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Unique developments in non-integer bases. (English) Zbl 0918.11006
Given a real number \(q\) with \(1< q\leq 2\), a \(q\)-development is a series \(\sum^\infty_{n=1}\varepsilon_n q^{-n}= 1\) where \(\varepsilon_n= 0\) or 1 for all \(n\). Then it is shown that there is a smallest such \(q\) for which there is only one \(q\)-development. This \(q\) is the unique positive solution of the equation \(1= \sum^\infty_{i=1} \delta_iq^{-i}\), where the sequence \(\{\delta_i\}\) of zeros and ones is defined recursively as follows: \(\delta_1= 1\); if \(n\geq 0\) and if \(\delta_1,\dots,\delta_{2^n}\) are already defined, then set \(\delta_{2^n+k}= 1-\delta_k\) for \(1\leq k< 2^n\) and \(\delta_{2^{n+1}}= 1\).

MSC:
11A67 Other number representations
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