Arithmetic Self-Similarity of Infinite Sequences
Abstract
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.
Keywords
Cite
@article{arxiv.1201.3786,
title = {Arithmetic Self-Similarity of Infinite Sequences},
author = {Dimitri Hendriks and Frits G. W. Dannenberg and Joerg Endrullis and Mark Dow and Jan Willem Klop},
journal= {arXiv preprint arXiv:1201.3786},
year = {2012}
}