The no-$\beta$ McMullen game and the perfect set property

TL;DR

This paper studies a variant of McMullen's game without the parameter $eta$, revealing its connection to the perfect set property and providing geometric and logical insights into the structure of uncountable sets in Euclidean spaces.

Contribution

It introduces the no-$eta$ McMullen game and establishes its equivalence to the perfect set game for certain convex sets, linking game theory, geometry, and logic.

Findings

Player I wins iff A contains a perfect set.

Player II wins iff A is countable.

Results apply to strictly convex sets, polytopes, and convex sets in $\

Abstract

Given a target set $A\subseteq \mathbb{R}^d$ and a real number $\beta\in (0,1)$, McMullen introduced the notion of $A$ being an absolutely $\beta$-winning set. This involves a two player game which we call the $\beta$-McMullen game. We consider the version of this game in which the parameter $\beta$ is removed, which we call the no-$\beta$ McMullen game. More generally, we consider the game with respect to arbitrary norms on $\mathbb{R}^d$, and even more generally with respect to general convex sets. We show that for strictly convex sets in $\mathbb{R}^d$, polytopes in $\mathbb{R}^d$, and general convex sets in $\mathbb{R}^2$, that player $\boldsymbol{I}$ wins the no-$\beta$ McMullen game iff $A$ contains a perfect set and player $\boldsymbol{I}\kern-0.05cm\boldsymbol{I}$ wins iff $A$ is countable. So, the no-$\beta$ McMullen game is equivalent to the perfect set game for $A$ in these cases. The proofs of these results use a connection between the geometry of the game and techniques from logic. Because of the geometry of this game, this result has strong implications for the geometry of uncountable sets in $\mathbb{R}^d$. We also present an example of a compact, convex set in $\mathbb{R}^3$ to which our methods do not apply, and also an example due to D.\ Simmons of a closed, convex set in $\ell_2(\mathbb{R})$ which illustrate the obstacles in extending the results further.