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Developments in non-integer bases. (English) Zbl 0906.11008
For a fixed real number $$q>1$$, the sequence $$0=y_0<y_1<y_2<\dots$$ of those real numbers $$y$$ is considered which have at least one representation of the form $y=\varepsilon_0+\varepsilon_1 q+\dots +\varepsilon_n q^n, \quad n\geq 0,\;\varepsilon_i\in \{0,1\}.$ It is shown that $$y_{k+1}-y_k$$ tends to zero for all $$q$$ between 1 and $$2^{1/4}$$, except possibly the square root of the second Pisot number $$\sqrt{p_2}\approx 1.175$$. Moreover, for each such $$q$$ there exists a sequence $$(\varepsilon_i)$$, $$\varepsilon_i\in \{0,1\}$$, such that $$\sum_{i>1} \varepsilon_i q^{-i}=1$$. In particular, for each such $$q$$ there even exists a development containing all possible finite variations of the digits 0 and 1.

##### MSC:
 11A67 Other number representations 11R06 PV-numbers and generalizations; other special algebraic numbers; Mahler measure
##### Keywords:
digital expansions; non-integer bases
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