Factors of characteristic words of irrational numbers

• Authors: Wai-Fong Chuan
• Venue: Theoretical Computer Science
• Year: 2005

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.