DETERMINACY OF SCHMIDT’S GAME AND OTHER INTERSECTION GAMES

Abstract Schmidt’s game and other similar intersection games have played an important role in recent years applications to number theory, dynamics, Diophantine approximation theory. These are real games, that is, which the players make moves from a complete separable metric space. The determinacy of these trivially follows axiom for $\mathsf {AD}_{\mathbb R}$ , is much stronger than asserting all integer determined, {AD}$ . One our main results general theorem under hypothesis implies property allowing strategies be simplified. In particular, we show $(\alpha ,\beta ,\rho )$ on $\mathbb R$ determined alone, but R^n$ $n \geq 3$ does not imply this game. We then give application simple prove winning player \beta \rho {R}$ has positional strategy, without appealing choice. also several specifically related highlight obstacles obtaining