正则 episturmian 词的初始非重复复杂度及其丢番图指数
Initial nonrepetitive complexity of regular episturmian words and their Diophantine exponents
AI总结：
本文提出评估正则 episturmian 词初始非重复复杂度的方法，发展广义 Ostrowski 记数系统理论，并以此刻画其丢番图指数的有限性与下界，进而发现无理指数严格大于2的新不可数超越数类。
摘要：
正则 episturmian 词是指其指示词具有正则且受限的形式，使得它们的行为更像 Sturmian 词而非一般的 episturmian 词。我们提出了一种评估正则 episturmian 词初始非重复复杂度的方法，扩展了 Wojcik 在 Sturmian 词上的工作。为此，我们发展了广义 Ostrowski 记数系统理论，并展示了如何将每个 episturmian 词与该记数系统下写成的唯一数字序列相关联。\n初始非重复复杂度的描述使我们能够获得关于正则 episturmian 词丢番图指数的新结果。我们证明，正则 episturmian 词的丢番图指数是有限的，当且仅当其指示词具有有界的部分商。此外，我们证明，如果部分商序列最终至少为 $3$，则正则 episturmian 词的丢番图指数严格大于 $2$。\n给定整数字母表上的无限词 $x$，我们可以考虑一个以 $x$ 为小数部分的实数 $ξ_x$。$x$ 的丢番图指数是 $ξ_x$ 的无理指数的下界。因此，我们的结果为小数部分为正则 episturmian 词的实数的无理指数提供了非平凡的下界。作为推论，我们确定了一类新的不可数超越数类，其无理指数严格大于 $2$。该类包含一个不可数的 Liouville 数子类。
英文摘要：
Regular episturmian words are episturmian words whose directive words have a regular and restricted form making them behave more like Sturmian words than general episturmian words. We present a method to evaluate the initial nonrepetitive complexity of regular episturmian words extending the work of Wojcik on Sturmian words. For this, we develop a theory of generalized Ostrowski numeration systems and show how to associate with each episturmian word a unique sequence of numbers written in this numeration system. The description of the initial nonrepetitive complexity allows us to obtain novel results on the Diophantine exponents of regular episturmian words. We prove that the Diophantine exponent of a regular episturmian word is finite if and only if its directive word has bounded partial quotients. Moreover, we prove that the Diophantine exponent of a regular episturmian word is strictly greater than $2$ if the sequence of partial quotients is eventually at least $3$. Given an infinite word $x$ over an integer alphabet, we may consider a real number $ξ_x$ having $x$ as a fractional part. The Diophantine exponent of $x$ is a lower bound for the irrationality exponent of $ξ_x$. Our results thus yield nontrivial lower bounds for the irrationality exponents of real numbers whose fractional parts are regular episturmian words. As a consequence, we identify a new uncountable class of transcendental numbers whose irrationality exponents are strictly greater than $2$. This class contains an uncountable subclass of Liouville numbers.
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