A note on a problem of J. Galambos

For any $x\epsilon(0, 1]$, let $x =\frac{1}{d_1}+\frac{a_1}{b_1}\frac{1}{d_2}+...+ \frac{a_1a_2...a_n}{b_1b_2...b_n}\frac{1}{d_{n+1}}+...$ be the Oppenheim series expansion of x. In this paper, we investigate the Hausdorff dimension of the set $B_m = {x : 1 < d_j/h_{j-1}(d_{j-1})\leq m,j\geq...

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Bibliographic Details
Published inTurkish journal of mathematics Vol. 32; no. 1; pp. 103 - 109
Main Authors LIU, Yue-hua, SHEN, Lu- ming, ZHOU, Yu-yuan
Format Journal Article
LanguageEnglish
Published TÜBİTAK 01.01.2008
Subjects
Hausdorff boyutu
Hausdorff dimension
Matematik
Mathematics
Seri açılımı
series expansion
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Summary:For any $x\epsilon(0, 1]$, let $x =\frac{1}{d_1}+\frac{a_1}{b_1}\frac{1}{d_2}+...+ \frac{a_1a_2...a_n}{b_1b_2...b_n}\frac{1}{d_{n+1}}+...$ be the Oppenheim series expansion of x. In this paper, we investigate the Hausdorff dimension of the set $B_m = {x : 1 < d_j/h_{j-1}(d_{j-1})\leq m,j\geq 1}$ which J. Galambos posed as an open question in 1976(see[6]). In [11], it has been considered with the condition $h_j (d)→∞as d→∞. In this note, we give a bound estimation of more general case without the former assumption.
Bibliography:TMUH
ISSN:1300-0098
1303-6149