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Erdős, Pál; Joó, István; Komornik, Vilmos

Characterization of the unique expansions \(1=\sum^{\infty}_{i=1}q^{-n_ i}\) and related problems. (English) Zbl 0721.11005

Bull. Soc. Math. Fr. 118, No. 3, 377-390 (1990).
Given a nonintegral base \(1<q<2\), the authors characterize the expansions of 1 base \(q\) for which uniqueness holds. Estimates for the number of consecutive zero digits in the greedy expansion of 1 are also obtained. These results are related to the sequence of real numbers which has finite base \(q\) expansions. Some open problems are given at the end of the article.
Reviewer: Arnold Knopfmacher (Wits)

Cited in 14 Reviews
Cited in 89 Documents

MSC:

11A67 Other number representations

Keywords:

nonintegral base; uniqueness; number of consecutive zero digits; greedy expansion; real numbers
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\textit{P. Erdős} et al., Bull. Soc. Math. Fr. 118, No. 3, 377--390 (1990; Zbl 0721.11005)
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References:

[1] RÉNYI (A.) . - Representations for real numbers and their ergodic properties , Acta Math. Hungar., t. 8, 1957 , p. 477-493. MR 20 #3843 | Zbl 0079.08901 · Zbl 0079.08901
[2] PARRY (W.) . - On the \beta -expansions of real numbers , Acta Math. Hungar., t. 11, 1960 , p. 401-416. MR 26 #288 | Zbl 0099.28103 · Zbl 0099.28103
[3] BERTRAND (A.) . - Développements en base de Pisot et répartition modulo 1 de la suite ({x\?n}) , Thèse de 3ème cycle, Université de Bordeaux I, n^\circ 1595, 1981 .
[4] ERDÖS (P.) and JOÓ (I.) . - On the expansions 1 = \Sigma q-ni, 1 &lt; q &lt; 2 , to appear.
[5] ERDÖS (P.) , HORVÁTH (M.) and JOÓ (I.) . - On the uniqueness of the expansions 1 = \Sigma q-ni , to appear.
[6] JOÓ (I.) . - On the distribution of the set {\Sigma Ni=1 \epsilon iqi : \epsilon i = 0,1} , to appear. Zbl 0766.11005 · Zbl 0766.11005
[7] JOÓ (I.) and JOÓ (M.) . - On an arithmetical property of \surd 2 , to appear. · Zbl 0766.11006
[8] BOGMÉR (A.) , HORVÁTH (M.) and SÖVEGJÁRTÓ (A.) . - On some problems of I. JOÓ , to appear. · Zbl 0766.11004
[9] RAUZY (G.) . - Sur les développements en base non entière , [Colloque de Théorie des nombres, Alger], 21-23 octobre 1989 , to appear.
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