Arithmetic Self-Similarity of Infinite Sequences

TL;DR

This paper investigates the arithmetic self-similarity of infinite sequences, providing characterizations for various sequence classes and revealing structural properties of Toeplitz and Morse sequences.

Contribution

It offers a complete characterization of the arithmetic self-similarity for completely additive sequences and classifies Toeplitz patterns that produce such sequences.

Findings

Complete characterization of AS for completely additive sequences

Classification of Toeplitz patterns yielding additive sequences

First difference of generalized Morse sequences is a Toeplitz word

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.