On continued fractions, substitutions and characteristic sequences \([nx+y]-[(n-1)x+y]\). (English) Zbl 0721.11009

Ito, Shunji; Yasutomi, Shin-ichi

Jap. J. Math., New Ser. 16, No. 2, 287-306 (1990).

Let x,y be real numbers with \(0<x<1\), \(0\leq y\) and \(x+y\leq 1\). Then the sequence \(c_ n(x,y):=[nx+y]-[nx-x+y]\), \(n=0,1,2,..\). is considered. For the special case \(y=0\) the sequence \(c_ n(x,0)\) can be expressed by the continued fraction expansion of x. A new characterization of this sequence is given which uses the map T: [0,1]\(\to [0,1]\), \(Tx=x/(1-x)\), \(0\leq x<1/2\), \(Tx=(2x-1)/x\), \(1/2\leq x<1\) and substitutions of 0-1- sequences. This method can be generalized to the case \(y\neq 0\). Here a suitable planar map is used.

Reviewer: F.Schweiger (Salzburg)

MSC:

11B83	Special sequences and polynomials
11A55	Continued fractions
11A63	Radix representation; digital problems

Keywords:

characteristic sequence; continued fraction expansion; substitutions of 0-1-sequences