Schmidt games and Cantor winning sets

David Samuel Simmons; Dzmitry Badziahin; Stephen Harrap; Erez Nesharim

Schmidt games and Cantor winning sets

David Samuel Simmons, Dzmitry Badziahin, Stephen Harrap, Erez Nesharim

Mathematics

Research output: Working paper › Preprint

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, and we suggest a prototypical example of a Cantor winning set to show the ubiquity of such sets in metric number theory and ergodic theory.

Original language	English
Publisher	arXiv
Publication status	Published - 25 May 2020

Access to Document

https://arxiv.org/abs/1804.06499

Cite this

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AB - 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, and we suggest a prototypical example of a Cantor winning set to show the ubiquity of such sets in metric number theory and ergodic theory.

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