Determinacy and indeterminacy of games played on complete metric spaces

TL;DR

This paper investigates the determinacy of Schmidt's game and related games on complete metric spaces, revealing they are generally undetermined on Bernstein sets except for specific cases, impacting their application in number theory and dynamics.

Contribution

It provides new insights into the conditions under which Schmidt's game and similar games are determined or undetermined on complex metric spaces.

Findings

Games are undetermined on Bernstein sets in most cases.

Exceptional cases where games are determined are identified.

Implications for number theory and dynamical systems are discussed.

Abstract

Schmidt's game is a powerful tool for studying properties of certain sets which arise in Diophantine approximation theory, number theory, and dynamics. Recently, many new results have been proven using this game. In this paper we address determinacy and indeterminacy questions regarding Schmidt's game and its variations, as well as more general games played on complete metric spaces (e.g. fractals). We show that except for certain exceptional cases, these games are undetermined on Bernstein sets.