A Representation Theorem of the Suffixes of Characteristic Sequences

Let α be an irrational number between 0 and 1 with continued fraction [0, a1, + 1, a2, …]. The characteristic sequence of α is by definition an infinite word f over the alphabet {0, 1}, whose nth letter is [(n + 1)α − nα], n ⩾ 1. A.A. Markov has proved that f has a representation f = u1u2…, where u0 = 0, u1 = 0a1 1, un = (un − 1)an − 1 un − 2un − 1, n ⩾ 2. On the other hand, A. de Luca has proved that f = (x0)a1 − 1(x2)a3 (x4)a5…, where x0 = 0, x1 = 0a1 1, xn = (xn − 1)an xn − 2, n ⩾ 2. Although these representations look different, they have one thing in common with each other — the un's and the xn's are α-words. The class of α-words associated with the irrational number α was introduced by the author in an earlier paper. In this paper, we prove a general representation theorem of the suffixes of f which involves α-words. Some known representations, including the above mentioned ones, are special cases or consequences of it.