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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

MSC:
 11A55 Continued fractions 11B83 Special sequences and polynomials
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