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Schmidt games and Cantor winning sets
- Part of
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- Journal:
- Ergodic Theory and Dynamical Systems / Volume 45 / Issue 1 / January 2025
- Published online by Cambridge University Press:
- 19 April 2024, pp. 71-110
- Print publication:
- January 2025
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- Article
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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.
