Rational sequences converging to left-c.e. reals of positive effective Hausdorff dimension
Hiroyuki Imai, Masahiro Kumabe, Kenshi Miyabe, Yuki Mizusawa, Toshio Suzuki
Computability Theory and Foundations of Mathematics
Proceedings of the 9th International Conference on Computability Theory and Foundations of Mathematics
The 9th International Conference on Computability Theory and Foundations of Mathematics, Wuhan, China, 21 – 27 March 2019
https://doi.org/10.1142/12917 | June 2022
Edited By: NingNing Peng (Wuhan University of Technology, China), Kazuyuki Tanaka (Tohoku University, Japan), Yue Yang (National University of Singapore, Singapore), Guohua Wu (Nanyang Technological University, Singapore) and Liang Yu (Nanjing University, China)
In our previous work, we characterized Solovay reducibility using Lipschitz condition,
and introduced quasi Solovay reducibility (qS-reducibility, for short) as a H ̈older condition counterpart.
In this paper, we investigate effective dimensions and ideals closely related to quasi Solovay reducibility by means of the rate of convergence.
We show that the qS-completeness among left-c.e. reals is equivalent to having a positive effective Hausdorff dimension.
The Solovay degrees of qS-complete left-c.e. reals form a filter. On the other hand, the Solovay degrees of non-qS-complete left-c.e. reals do not form an ideal.
Based on observations on the relationships between rational sequences and reducibility, we introduce a stronger version of qS-reducibility.
Given a degree of this reducibility, the lower cone (including the given degree) forms an ideal.
By developing these investigations, we characterize the effective dimensions by means of the rate of convergence.
We give a variation of the first incompleteness theorem based on Solovay reducibility.