Lightface \(\mathop {\varPi }\nolimits _{3}^{0}\)-Completeness of Density Sets Under Effective Wadge Reducibility

  • Gemma Carotenuto
  • André Nies
Conference paper
Part of the Lecture Notes in Computer Science book series (LNCS, volume 9709)


Let \(\mathcal {A} \subseteq {}^\omega {2}\) be measurable. The density set \(D \mathcal {A}\) is the set of \(Z \in {}^\omega {2}\) such that the local measure of \(\mathcal {A}\) along Z tends to 1. Suppose that \(\mathcal {A}\) is a \(\varPi ^0_{1}\) set with empty interior and the uniform measure of \(\mathcal {A}\) is a positive computable real. We show that \(D \mathcal {A}\) is lightface \(\varPi ^0_3\) complete for effective Wadge reductions. This is an algorithmic version of a result in descriptive set theory by Andretta and Camerlo [1]. They show a completeness result for boldface \(\varPi ^0_3\) sets under plain Wadge reductions.


  1. 1.
    Andretta, A., Camerlo, R.: The descriptive set theory of the Lebesgue density theorem. Adv. Math. 234, 1–42 (2013)MathSciNetCrossRefzbMATHGoogle Scholar
  2. 2.
    Bienvenu, L., Day, A., Greenberg, N., Kučera, A., Miller, J., Nies, A., Turetsky, D.: Computing \(K\)-trivial sets by incomplete random sets. Bull. Symb. Logic 20, 80–90 (2014)MathSciNetCrossRefzbMATHGoogle Scholar
  3. 3.
    Bienvenu, L., Greenberg, N., Kučera, A., Nies, A., Turetsky, D.: Coherent randomness tests and computing the K-trivial sets. J. Eur. Math. Soc. 18, 773–812 (2016)MathSciNetCrossRefzbMATHGoogle Scholar
  4. 4.
    Carotenuto, C.: On the topological complexity of the density sets of the real line. Ph.D. thesis, Università di Salerno (2015).
  5. 5.
    Day, A.R., Miller, J.S.: Density, forcing and the covering problem. Math. Res. Lett. 22(3), 719–727 (2015)MathSciNetCrossRefzbMATHGoogle Scholar
  6. 6.
    Khan, M.: Lebesgue density and \(\Pi ^0_1\)-classes. J. Symb. Logic, to appearGoogle Scholar
  7. 7.
    Miyabe, K., Nies, A., Zhang, J.: Using almost-everywhere theorems from analysis to study randomness. Bull. Symb. Logic, (2016, to appear)Google Scholar
  8. 8.
    Montagna, F., Sorbi, A.: Creativeness and completeness in recursion categories of partial recursive operators. J. Symb. Logic 54, 1023–1041 (1989)MathSciNetCrossRefzbMATHGoogle Scholar
  9. 9.
    Nies, A.: Computability and randomness. Oxford Logic Guides, vol. 51. Oxford University Press, Oxford (2009). 444 pp. Paperback version 2011CrossRefzbMATHGoogle Scholar
  10. 10.
    Odifreddi, P.: Classical Recursion Theory, vol. 1. NorthHolland Publishing Co., Amsterdam (1989)zbMATHGoogle Scholar

Copyright information

© Springer International Publishing Switzerland 2016

Authors and Affiliations

  1. 1.Dipartimento di MatematicaUniversita di SalernoFiscianoItaly
  2. 2.Department of Computer ScienceUniversity of AucklandAucklandNew Zealand

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