Winning sets, quasiconformal maps and Diophantine approximation
type: Item
instance of: scholarly article
title: Winning sets, quasiconformal maps and Diophantine approximation
zbMATH Open document ID: 1242.11054
author: Curtis T. McMullen
published in: Geometric and Functional Analysis. GAFA
publication date: 2010-10-08T00:00:00Z
review text: 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 i…
reviewed by: Simon Kristensen
Mathematics Subject Classification ID: 11J83
Mathematics Subject Classification ID: 37F30
zbMATH DE Number: 5797329
zbMATH Keywords: Diophantine numbers
zbMATH Keywords: Schmidt's game
zbMATH Keywords: Kleinian groups
zbMATH Keywords: Hausdorff dimension
zbMATH Keywords: porosity
zbMATH Keywords: quasiconformal maps
MaRDI profile type: Publication
full work available at URL: s00039-010-0078-3
OpenAlex ID: W2149057882
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DOI: 10.1007/S00039-010-0078-3
description: scientific article; zbMATH DE number 5797329