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Strong measure zero and infinite games

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Abstract

We show that strong measure zero sets (in a \(\sigma \)-totally bounded metric space) can be characterized by the nonexistence of a winning strategy in a certain infinite game. We use this characterization to give a proof of the well known fact, originally conjectured by K. Prikry, that every dense \(G_\delta \) subset of the real line contains a translate of every strong measure zero set. We also derive a related result which answers a question of J. Fickett.

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Acknowledgements

We thank Karel Prikry and James Fickett for communicating their conjectures and questions to us. We are grateful to John C. Morgan II for informing us of the relevant work of Bagemihl, Scheeffer, and Sierpiński and providing references. The first author, taking full responsibility for all errors and infelicities contained herein, would like to thank Marion Scheepers for his assistance in preparing this paper for publication.

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Authors and Affiliations

  1. University of Kansas, Lawrence, KS, USA

    Fred Galvin

  2. University of Colorado, Boulder, CO, USA

    Jan Mycielski

  3. University of California at Berkeley, Berkeley, CA, USA

    Robert M. Solovay

Authors
  1. Fred Galvin
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  2. Jan Mycielski
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  3. Robert M. Solovay
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Correspondence to Fred Galvin.

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Galvin, F., Mycielski, J. & Solovay, R.M. Strong measure zero and infinite games. Arch. Math. Logic 56, 725–732 (2017). https://doi.org/10.1007/s00153-017-0541-z

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  • DOI: https://doi.org/10.1007/s00153-017-0541-z

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