论文标题
关于与斯特里亚语单词有关的系列的超越
On the transcendence of a series related to Sturmian words
论文作者
论文摘要
令$ b $为一个代数号码，$ | b |> 1 $和$ \ MATHCAL {h} $一组有限的代数数字。我们研究表单$ \ sum_ {n = 0}^\ infty \ frac {a_n} {b^n} $的超越性，其中$ a_n \ in \ mathcal {h h} $ in \ mathbbbb {n} $中的所有$ n \ in \ mathcal {h} $。我们假设序列$（a_n）_ {n = 0}^\ infty $是通过在单位圆的非理性旋转下编码点的轨道来生成的。特别是，每当序列是Sturmian时，此假设就会存在。我们的主要结果表明，上述形式的所有数字都是先验的。此外，我们提供了足够的条件，可以在〜$ \ bar {\ mathbb {q}} $上线性独立的有限数字集。
Let $b$ be an algebraic number with $|b|>1$ and $\mathcal{H}$ a finite set of algebraic numbers. We study the transcendence of numbers of the form $\sum_{n=0}^\infty \frac{a_n}{b^n}$ where $a_n \in \mathcal{H}$ for all $n\in\mathbb{N}$. We assume that the sequence $(a_n)_{n=0}^\infty$ is generated by coding the orbit of a point under an irrational rotation of the unit circle. In particular, this assumption holds whenever the sequence is Sturmian. Our main result shows that all numbers of the above form are transcendental. We moreover give sufficient conditions for a finite set of such numbers to be linearly independent over~$\bar{\mathbb{Q}}$.
