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Classification of Computably Approximable Real Numbers

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Abstract

A real number is called computably approximable if it is the limit of a computable sequence of rational numbers. Therefore the complexity of these real numbers can be classified by considering the convergence speed of computable sequences. In this paper we introduce a natural way to measure the convergence speed by counting the number of jumps of given sizes that appear after certain stages. Bounding the number of such kind of big jumps by different bounding functions, we introduce various classes of real numbers with different levels of approximation quality. We discuss further their mathematical properties as well as computability theoretical properties.

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Authors and Affiliations

  1. Department of Computer Science, Jiangsu University, Zhenjiang, 212013, China

    Xizhong Zheng

  2. Theoretische Informatik, BTU Cottbus, 03044, Cottbus, Germany

    Xizhong Zheng

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  1. Xizhong Zheng
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Correspondence to Xizhong Zheng.

Additional information

This work is supported by DFG (446 CHV 113/240/0-1) and NSFC (10420130638).

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Zheng, X. Classification of Computably Approximable Real Numbers. Theory Comput Syst 43, 603–624 (2008). https://doi.org/10.1007/s00224-007-9026-5

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  • DOI: https://doi.org/10.1007/s00224-007-9026-5

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