Determinacy of Schmidt's Game and Other Intersection Games

TL;DR

This paper investigates the determinacy of Schmidt's game and similar intersection games within set theory, showing that determinacy can be derived from the axiom of determinacy for reals in some cases, but not in higher dimensions.

Contribution

It provides a general theorem linking the axiom of determinacy to the determinacy of intersection games, and clarifies when Schmidt's game is determined based on dimension and set-theoretic assumptions.

Findings

Schmidt's $( ho,eta, ho)$ game on $ extbf{R}$ is determined from $ extbf{AD}$.

On $ extbf{R}^n$ for $n extgreater 2$, $ extbf{AD}$ does not imply game determinacy.

The paper identifies obstacles to establishing determinacy of Schmidt's game from $ extbf{AD}$.

Abstract

Schmidt's game, and other similar intersection games have played an important role in recent years in applications to number theory, dynamics, and Diophantine approximation theory. These games are real games, that is, games in which the players make moves from a complete separable metric space. The determinacy of these games trivially follows from the axiom of determinacy for real games, $\mathsf{AD}_\mathbb{R}$, which is a much stronger axiom than that asserting all integer games are determined, $\mathsf{AD}$. One of our main results is a general theorem which under the hypothesis $\mathsf{AD}$ implies the determinacy of intersection games which have a property allowing strategies to be simplified. In particular, we show that Schmidt's $(\alpha,\beta,\rho)$ game on $\mathbb{R}$ is determined from $\mathsf{AD}$ alone, but on $\mathbb{R}^n$ for $n \geq 3$ we show that $\mathsf{AD}$ does not imply the determinacy of this game. We also prove several other results specifically related to the determinacy of Schmidt's game. These results highlight the obstacles in obtaining the determinacy of Schmidt's game from $\mathsf{AD}$.