Atlantic Graph Theory Seminar: Lucas Mol, Thomson Rivers University

A word is a sequence of symbols taken from some finite alphabet. A square is a word of the form xx, where x is a nonempty word. It is well-known that there are infinite words over an alphabet of size 3 that contain no squares. Suppose now that the alphabet is some finite subset of the integers. An additive square is a word of the form xx’, where x and x’ have the same nonzero length and the same sum. Additive cubes, fourth powers, etc., are defined similarly. We present a method for proving that certain types of infinite words contain no additive k-powers. This is joint work with James Currie, Narad Rampersad, and Jeffrey Shallit.