Hausdorff dimension of some kind of alternating Oppenheim series expansion

Abstract

For Oppenheim series epansions, the authors of [7] discussed the exceptional sets

$$B_m = \{ x \in (0,1]:1 < \frac{{d_j (x)}}{{h_{j - 1} (d_{j - 1} (x))}} \leqslant m for any j \geqslant 2\} $$

. In this paper, we investigate the Hausdorff dimension of a kind of exceptional sets occurring in alternating Oppenheim series expansion. As an application, we get the exact Hausdorff dimension of the set in Lüroth series expansion, also we give an estimate of such dimensional number.