Arithmetic Self-Similarity of Infinite Sequences
  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. 
Dimitri Hendriks and Frits G. W. Dannenberg and Joerg Endrullis and   Mark Dow and Jan Willem Klop
math.CO cs.LO math.NT