On a sequence related to that of Thue–Morse and its applications Original Research Article

It is known that the sequence image of lengths of blocks of identical symbols in the Thue–Morse sequence has several extremal properties among all non-periodic sequences of the symbols 1 and image. Its generating function image is equal to image. In terms of combinatorics on words, for any given image and image, we prove that every non-periodic word of an alphabet image has a suffix image whose generating function image satisfies the inequality image. Using this, we prove several bounds for the largest and the smallest limit points of the sequence of fractional parts image, image, where image is a negative rational number and image is a real number. Our results show, for example, that, for any real number image, the sequence of fractional parts image, image, has a limit point greater than image. Furthermore, for each integer image and each real number image, we prove that image and show that this inequality is sharp.