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DETERMINACY AND INDETERMINACY OF GAMES PLAYED ON COMPLETE METRIC SPACES

Part of: Game theory

Published online by Cambridge University Press:  12 May 2014

LIOR FISHMAN ,
TUE LY  and
DAVID SIMMONS
LIOR FISHMAN
Affiliation:
University of North Texas, Department of Mathematics, 1155 Union Circle #311430, Denton, TX 76203-5017, USA email Lior.Fishman@unt.edu
TUE LY
Affiliation:
Brandeis University, Department of Mathematics, 415 South Street, Waltham, MA 02454-9110, USA email lntue@brandeis.edu
DAVID SIMMONS*
Affiliation:
Ohio State University, Department of Mathematics, 231 W. 18th Avenue, Columbus, OH 43210-1174, USA email simmons.465@osu.edu
*
✉simmons.465@osu.edu
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Abstract

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Schmidt’s game is a powerful tool for studying properties of certain sets which arise in Diophantine approximation theory, number theory and dynamics. Recently, many new results have been proven using this game. In this paper we address determinacy and indeterminacy questions regarding Schmidt’s game and its variations, as well as more general games played on complete metric spaces (for example, fractals). We show that, except for certain exceptional cases, these games are undetermined on certain sets. Judging by the vast numbers of papers utilising these games, we believe that the results in this paper will be of interest to a large audience of number theorists as well as set theorists and logicians.

Keywords

determinacy of gamesBorel determinacyGale–Stewart gamesSchmidt’s game

MSC classification

Secondary: 91A44: Games involving topology or set theory 91A05: 2-person games 91A18: Games in extensive form 91A50: Discrete-time games
Type
Research Article
Information
Bulletin of the Australian Mathematical Society , Volume 90 , Issue 2 , October 2014 , pp. 339 - 351
Copyright
Copyright © 2014 Australian Mathematical Publishing Association Inc. 

References

Fishman, L., Simmons, D. S. and Urbański, M., ‘Diophantine approximation and the geometry of limit sets in Gromov hyperbolic metric spaces (extended version)’, Preprint, 2013. arXiv:1301.5630.Google Scholar
Gale, D. and Stewart, F. M., ‘Infinite games with perfect information’, in: Contributions to the Theory of Games, Vol. 2, Annals of Mathematics Studies 28 (Princeton University Press, Princeton, NJ, 1953), 245–266.Google Scholar
Hutchinson, J. E., ‘Fractals and self-similarity’, Indiana Univ. Math. J. 30(5) (1981), 713–747.CrossRefGoogle Scholar
Kechris, A. S., Classical Descriptive Set Theory, Graduate Texts in Mathematics, 156 (Springer-Verlag, New York, 1995).Google Scholar
Kysiak, M. and Zoli, E., ‘Set-theoretic properties of Schmidt’s ideal’, Georgian Math. J. 12(3) (2005), 493–504.Google Scholar
Martin, D. A., ‘A purely inductive proof of Borel determinacy’, in: Recursion Theory (Ithaca, N.Y., 1982), Proceedings of Symposia in Pure Mathematics, 42 (American Mathematical Society, Providence, RI, 1985), 303–308.Google Scholar
McMullen, C. T., ‘Winning sets, quasiconformal maps and Diophantine approximation’, Geom. Funct. Anal. 20(3) (2010), 726–740.CrossRefGoogle Scholar
Oxtoby, J. C., ‘Measure and category’, A Survey of the Analogies Between Topological and Measure Spaces, Graduate Texts in Mathematics, 2 (Springer, New York, Berlin, 1971).Google Scholar
Schmidt, W. M., ‘On badly approximable numbers and certain games’, Trans. Amer. Math. Soc. 123 (1966), 27–50.CrossRefGoogle Scholar