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Badly approximable points on planar curves and a problem of Davenport

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JournalMathematische Annalen
DateE-pub ahead of print - 1 Mar 2014
DatePublished (current) - Aug 2014
Issue number3-4
Volume359
Number of pages55
Pages (from-to)969-1023
Early online date1/03/14
Original languageEnglish

Abstract

Let C be two times continuously differentiable curve in R^2 with at least one point at which the curvature is non-zero. For any i,j > 0 with i+j =1, let Bad(i,j) denote the set of points (x,y) in R^2 for which max {||qx ||^{1/i}, ||qy||^{1/j}} > c/q for all integers q >0. Here c = c(x,y) is a positive constant. Our main result implies that any finite intersection of such sets with C has full Hausdorff dimension. This provides a solution to a problem of Davenport dating back to the sixties.

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44 pages

    Research areas

  • math.NT, 11J83, 11J13, 11K60

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