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arXiv · 1201.3786

Arithmetic Self-Similarity of Infinite Sequences

Dimitri Hendriks·Frits G. W. Dannenberg·Joerg Endrullis·Mark Dow·Jan Willem Klop
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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.

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Dimitri Hendriks, Frits G. W. Dannenberg, Joerg Endrullis, Mark Dow, Jan Willem Klop. 2012-01-18. Arithmetic Self-Similarity of Infinite Sequences. https://arxiv.org/abs/1201.3786

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