeaud, Yann
On the complexity of algebraic numbers. I: Expansions in integer bases. (English)
[J] Ann. Math. (2) 165, No. 2, 547-565 (2007). ISSN 0003-486X; ISSN 1939-0980/e

From the text: Let $b\geq 2$ be an integer. In the present paper, we prove new results concerning both notions of complexity. Our Theorem 1 provides a sharper lower estimate for the complexity of the $b$-adic expansion of every irrational algebraic number. We are still far away from proving that such an expansion is normal, but we considerably improve upon the earlier known results. We further establish (Theorem 2) the conjecture of Loxton and van der Poorten, namely that irrational automatic numbers are transcendental. Our proof yields more general statements and allows us to confirm that irrational morphic numbers are transcendental, for a wide class of morphisms (Theorems 3 and 4). We derive Theorems 1 to 4 from a refinement (Theorem 5) of the combinatorial criterion from [{\it S. Ferenczi} and {\it C. Mauduit}, J. Number Theory 67, 146--161 (1997; Zbl 0895.11029)], that we obtain as a consequence of the Schmidt Subspace Theorem.
[Olaf Ninnemann (Berlin)]

MSC 2000:: *11J81 Transcendence (general theory)
11J87
11K16 Normal numbers, etc.
11A63 Radix representation
11B85 Automata sequences

Citations: Zbl 0895.11029

Cited in: Zbl 1260.11001 Zbl 1255.11037 Zbl 1200.11053 Zbl 1236.11062 Zbl 1113.11008 Zbl 1134.11011 Zbl 1113.11041 Zbl 1195.11093 Zbl 1119.11020

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