Let q be a positive integer. Consider an infinite word ω = w_0w_1w_2 … over an alphabet of cardinality q. A finite word u is called an arithmetic factor of ω if u = w_cw_(c+d)w_(c+2d)…w_(c+(|u|-1)d) for some choice of positive integers c and d. We call c the initial number and d the difference of u. For each such u we define its arithmetic index by [log_q d] where d is the least positive integer such that u occurs in w as an arithmetic factor with difference d. In this paper we study the rate of growth of the arithmetic index of arithmetic factors of a generalization of the Thue-Morse word defined over an alphabet of prime cardinality. More precisely, we obtain upper and lower bounds for the maximum value of the arithmetic index in ω among all its arithmetic factors of length n.
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