Abstract
Suppose p is a computable real so that \(p \ge 1\). We define the isometry degree of a computable presentation of \(\ell ^p\) to be the least powerful Turing degree \(\mathbf {d}\) by which it is \(\mathbf {d}\)-computably isometrically isomorphic to the standard presentation of \(\ell ^p\). We show that this degree always exists and that when \(p \ne 2\) these degrees are precisely the c.e. degrees.
D. Stull—Research of the first author supported in part by a Simons Foundation grant # 317870. Research of the second author supported in part by National Science Foundation Grants 1247051 and 1545028.
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Acknowledgments
We thank U. Andrews, R. Kuyper, S. Lempp, J. Miller, and M. Soskova for very helpful conversations during the first author’s visit to the University of Wisconsin; in particular for suggesting the use of enumeration reducibility. We also thank Diego Rojas for proofreading and making several very useful suggestions. Finally, we thank the reviewers for helpful comments and suggestions.
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McNicholl, T.H., Stull, D. (2018). The Isometry Degree of a Computable Copy of \(\ell ^p\). In: Manea, F., Miller, R., Nowotka, D. (eds) Sailing Routes in the World of Computation. CiE 2018. Lecture Notes in Computer Science(), vol 10936. Springer, Cham. https://doi.org/10.1007/978-3-319-94418-0_28
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DOI: https://doi.org/10.1007/978-3-319-94418-0_28
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