On Nontrivial Winning and Losing Parameters of Schmidt Games

TL;DR

This work analyzes the classical Schmidt game on two nontrivial set families to uncover nontrivial winning and losing parameter regions. It derives explicit $(\alpha,\beta)$-winning/losing criteria for digit-frequency sets tied to base-2 expansions and for d-Badly Approximable numbers, including refined conditions under irrationality assumptions and tilde-variants that yield new Schmidt diagrams. Central contributions include a detailed strategy framework for general Schmidt-game moves, a proof structure for the main diagram-theoretic results (Theorem $\text{thmdf}$), and complete proofs for the d-BA related statements using center-control and symmetry arguments. The results deepen understanding of how digit-frequency properties and Diophantine approximation interact with Schmidt-game dynamics, linking winning zones to digit-frequency thresholds and to Hausdorff-dimension bounds, with conjectures about the exact shape of the Schmidt diagrams for these families.

Abstract

In this paper we study the classical Schmidt game on two families of sets: one related to frequencies of digits in base-$2$ expansions, and one connected to the set of the badly approximable numbers. Namely, we describe some nontrivial winning and losing parameters $(α, β)$ for these sets.

Figures (7)

• Figure 1: $\check D$ (blue)
• Figure 2: $\hat{D}$ (blue)
• Figure 3: Bounds for $D(F^+_c)$ with $c = 0.1$
• Figure 4: Bounds for $D(F^+_c)$ with $c = 0.55$
• Figure 5: Bounds for $D(F^-_c)$ with $c = 0.9$

Theorems & Definitions (32)

• Lemma 1.1
• Proposition 2.1
• Theorem 2.2
• Proposition 2.3
• Conjecture 2.4
• Theorem 2.5
• Theorem 2.6
• Proposition 3.1
• Proposition 3.2
• Lemma 3.3