Geodesic
Sums of Digits, Overlaps, and Palindromes
Discrete mathematics & theoretical computer science, Tome 4 (2000-2001) no. 1.

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Let s_k(n) denote the sum of the digits in the base-k representation of n. In a celebrated paper, Thue showed that the infinite word (s_2(n) \bmod 2)_n≥ 0 is \emphoverlap-free, i.e., contains no subword of the form axaxa where x is any finite word and a is a single symbol. Let k,m be integers with k>2, m≥ 1. In this paper, generalizing Thue's result, we prove that the infinite word t_k,m := (s_k(n) \bmod m)_n≥ 0 is overlap-free if and only if m≥ k. We also prove that t_k,m contains arbitrarily long squares (i.e., subwords of the form xx where x is nonempty), and contains arbitrarily long palindromes if and only if m≤ 2. 
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     author = {Allouche, Jean-Paul and Shallit, Jeffrey},
     title = {Sums of {Digits,} {Overlaps,} and {Palindromes}},
     journal = {Discrete mathematics & theoretical computer science},
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Allouche, Jean-Paul; Shallit, Jeffrey. Sums of Digits, Overlaps, and Palindromes. Discrete mathematics & theoretical computer science, Tome 4 (2000-2001) no. 1. doi : 10.46298/dmtcs.282. http://geodesic.mathdoc.fr/articles/10.46298/dmtcs.282/

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