Schmidt games and Cantor winning sets

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DatePublished - 25 May 2020
PublisherarXiv
Original languageEnglish

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.

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