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
Full Text: DOI Numdam EuDML


[1] Brown, T.C., A characterisation of the quadratic irrationals, Canad. Math. Bull.34 (1991), 36-41. · Zbl 0688.10007
[2] Cohn, H., Some direct limits of primitive homotopy words and of Markoff geodesics, Discontinuous groups and Riemann surfaces, Ann. of Math. Studies No. 79, Princeton Univ. Press, Princeton, N.J., (1974), 81-98. · Zbl 0294.20044
[3] Fraenkel, A.S., Determination of [nθ] by its sequence of differences, Canad. Math. Bull.21 (1978), 441-446. · Zbl 0401.10018
[4] Sh., Ito and Yasutomi, S., On continued fractions, substitutions and characteristic sequences [nx + y] - [(n - 1)x + y], Japan J. Math.16 (1990), 287-306. · Zbl 0721.11009
[5] Series, C., The geometry of Markoff numbers, Math. Intelligencer7 (1985), 20-29. · Zbl 0566.10024
[6] Stolarsky, K.B., Beatty sequences, continued fractions, and certain shift operators, Canad. Math. Bull.19 (1976), 473-482. · Zbl 0359.10028
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.