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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
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