Sum of digits sequences modulo m

Let s_k(n) denote the sum of the digits of the base k representation of n. Define the sequence (or word) t_k,m=s _k(n)(mod m)_n≥0, which generalizes the well-known ThueMorse sequence t_2,2. We give a much shorter proof of the main result in Allouche and Shallit (2000) [1], which says that t_k,m has no overlaps (that is, contains no subword of the form axaxa, where x is any finite word and a is a single symbol), using techniques from Cusick and Stǎnicǎ (2009) [2]. We also give different proofs of some other results from Allouche and Shallit (2000) [1] and one result from Morton and Mourant (1991) [3], using the same techniques.