2012/01/18 by Dimitri Hendriks, Frits G. W. Dannenberg, Hendriks, Dimitri +7 · 1 citation
Computer Science · Mathematics · #Combinatorics (math.CO) #FOS: Computer and information sciences #FOS: Mathematics #Logic in Computer Science (cs.LO) #Number Theory (math.NT) #cs.LO #math.CO #math.NT
paper · pdf · doi:10.48550/arxiv.1201.3786
arxiv created 2012/05/21 · arxiv updated 2012/05/22
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.