Factors of characteristic words of irrational numbers

Let β be an irrational number between 0 and 1. The characteristic word f(β) of β is defined to be the infinite word over {0,1} whose nth letter is [(n+1)β]-[nβ], n⩾1. It is well known that, for each m⩾1, f(β) 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 x0(m),x1(m),…,xm(m) are determined, these factors are increasing in the lexicographic order and their moments M(x0(m)),M(x1(m)),…,M(xm(m)) form an increasing sequence of m+1 consecutive integers. Some known results about generating factors of f(β) using the unbordered α-words and their conjugates turn out to be consequences of our main theorem.