Authors: LU-MING SHEN, YUE-HUA LIU, YU-YUAN ZHOU
Abstract: For any x\in (0,1], letx = \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) \to \infty as d \to \infty. In this note, we give a bound estimation of more general case without the former assumption.
Keywords: Oppenheim series expansion; Restricted Oppenheim series expansion; Lüroth series; Hausdorff dimension
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