On the complexity of algebraic numbers I. Expansions in integer bases

• 成果类型: Article
• 署名作者: Adamczewski, Boris; Bugeaud, Yann
• 刊物名称: ANNALS OF MATHEMATICS
• ISSN/ISSBN: 0003-486X
• DOI: 10.4007/annals.2007.165.547
• 发表日期: 2007
• 页码: 547-565
• 关键词: quantitative subspace theorem; arithmetic characteristics; transcendental-numbers; functional-equations; symbolic dynamics; real numbers; series; SEQUENCES; morphisms; constant

摘要

Let b > 2 be an integer. We prove that the b-ary expansion of every irrational algebraic number cannot have low complexity. Furthermore, we establish that irrational morphic numbers are transcendental, for a wide class of morphisms. In particular, irrational automatic numbers are transcendental. Our main tool is a new, combinatorial transcendence criterion.