## On the distribution of the set $$\{\sum^ n_{i=1}\varepsilon _ i q^ i : \varepsilon_ i \in\{0,1\}, n \in \mathbb{N}\}$$.(English)Zbl 0766.11005

Fix $$1<q<\sqrt{2}$$ and let $$\{y_ n\}=\{\sum_{i=1}^ n \varepsilon_ i q^{2(n-i)}$$: $$\varepsilon_ i\in\{0,1\}\}$$, $$n=1,2,3,\dots\;$$. The author shows that if $$y_{n+1}-y_ n\to 0$$ as $$n\to\infty$$ then there exists an expansion $$1=\sum_{i=1}^ \infty q^{-n_ i}$$ such that $$\sup_ i(n_{i+1}-n_ i)=\infty$$.

### MSC:

 11A67 Other number representations

### Keywords:

expansions; Pisot numbers
Full Text:

### References:

 [1] A. Bogmér, M. Horváth, A. Sövegjártó, On some problems of I. Joó,Acta Math. Hung.,58 (1991), 153–155. · Zbl 0766.11004
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