On the complexity of algebraic numbers. I: Expansions in integer bases.(English)Zbl 1195.11094

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 [S. Ferenczi and C. Mauduit, J. Number Theory 67, 146–161 (1997; Zbl 0895.11029)], that we obtain as a consequence of the Schmidt Subspace Theorem.

MSC:

 11J81 Transcendence (general theory) 11J87 Schmidt Subspace Theorem and applications 11K16 Normal numbers, radix expansions, Pisot numbers, Salem numbers, good lattice points, etc. 11A63 Radix representation; digital problems 11B85 Automata sequences

Zbl 0895.11029
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