On Arithmetic Progressions in the Generalized Thue-Morse Word

摘要

We consider the generalized Thue-Morse word on the alphabet E = {0,1,2}. The object of the research is arithmetic progressions in this word, which are sequences consisting of identical symbols. Let A(d) be a function equal to the maximum length of an arithmetic progression with the difference d in the generalized Thue-Morse word. If n, d are positive integers and d is less than 3~n, then the upper bound for A(d) is 3~n + 6 and is attained with the difference d = 3~n - 1.