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

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.

### MSC:

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