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Schmidt games and Cantor winning sets

Part of: Ergodic theory Diophantine approximation, transcendental number theory Game theory

Published online by Cambridge University Press:  19 April 2024

DZMITRY BADZIAHIN ,
STEPHEN HARRAP ,
EREZ NESHARIM  and
DAVID SIMMONS
DZMITRY BADZIAHIN
Affiliation:
School of Mathematics and Statistics, University of Sydney, NSW 2006, Australia (e-mail: dzmitry.badziahin@sydney.edu.au)
STEPHEN HARRAP
Affiliation:
Department of Mathematical Sciences, Durham University, Durham DH1 3LE, UK (e-mail: s.g.harrap@durham.ac.uk)
EREZ NESHARIM*
Affiliation:
Faculty of Mathematics, Technion – Israel Institute of Technology, Haifa 3200003, Israel
DAVID SIMMONS
Affiliation:
Department of Mathematics, University of York, Heslington, York YO10 5DD, UK (e-mail: david9550@gmail.com)
*
e-mail: nesharim@technion.ac.il
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Abstract

Schmidt games and the Cantor winning property give alternative notions of largeness, similar to the more standard notions of measure and category. Being intuitive, flexible, and applicable to recent research made them an active object of study. We survey the definitions of the most common variants and connections between them. A new game called the Cantor game is invented and helps with presenting a unifying framework. We prove surprising new results such as the coincidence of absolute winning and $1$ Cantor winning in metric spaces, and the fact that $1/2$ winning implies absolute winning for subsets of $\mathbb {R}$. We also suggest a prototypical example of a Cantor winning set to show the ubiquity of such sets in metric number theory and ergodic theory.

Keywords

Schmidt gamesCantor-winningDiophantine approximation

MSC classification

Primary: 11J83: Metric theory 37A05: Measure-preserving transformations 91A44: Games involving topology or set theory

Information

Type
Original Article
Information
Ergodic Theory and Dynamical Systems , Volume 45 , Issue 1 , January 2025 , pp. 71 - 110
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Copyright
© The Author(s), 2024. Published by Cambridge University Press

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