Winning sets, quasiconformal maps and Diophantine approximation
The Schmidt game was introduced by \textit{W. M. Schmidt} in [Trans. Am. Math. Soc. 123, 178--199 (1966; Zbl 0232.10029)]. The game has two players, \(A\) and \(B\), who successively choose euclidean balls \[ B_1 \supset A_1 \supset B_2 \supset A_2 \supset \cdots, \] with diameters satisfying \(| A_i | = \alpha | B_i |\) and \(| B_{i+1} | = \beta | A_i|\). Here, \(\alpha, \beta \in (0,1)\) are fixed parameters of the game. A set \(E \subseteq {\mathbb R}^n\) is said to be \(\alpha\)-winning if player \(A\) can ensure that \(\cap B_i\) intersects \(E\) for any \(\beta \in (0,1)\). An \(\alpha\)-winning set has maximal Hausdorff dimension. In the present paper, two modifications to this game are studied. In the strong game, both players are allowed to choose larger balls, \textit{i.e.}, the equalities in the conditions on diameters are replaced by inequalities \(\geq\). In the absolute game, only one parameter \(\beta \in (0,1/3)\) is required. Player \(A\) can block a ball of radius at most \(\beta | B_i |\), so that player \(B\) must choose \(B_{i+1} \subset B_i \setminus A_i\). Player \(B\) is still allowed to choose larger balls. A set is absolutely winning if player \(A\) can win the absolute game for any \(\beta\). An absolutely winning set is \(\alpha\)-strong winning for any \(\alpha \in (0,1/2)\). An \(\alpha\)-strong winning set is \(\alpha\)-winning. In the paper it is shown that \(\alpha\)-strong winning sets and absolutely winning sets are preserved under quasi-symmetric homeomorphisms of \({\mathbb R}^n\). This specialises to quasiconformal homeomorphisms when \(n=2\). However, this stability property does not hold for \(\alpha\)-winning sets. The results are applied to so-called Diophantine sets. If \(\Gamma \subset \text{Isom}({\mathbb H}^{n+1})\) is a lattice in the isometry group of hyperbolic space with a cusp at infinity, the Diophantine set of \(\Gamma\) consists of the endpoints in \(\partial{\mathbb H}^{n+1}\) of lifts of all bounded geodesics in \({\mathbb H}^{n+1}/\Gamma\). In the case \(n=1\) and \(\Gamma = \text{SL}_2({\mathbb Z})\), the boundary of \({\mathbb H}^2\) is naturally identified with \({\mathbb R} \cup \{\infty\}\), and the Diophantine set of \(\Gamma\) becomes the usual set of badly approximable numbers. In the paper, it is shown that for any such \(\Gamma\), the Diophantine set is absolutely winning. Finally, it is shown that such sets enjoy certain porosity properties, implying that such Diophantine sets are of Lebesgue measure zero.
scientific article; zbMATH DE number 5797329
scientific article