Arithmetic Self-Similarity of Infinite Sequences

We define the arithmetic self-similarity (AS) of a one-sided infinite sequence σ to be the set of arithmetic subsequences of σ which are a vertical shift of σ. We study the AS of several families 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.