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Intersection Games and Bernstein Sets

James Atchley, Lior Fishman, Saisneha Ghatti

TL;DR

The paper investigates the determinacy of three classic intersection games on the real line when the target set is a Bernstein set, a non-Lebesgue measurable set constructed via the axiom of choice. By showing that any purported winning strategy would force the target (or its complement) to contain a perfect set, it proves that Bernstein sets cannot host a winning strategy for Banach-Mazur, McMullen's Absolute Winning, or Schmidt's game under the stated parameter regimes. The results illuminate the tension between axiom-of-choice constructions and determinacy, highlighting how Bernstein sets obstruct determinacy even in prominent, dimension-theoretic game scenarios. These insights have implications for understanding the limits of determinacy axioms and the geometric structure of exceptional sets in real analysis.

Abstract

The Banach-Mazur game, Schmidt's game and McMullen's absolute winning game are three quintessential intersection games. We investigate their determinacy on the real line when the target set for either player is a Bernstein set, a non-Lebesgue measurable set whose construction depends on the axiom of choice.

Intersection Games and Bernstein Sets

TL;DR

The paper investigates the determinacy of three classic intersection games on the real line when the target set is a Bernstein set, a non-Lebesgue measurable set constructed via the axiom of choice. By showing that any purported winning strategy would force the target (or its complement) to contain a perfect set, it proves that Bernstein sets cannot host a winning strategy for Banach-Mazur, McMullen's Absolute Winning, or Schmidt's game under the stated parameter regimes. The results illuminate the tension between axiom-of-choice constructions and determinacy, highlighting how Bernstein sets obstruct determinacy even in prominent, dimension-theoretic game scenarios. These insights have implications for understanding the limits of determinacy axioms and the geometric structure of exceptional sets in real analysis.

Abstract

The Banach-Mazur game, Schmidt's game and McMullen's absolute winning game are three quintessential intersection games. We investigate their determinacy on the real line when the target set for either player is a Bernstein set, a non-Lebesgue measurable set whose construction depends on the axiom of choice.
Paper Structure (9 sections, 4 theorems, 9 equations)

This paper contains 9 sections, 4 theorems, 9 equations.

Table of Contents

  1. Introduction
  2. Description of the Games
  3. The Banach-Mazur game
  4. McMullen's Absolute Winning game
  5. Schmidt's game
  6. Bernstein Sets and the determinacy of games on a Bernstein set
  7. The Banach-Mazur Game
  8. McMullen's game
  9. Schmidt's Game

Key Result

Theorem 1

The Banach-Mazur Game is not determined on a Bernstein set.

Theorems & Definitions (13)

  • Definition 1
  • Definition 2
  • Definition 3
  • Definition 4
  • Definition 5
  • Theorem 1
  • proof
  • Theorem 2
  • proof
  • Theorem 3
  • ...and 3 more