Schmidt games and Cantor winning sets

Badziahin, D.; Harrap, S.; Nesharim, E.; Simmons, D.

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 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.

Citation

Badziahin, D., Harrap, S., Nesharim, E., & Simmons, D. (2025). Schmidt games and Cantor winning sets. Ergodic Theory and Dynamical Systems, 45(1), 71-110. https://doi.org/10.1017/etds.2024.23

• Journal Article Type: Article
• Acceptance Date: Mar 19, 2024
• Online Publication Date: Apr 19, 2024
• Publication Date: 2025-01
• Deposit Date: Apr 19, 2018
• Publicly Available Date: May 15, 2024
• Journal: Ergodic Theory and Dynamical Systems
• Print ISSN: 0143-3857
• Electronic ISSN: 1469-4417
• Publisher: Cambridge University Press
• Peer Reviewed: Peer Reviewed
• Volume: 45
• Issue: 1
• Pages: 71-110
• DOI: https://doi.org/10.1017/etds.2024.23
• Public URL: https://durham-repository.worktribe.com/output/1334174