Substitution invariant cutting sequences. (English) Zbl 0786.11041

Let \(\alpha\) be a real number. The sequence \(f_ \alpha\) of 0’s and 1’s arising from the formula \(f_ \alpha(n)= [(n+1)\alpha]- [n\alpha]\) can be transformed by substituting the 0’s and 1’s by finite strings of 0’s and 1’s respectively. A complete characterisation of those \(\alpha\) for which there is an invariant non-trivial substitution is obtained by relating the \(f_ \alpha\) to cutting sequences and continued fraction expansions. For example, if \(\alpha\) is an irrational in \((0,{1\over 2})\), then \(f_ \alpha\) is invariant under some non-trivial substitution if and only if \(\alpha=[0,a_ 1,\overline{a_ 2,\dots, a_ n}]\), where \(a_ n+1\geq a_ 1\geq 2\).


11J70 Continued fractions and generalizations
11B99 Sequences and sets
11A55 Continued fractions
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