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Badly approximable points on planar curves and winning

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Publication details

JournalAdvances in Mathematics
DateAccepted/In press - 6 Nov 2017
DateE-pub ahead of print - 21 Nov 2017
DatePublished (current) - 14 Jan 2018
Number of pages55
Pages (from-to)148–202
Early online date21/11/17
Original languageEnglish


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.

Bibliographical note

© 2017 The Author(s).

    Research areas

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


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