Modified Schmidt games and Diophantine approximation with weights

By Dmitry Kleinbock, Barak Weiss. 2008

85 citations recorded by the source corpus.

Abstract We show that the sets of weighted badly approximable vectors in R n are winning sets of certain games, which are modifications of ( α , β ) -games introduced by W.M. Schmidt in 1966. The latter winning property is stable with respect to countable intersections, and is shown to imply full Hausdorff dimension.

Subfields: Computational Theory and Mathematics, Geometry and Topology, Mathematical Physics

View Paper: https://www.semanticscholar.org/paper/6d34926aa1e4bf89a237b03868d3e7485f9d13a8

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In Computational Theory and Mathematics

In Geometry and Topology

In Mathematical Physics

Citations in the corpus

Two-dimensional badly approximable vectors and Schmidt's game (2012, background): “This expectation was raised explicitly by Kleinbock [9] (see also [12, 15]).”
Badly approximable points on manifolds (2013, background): “Generalising the techniques of [BPV11] in yet another direction Nesharim [Nes], independently from An, proves that the set in the left hand side of (6) intersected with naturally occurring fractals embedded in Lθ is uncountable for any sequence (rk)k∈N. Subsequently, Nesharim jointly with Weiss esta”
Metric Diophantine Approximation : aspects of recent work (2016, background): “For details see [50, 67] and references within.”
On a problem in simultaneous Diophantine approximation: Schmidt’s conjecture (2010, background): “For further background and various strengthenings of this full dimension statement the reader is referred to [4, 5, 7].”
Badly approximable systems of affine forms, fractals, and Schmidt games (2009, background): “Moreover, Kleinbock conjectured that the set is winning [10] (or winning in the modified sense of [12], as mentioned in personal communication).”
Schmidt’s game, fractals, and orbits of toral endomorphisms (2010, methodology): “Schmidt’s game is played by two players, whom we will call Alice and Bob, following a convention used previously in [1, 18].”