Komornik, Vilmos; Loreti, Paola Unique developments in non-integer bases. (English) Zbl 0918.11006 Am. Math. Mon. 105, No. 7, 636-639 (1998). 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 \(\delta_{2^{n+1}}= 1\). Reviewer: Joachim Piehler (Merseburg) Cited in 4 ReviewsCited in 70 Documents MSC: 11A67 Other number representations Keywords:unique developments; non-integer bases; \(q\)-development PDFBibTeX XMLCite \textit{V. Komornik} and \textit{P. Loreti}, Am. Math. Mon. 105, No. 7, 636--639 (1998; Zbl 0918.11006) Full Text: DOI Online Encyclopedia of Integer Sequences: Decimal expansion of Komornik-Loreti constant.