Determinacy and indeterminacy of games played on complete metric spaces (Q2922948)

In contexts of game theory on complete metric spaces, 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 the authors address determinacy and indeterminacy questions regarding Schmidt's game and its variations, as well as more general games (Gale-Stewart games, and then games played on complete metric spaces -- for example, fractals). The authors show that, except for certain exceptional cases, these games are undetermined on certain sets: determinacy on Borel sets as well as indeterminacy on Berstein sets is analyzed at this stage. Judging by the vast numbers of papers utilising these games, the results in the manuscript will be of interest to a large audience of number theorists as well as set theorists and logicians.