无穷序列的算术自相似性
Arithmetic Self-Similarity of Infinite Sequences
摘要：
定义了单边无穷序列sigma的算术自相似性为通过sigma的等差数列的集合，这些等差数列是sigma的垂直移动。本文研究了几类序列，即完全可加序列、Toeplitz字和Keane广义Morse序列的AS。我们给出了完全加性序列的AS的一个完整的特征，并对产生完全加性Toeplitz词的单间隙Toeplitz模式集进行了分类。我们表明，每一个算术子序列的Toeplitz字所产生的一个缺口模式再次Toeplitz字。最后，我们建立了广义Morse序列是特定的数字和序列，并证明了它们的第一差是Toeplitz字。
英文摘要：
We define the arithmetic self-similarity (AS) of a one-sided infinite sequence sigma to be the set of arithmetic progressions through sigma which are a vertical shift of sigma. We study the AS of several famlies of sequences, viz. completely additive sequences, Toeplitz words and Keane's generalized Morse sequences. We give a complete characterization of the AS of completely additive sequences, and classify the set of single-gap Toeplitz patterns that yield completely additive Toeplitz words. We show that every arithmetic subsequence of a Toeplitz word generated by a one-gap pattern is again a Toeplitz word. Finally, we establish that generalized Morse sequences are specific sum-of-digits sequences, and show that their first difference is a Toeplitz word.
机构、总结、中文摘要均由AI分析而来，可能存在错误，请以论文正文为准。
