  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.
