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Effective Versions of Strong Measure Zero

Author Matthew Rayman

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LIPIcs.STACS.2026.75.pdf
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Subject Classification

ACM Subject Classification
  • Theory of computation
Keywords
  • Strong measure zero
  • NCR
  • Effective fractal dimensions
  • Borel’s Conjecture
  • Hausdorff dimension
  • Packing dimension

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Abstract

Effective versions of strong measure zero sets are developed for various levels of complexity and computability. It is shown that the sets can be equivalently defined using a generalization of supermartingales called odds supermartingales, success rates on supermartingales, predictors, and coverings. We show Borel’s conjecture that a set has strong measure zero if and only if it is countable holds in the time and space bounded setting. At the level of computability this does not hold. We show the computable level contains sequences at arbitrary levels of the hyperarithmetical hierarchy. This is done by proving a correspondence principle yielding a condition for the sets of computable strong measure zero to agree with the classical sets of strong measure zero.
An algorithmic version of strong measure zero using lower semicomputability is defined. We show that this notion is equivalent to the set of NCR reals studied by Reimann and Slaman, thereby giving new characterizations of this set.
Effective strong packing dimension zero is investigated by requiring success with respect to the limit inferior instead of the limit superior. It is proven that every sequence in the corresponding algorithmic class is decidable. At the level of computability, the sets coincide with a notion of weak countability that we define.

Cite As Get BibTex

Matthew Rayman. Effective Versions of Strong Measure Zero. In 43rd International Symposium on Theoretical Aspects of Computer Science (STACS 2026). Leibniz International Proceedings in Informatics (LIPIcs), Volume 364, pp. 75:1-75:18, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2026) https://doi.org/10.4230/LIPIcs.STACS.2026.75

Author Details

Matthew Rayman
  • Department of Computer Science, Iowa State University, Ames, IA, USA

Funding

This research was supported in part by National Science Foundation research grant 1900716.

Acknowledgements

The author thanks Jack Lutz for many helpful conversations.

References

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