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

Adamczewski, Boris; Bugeaud, Yann

Ann. Math. (2) 165, No. 2, 547-565 (2007).

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.

Reviewer: Olaf Ninnemann (Uffing am Staffelsee)

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

Citations:

Zbl 0895.11029