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On the sequence of numbers of the form $$\varepsilon_0+\varepsilon_1q+\dots+\varepsilon_n q^n$$, $$\varepsilon \in\{0,1\}$$. (English) Zbl 0896.11006
Let $$k=\varepsilon_0+ 2\varepsilon_1+\dots+2^n\varepsilon_n$$ be the dyadic expansion of the nonnegative integer $$k$$ and set $$x_k= \varepsilon_0+\varepsilon_1q+\dots+\varepsilon_nq^n$$, where $$q$$ is a fixed real number with $$1<q<2$$. Denote by $$y_0<y_1<y_2<\dots$$ the increasing rearrangement of the sequence $$(x_k)$$ without repetitions. Furthermore, let $$l(q)=\liminf (y_{k+1}-y_k)=\inf (y_{k+1}-y_k)$$ and $$L(q)=\limsup (y_{k+1}-y_k)$$.
Continuing earlier investigations of P. Erdős, I. Jó and V. Komornik [Bull. Soc. Math. Fr. 118, 377–390 (1990; Zbl 0721.11005)] and P. Erdős, I. Joó and M. Joó [ibid. 120, 507–521 (1992; Zbl 0787.11002)], the authors obtain new estimates of $$l(q)$$ and $$L(q)$$ for certain special classes of numbers $$1<q<2$$. In particular, they prove the following: $$l(q)>0$$ for all Pisot numbers and $$L(q)=0$$, i.e. $$y_{k+1}-y_k \to 0$$ for all transcendental numbers $$1<q<\sqrt 2$$.

MSC:
 11A67 Other number representations 11B83 Special sequences and polynomials 11R06 PV-numbers and generalizations; other special algebraic numbers; Mahler measure
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