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PreprintCC BY 4.0arXiv

Some variants of function domination

  • Xuanheng Zhao
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Abstract

We investigate whether classical recursion-theoretic notions defined by function domination remain unchanged when success is required only on sets of prescribed asymptotic density. We establish robustness for many such notions and uncover unexpected connections between familiar degree properties by comparing upper and lower density. In contrast, pdomination, as defined by Chong, Hoi, Stephan and Turetsky, is not robust under density weakenings. We also answer a question of Kjos-Hanssen and Nies by characterizing superhighness as domination with a threshold boundedly limit recursive in the oracle. As an application, we introduce supermaximal sets, a strengthening of maximality whose degrees are exactly the superhigh r.e. degrees.

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Cite

BibTeXRIS
BibTeX
@misc{zhao2026some,
  title = {{Some variants of function domination}},
  author = {Xuanheng Zhao},
  howpublished = {arXiv},
  year = {2026},
  month = nov,
  doi = {10.48550/arxiv.2611.01420},
  eprint = {2611.01420},
  archivePrefix = {arXiv},
  url = {https://arxiv.org/abs/2611.01420},
  note = {CC BY 4.0},
}
RIS
TY  - UNPB
TI  - Some variants of function domination
AU  - Xuanheng Zhao
PY  - 2026
DA  - 2026/11/01
PB  - arXiv
DO  - 10.48550/arxiv.2611.01420
UR  - https://arxiv.org/abs/2611.01420
AB  - We investigate whether classical recursion-theoretic notions defined by function domination remain unchanged when success is required only on sets of prescribed asymptotic density. We establish robustness for many such notions and uncover unexpected connections between familiar degree properties by comparing upper and lower density. In contrast, pdomination, as defined by Chong, Hoi, Stephan and Turetsky, is not robust under density weakenings. We also answer a question of Kjos-Hanssen and Nies by characterizing superhighness as domination with a threshold boundedly limit recursive in the oracle. As an application, we introduce supermaximal sets, a strengthening of maximality whose degrees are exactly the superhigh r.e. degrees.
ER  -