Analytic sets of reals and the density function in the Cantor space

Research Article
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Abstract

We study the density function of measurable subsets of the Cantor space. Among other things, we identify a universal set Open image in new window for \(\varvec{\varSigma }^{1}_{1}\) subsets of Open image in new window in terms of the density function; specifically Open image in new window is the set of all pairs (Kr) with K compact and Open image in new window being the density of some point with respect to K. This result yields that the set of all K such that the range of their density function is Open image in new window , for some fixed uncountable analytic set \(S\subseteq (0;1)\), is \(\varvec{\varPi }^{1}_{2}\)-complete.

Keywords

Density function Cantor space Analytic sets 

Mathematics Subject Classification

03E15 28A05 

Notes

Acknowledgements

The authors would like to thank Vassilios Gregoriades, Alain Louveau, and John Steel for illuminating discussions.

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Copyright information

© Springer International Publishing AG, part of Springer Nature 2018

Authors and Affiliations

  1. 1.Dipartimento di MatematicaUniversita di TorinoTurinItaly
  2. 2.Dipartimento di MatematicaPolitecnico di TorinoTurinItaly

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