Factors of characteristic words of irrational numbers

Let P be an irrational number between 0 and 1. The characteristic word f(beta) of beta is defined to be the infinite word over {0, 1} whose nth letter is [(n + 1)beta] - [n beta], n >= 1. It is well known that, for each m >= 1, f (beta) has exactly m + 1 distinct factors of length m. In this paper, we shall develop a method to construct these factors. Under our construction, the 1-sets of these m + 1 factors x(0)((m)), x(1)((m)) x(m)((m)) are determined, these factors are increasing in the lexicographic order and their moments M(x(0)((m))), M(x(1)((m))),...,M(x(m)((m))) form an increasing sequence of m + 1 consecutive integers. Some known results about generating factors of f (beta) using the unbordered alpha-words and their conjugates turn out to be consequences of our main theorem. (c) 2005 Elsevier B.V. All rights reserved.