A representation theorem of the suffixes of characteristic sequences

✍ Scribed by Chuan Wai-Fong

• Publisher: Elsevier Science
• Year: 1998
• Tongue: English
• Weight: 574 KB
• Volume: 85
• Category: Article
• ISSN: 0166-218X
• DOI: 10.1016/s0166-218x(97)00142-x

✦ Synopsis

Let CL be an irrational number between 0 and 1 with continued fraction [0, al + 1,~.

.]. The characteristic sequence of x is by definition an infinite word f over the alphabet (0. 1 }, whose nth letter is [(n + 1)cx -ncr], n > 1. A.A. Markov has proved that f has a representation ~'=uIu~..., where uo=O, UI =O"'l, u~=(u~-I)~~~'u~~-~u~-_I, n32. On the other hand, A. de Luca has proved that f =(&)'I-(F2)03 (F4)a5 . , where x0 = 0, XI = 0"' 1, xn = (x,_ I )"'I ~~-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 g-words. The class of cc-words associated with the irrational number a 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 a-words. Some known representations, including the above mentioned ones, are special cases or consequences of it.