Winning Sets, Quasiconformal Maps and Diophantine Approximation

By Curtis T. McMullen. 2010

94 citations recorded by the source corpus.

This paper describes two new types of winning sets in $${\mathbb{R}^n}$$, defined using variants of Schmidt’s game. These strong and absolute winning sets include many Diophantine sets of measure zero and first category, and have good behavior under countable intersections. Most notably, they are invariant under quasiconformal maps, while classical winning sets are not.

Subfields: Geometry and Topology, Mathematical Physics

View Paper: https://www.semanticscholar.org/paper/5b02e630ce1aeb30b5492c7bbfe31764a9e6073f

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In Geometry and Topology

In Mathematical Physics

Citations in the corpus

Diophantine approximation and the geometry of limit sets in Gromov hyperbolic metric spaces (2013, methodology): “Several examples of absolute winning sets were exhibited by McMullen in [56], most notably the set of badly approximable numbers in R.”
The set of badly approximable vectors is strongly C1 incompressible (2011, background): “One of the main interesting examples of incompressible sets is the set BAd of badly approximable vectors, namely those x ∈ R d satisfying (1.3) ∥ ∥ ∥ ∥ x− p q ∥ ∥ ∥ ∥ ≥ c q1+1/d for some c = c(x) > 0 and for any p ∈ Zd, q ∈ N.”
Badly approximable systems of affine forms, fractals, and Schmidt games (2009, background): “Strong winning implies winning and is preserved by quasisymmetric homeomorphisms [13].”
Schmidt’s game, fractals, and orbits of toral endomorphisms (2010, background): “mplies winning (see [11] for a proof), and that a countable intersection of α-strong winning sets is α-strong winning. Furthermore, this class has stronger invariance properties, e.g. it is proved in [21] that strong winning subsets of Rn are preserved by quasisymmetric homeomorphisms. It is not har”
Two-dimensional badly approximable vectors and Schmidt's game (2012, methodology): “ad(sn,tn)) has full Hausdorﬀ dimension in R2. It should be noted that several stronger variants of Schmidt’s game have been deﬁned and used to problems in Diophantine approximation (see, for example, [14, 3]). By using the main lemma in a previous version of this paper (a weaker form of Corollary 4.”