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On some problems of I. Joó. (English) Zbl 0766.11004

The authors provide negative answers to two problems raised bo I. Joó: Does there exist an expansion \(1=\sum_{i=1}^ \infty q^{-n_ i}\) satisfying \(\sup(n_{i+1}-n_ i)=\infty\) for every \(1<q<(1+\sqrt{5})/2\) and, for every \(1<q<2\) is the existence of such an expansion equivalent to the existence of \(2^{\aleph_ 0}\) such different expansions?

MSC:

11A67 Other number representations
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References:

[1] J. W. S. Cassels,An Introduction to Diophantine Approximation, Cambridge University Press, 1957. · Zbl 0077.04801
[2] I. N. Stewart and D. O. Tall,Algebraic Number Theory, Chapman and Hall (London, 1979). · Zbl 0413.12001
[3] I. Joó, On Riesz bases,Annales Univ. Sci. Budapest., Sectio Math.,31 (1988), 141–153. · Zbl 0674.46006
[4] P. Erdos and I. Joó, On the expansion \(1 = \sum {q^{ - n_i } } \) ,Per. Math. Hung.,23 (1991).
[5] P. Erdos, M. Joó and I. Joó, On a problem of Tamás Varga (to appear).
[6] J. Dufresnoy, C. Pisot, Sur un ensemble fermé d’entiers algébriques,Ann. Sci. Éc. Norm. Sup. Paris,70 (1953), 105–134. · Zbl 0051.02904
[7] I. Joó and M. Joó, On an arithmetical property of ,Publ. Math. Debrecen,37 (1990).
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