Badly approximable points on planar curves and winning

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For any i,j>0 with i+j=1, let Bad(i,j) denote the set of points (x,y)∈R 2 such that max⁡{‖qx‖ 1/i,‖qy‖ 1/j}>c/q for some positive constant c=c(x,y) and all q∈N. We show that Bad(i,j)∩C is winning in the sense of Schmidt games for a large class of planar curves C, namely, everywhere non-degenerate planar curves and straight lines satisfying a natural Diophantine condition. This strengthens recent results solving a problem of Davenport from the sixties. In short, within the context of Davenport's problem, the winning statement is best possible. Furthermore, we obtain the inhomogeneous generalisations of the winning results for planar curves and lines and also show that the inhomogeneous form of Bad(i,j) is winning for two dimensional Schmidt games.

Original languageEnglish
Pages (from-to)148–202
Number of pages55
JournalAdvances in Mathematics
Early online date21 Nov 2017
Publication statusPublished - 14 Jan 2018

Bibliographical note

© 2017 The Author(s).


  • Inhomogeneous Diophantine approximation
  • Non-degenerate curves
  • Schmidt games

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