[Paper Review] Schmidt's Game on Certain Fractals

Lior Fishman|arXiv (Cornell University)|Jun 13, 2006

Mathematical Dynamics and Fractals6 references3 citations

TL;DR

This paper establishes that the intersection of a self-similar fractal $K$—such as the Cantor set, Koch curve, or Sierpinski gasket—with a countable intersection of affine transforms of the set of badly approximable vectors (BA) is an $\alpha$-winning set on $K$, and thus has full Hausdorff dimension equal to $\dim K$. The result extends Schmidt's classical game-theoretic approach to fractal subsets of $\mathbb{R}^N$, proving that BA sets remain 'large' under affine transformations even when restricted to fractals satisfying the open set condition.

ABSTRACT

We construct (α,β) and α-winning sets in the sense of Schmidt's game, played on the support of certain measures (very friendly and awfully friendly measures) and show how to derive the Hausdorff dimension for some. In particular we prove that if K is the attractor of an irreducible finite family of contracting similarity maps of R^N satisfying the open set condition then for any countable collection of non-singular affine transformations Λ_i:R^N o R^N, dimK=dimK\cap (\cap ^{\infty}_{i=1}(Λ_i(BA))) where BA is the set of badly approximable vectors in R^N.

Motivation & Objective

• To extend Schmidt's game theory to fractal subsets of $\mathbb{R}^N$ by analyzing the dimension of intersections between fractals and affine transforms of the set of badly approximable vectors.
• To establish that such intersections are $\alpha$-winning sets on the fractal support, implying full Hausdorff dimension.
• To generalize prior results on $\dim(K \cap \text{BA}) = \dim K$ to intersections with countably many affine-transformed BA sets.
• To show that the $\delta$-dimensional Hausdorff measure on self-similar fractals satisfying the open set condition is absolutely friendly, enabling application of Schmidt's game.

Proposed method

• Constructs $(\alpha, \beta)$-winning sets on the support of absolutely friendly measures, which include the $\delta$-dimensional Hausdorff measure on self-similar fractals.
• Applies a modified version of the simplex lemma from Schmidt’s original game to handle the fractal setting and ensure game-winning strategies exist.
• Uses the power law property of the Hausdorff measure on self-similar sets to verify the necessary decay and doubling conditions for absolute friendliness.
• Applies Theorem 2.2 and Corollary 2.1 to show that $K \cap \Lambda_i(\text{BA})$ is $\alpha$-winning for each affine transformation $\Lambda_i$.
• Establishes that the winning condition holds uniformly across countably many affine transforms by defining a sequence of nested sets $U_k$ based on rational approximation levels.
• Leverages the irreducibility and open set condition of the generating similarity maps to ensure the fractal is uniformly distributed and supports a regular measure.

Experimental results

Research questions

• RQ1 Is the intersection of a self-similar fractal $K$ with a countable intersection of affine transforms of the set of badly approximable vectors non-empty?
• RQ2 Does the intersection $K \cap \bigcap_{i=1}^\infty \Lambda_i(\text{BA})$ retain the full Hausdorff dimension of $K$?
• RQ3 Can Schmidt’s game be adapted to prove that such intersections are $\alpha$-winning sets on the fractal support?
• RQ4 Under what measure-theoretic conditions on the fractal is the set of badly approximable vectors a winning set?
• RQ5 What is the relationship between the power law of the Hausdorff measure on $K$ and the winning property of $K \cap \Lambda_i(\text{BA})$?

Key findings

• The set $\mathcal{S} = K \cap \bigcap_{i=1}^\infty \Lambda_i(\text{BA})$ is an $\alpha$-winning set on $K$ for any $0 < \alpha < \frac{1}{12}\alpha'$, where $\alpha'$ is a constant derived from the similarity ratios of the generating maps.
• The Hausdorff dimension of $\mathcal{S}$ equals the Hausdorff dimension of $K$, i.e., $\dim \mathcal{S} = \dim K$, even when intersected with countably many affine transforms of the BA set.
• The $\delta$-dimensional Hausdorff measure $\mu$ restricted to $K$ is absolutely friendly and satisfies condition 3.6 with exponent $\delta = \dim K$, enabling the application of Schmidt’s game.
• The proof relies on verifying that the measure $\mu$ is absolutely $(C,a)$-decaying and satisfies the power law, which ensures the necessary regularity for game-theoretic constructions.
• The result strengthens prior work by showing that the winning property is preserved not only for $K \cap \text{BA}$ but also for $K \cap \bigcap_i \Lambda_i(\text{BA})$, under mild conditions on the transformations.
• The construction generalizes Schmidt’s original result in $\mathbb{R}^N$ to fractal subsets, showing that the 'large' dimensionality of BA sets is preserved under affine images and fractal restrictions.