A characterization of the quadratic irrationals. (English) Zbl 0688.10007

Let \(\alpha\) be a positive irrational real number, and let \(f_{\alpha}(n)=[(n+1)\alpha]-[n\alpha]-[\alpha]\), \(n\geq 1\), where [x] denotes the greatest integer not exceeding x. It is shown that the sequence \(f_{\alpha}\) has a certain ‘substitution property’ if and only if \(\alpha\) is the root of a quadratic equation over the rationals. In the simplest case (which depends on the simple continued fraction for \(\alpha)\), this means that there are blocks \(B_ 1\) and \(B_ 2\) of of 0’s and 1’s such that if every 0 in the sequence \(f_{\alpha}\) is replaced by \(B_ 1\), and every 1 is replaced by \(B_ 2\), then the resulting sequence is identical to \(f_{\alpha}\).
Reviewer: T.C.Brown


11A55 Continued fractions
11B83 Special sequences and polynomials
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