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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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